Lectures on

QUANTUM GROUPS AND NONCOMMUTATIVE GEOMETRY

Prof. Dr. B. Pareigis

Summer Semester 2002

Table of Contents

Chapter 1. Commutative and Noncommutative Algebraic Geometry 1 Introduction 1 1. The Principles of Commutative Algebraic Geometry 2 2. Quantum Spaces and Noncommutative Geometry 12 3. Quantum Monoids and their Actions on Quantum Spaces 17

Chapter 2. Hopf Algebras, Algebraic, Formal, and Quantum Groups 31 Introduction 31 1. Hopf Algebras 32 2. Monoids and Groups in a Category 38 3. Affine Algebraic Groups 42 4. Formal Groups 48 5. Quantum Groups 50 6. Quantum Automorphism Groups 56 7. Duality of Hopf Algebras 62

Chapter 3. Representation Theory, Reconstruction and Tannaka Duality 67 Introduction 67 1. Representations of Hopf Algebras 68 2. Monoidal Categories 74 3. Dual Objects 82 4. Finite reconstruction 90 5. The coalgebra coend 96 6. The bialgebra coend 98 7. The quantum monoid of a quantum space 100 8. Reconstruction and C-categories 103

Chapter 4. The Infinitesimal Theory 113 1. Integrals and Fourier Transforms 113 2. Derivations 126 3. The Lie Algebra of Primitive Elements 131 4. Derivations and Lie Algebras of Affine Algebraic Groups 134

Bibliography 139

CHAPTER 1 Commutative and Noncommutative Algebraic Geometry

Introduction

Throughout we will fix a base field 𝕂. The reader may consider it as real numbers or complex numbers or any other of his most favorite fields.

A fundamental and powerful tool for geometry is to associate with each space X the algebra of functions O(X) from X to the base field (of coefficients). The dream of geometry is that this construction is bijective, i.e. that two different spaces are mapped to two different function algebras and that each algebra is the function algebra of some space.

Actually the spaces and the algebras will form a category. There are admissible maps. For algebras it is quite clear what these maps will be. For spaces this is less obvious, partly due to the fact that we did not say clearly what spaces exactly are. Then the dream of geometry would be that these two categories, the category of (certain) spaces and the category of (certain) algebras, are dual to each other.

Algebraic geometry, noncommutative geometry, and theoretical physics have as a basis this fundamental idea, the duality of two categories, the category of spaces (state spaces in physics) and the category of function algebras (algebras of observables) in physics. We will present this duality in the 1. chapter. Certainly the type of spaces as well as the type of algebras will have to be specified.

Theoretical physics uses the categories of locally compact Hausdorff spaces and of commutative C^{∗}-algebras. A famous theorem of Gelfand-Naimark says that these categories are duals of each other.

(Affine) algebraic geometry uses a duality between the categories of affine algebraic schemes and of (reduced) finitely generated commutative algebras.

To get the whole framework of algebraic geometry one needs to go to more general spaces by patching affine spaces together. On the algebra side this amounts to considering sheaves of commutative algebras. We shall not pursue this more general approach to algebraic geometry, since generalizations to noncommutative geometry are still in the state of development and incomplete.

Noncommutative geometry uses either (imaginary) noncommutative spaces and not necessarily commutative algebras or (imaginary) noncommutative spaces and not necessarily commutative C^{∗}-algebras.

We will take an approach to the duality between geometry and algebra that heavily uses functorial tools, especially representable functors. The affine (algebraic) spaces we use will be given in the form of sets of common zeros of certain polynomials, where the zeros can be taken in arbitrary (commutative) 𝕂-algebras B. So an affine space will consist of many different sets of zeros, depending on the choice of the coefficient algebra B.

We first give a short introduction to commutative algebraic geometry in this setup and develop a duality between the category of affine (algebraic) spaces and the category of (finitely generated) commutative algebras.

Then we will transfer it to the noncommutative situation. The functorial approach to algebraic geometry is not too often used but it lends itself particularly well to the study of the noncommutative situation. Even in that situation one obtains space-like objects.

The chapter will close with a first step to construct automorphism “groups” of noncommutative spaces. Since the construction of inverses presents special problems we will only construct endomorphism “monoids” in this chapter and postpone the study of invertible endomorphisms or automorphisms to the next chapter.

At the end of the chapter you should

• know how to construct an affine scheme from a commutative algebra,

• know how to construct the function algebra of an affine scheme,

• know what a noncommutative space is and know examples of such,

• understand and be able to construct endomorphism quantum monoids of certain noncommutative spaces,

• understand, why endomorphism quantum monoids are not made out of endomorphisms of a noncommutative space.

1. The Principles of Commutative Algebraic Geometry

We will begin with simplest form of (commutative) geometric spaces and see a duality between these very simple “spaces” and certain commutative algebras. This example will show how the concept of a function algebra can be used to fulfill the dream of geometry in this situation. It will also show the functorial methods that will be applied throughout this text. It is a particularly simple example of a duality as mentioned in the introduction. This example will not be used later on, so we will only sketch the proofs for some of the statements.

Example 1.1.1. Consider a set of points without any additional geometric structure. So the geometric space is just a set. We introduce the notion of its algebra of functions.

Let X be a set. Then 𝕂^{X} := Map(X, 𝕂) is a 𝕂-algebra with componentwise addition and multiplication: (f + g)(x) := f(x) + g(x) and (fg)(x) := f(x)g(x). We study this fact in more detail.

The set 𝕂^{X} considered as a vector space with the addition (f +g)(x) := f(x)+g(x) and the scalar multiplication (αf)(x) := αf(x) defines a representable contravariant functor

𝕂- : Set ⟶ Vec .

This functor is a representable functor represented by 𝕂. In fact 𝕂^{h} : 𝕂^{Y} ⟶ 𝕂^{X} is a linear map for every map h : X ⟶ Y since 𝕂^{h}(αf + βg)(x) = (αf + βg)(h(x)) = αf(h(x))+βg(h(x)) = (αfh+βgh)(x) = (α𝕂^{h}(f)+β𝕂^{h}(g))(x) hence 𝕂^{h}(αf +βg) = α𝕂^{h}(f) + β𝕂^{h}(g).

Consider the homomorphism τ : 𝕂^{X} ⊗ 𝕂^{Y} ⟶ 𝕂^{X×Y}, defined by τ(f ⊗ g)(x, y) := f(x)g(y). In order to obtain a unique homomorphism τ defined on the tensor product we have to show that τ′ : 𝕂^{X} × 𝕂^{Y} ⟶ 𝕂^{X×Y} is a bilinear map : τ′(f + f′, g)(x, y) = (f +f′)(x)g(y) = (f(x)+f′(x))g(y) = f(x)g(y)+f′(x)g(y) = (τ′(f, g)+τ′(f′, g))(x, y) gives the additivity in the left hand argument. The additivity in the right hand argument and the bilinearity is checked similarly. One can check that τ is always injective. If X or Y are finite then τ is bijective.

As a special example we obtain a multiplication ∇ : 𝕂^{X} ⊗ 𝕂^{X} ⟶^{τ} 𝕂^{X×X} ⟶^{𝕂^{Δ}} 𝕂^{X} where Δ : X ⟶ X × X in Set is the diagonal map Δ(x) := (x, x). Furthermore we get a unit η : 𝕂^{{∗}} ⟶^{𝕂^{ϵ}} 𝕂^{X} where ϵ : X ⟶ {∗} is the unique map into the one element set. One verifies easily that (𝕂^{X}, η, ∇) is a 𝕂-algebra. Two properties are essential here, the associativity and the unit of 𝕂 and the fact that (X, Δ, ϵ) is a “comonoid” in the category Set :

[figure: X × X; X × X × X; Δ×1; X; X × X; Δ; Δ; 1×Δ]

[figure: X; X × X; Δ; Δ; 1_{X}; 1×ϵ; X × X; {∗} × X ≅ X ≅ X × {∗}.; ϵ×1]

Since 𝕂- is a functor these two diagrams carry over to the category Vec and produce the required diagrams for a 𝕂-algebra.

For a map f : X ⟶ Y we obtain a homomorphism of algebras 𝕂^{f} : 𝕂^{Y} ⟶ 𝕂^{X} because the diagrams

[figure: 𝕂^{Y} ⊗ 𝕂^{Y}; 𝕂^{Y ×Y}; τ; 𝕂^{X} ⊗ 𝕂^{X}; 𝕂^{X×X}; τ; 𝕂^{Y}; 𝕂^{Δ}; 𝕂^{X}; 𝕂^{Δ}; 𝕂^{f} ⊗𝕂^{f}; 𝕂^{f×f}; 𝕂^{f}]

and

[figure: 𝕂^{{∗}} ≅ 𝕂; η; η; 𝕂^{Y}; 𝕂^{X}; 𝕂^{f}]

commute.

Thus

𝕂- : Set ⟶ 𝕂-cAlg

is a contravariant functor.

By the definition of the set-theoretic (cartesian) product we know that 𝕂^{X} = ∏_{X} 𝕂. This identity does not only hold on the set level, it holds also for the algebra structures on 𝕂^{X} resp. ∏_{X} 𝕂.

We now construct an inverse functor

Spec : 𝕂-cAlg ⟶ Set .

For each point x ∈ X there is a maximal ideal m_{x} of ∏_{X} 𝕂 defined by m_{x} := {f ∈ Map(X, 𝕂)|f(x) = 0}. If X is a finite set then these are exactly all maximal ideals of ∏_{X} 𝕂. To show this we observe the following. The surjective homomorphism p_{x} : ∏_{X} 𝕂 ⟶ 𝕂 has kernel m_{x} hence m_{x} is a maximal ideal. If m ⊆ ∏_{X} 𝕂 is a maximal ideal and a = (α_{1}, …, α_{n}) ∈ m then for any α_{i} ≠ 0 we get (0, …, 0, 1_{i}, 0, …, 0) = (0, …, 0, α_{i}^{−1}, 0, …, 0)(α_{1}, …, α_{n}) ∈ m hence the i-th factor 0 × … × 𝕂 × … × 0 of ∏_{X} 𝕂 is in m. So the elements a ∈ m must have at least one common component α_{j} = 0 since m ≠ 𝕂. But more than one such a component is impossible since we would get zero divisors in the residue class algebra. Thus m = m_{x} where x ∈ X is the j-th elements of the set.

One can easily show more namely that the ideals m_{x} are precisely all prime ideals of Map(X, 𝕂).

With each commutative algebra A we can associate the set Spec(A) of all prime ideals of A. That defines a functor Spec: 𝕂-Alg ⟶ Set . Applied to algebras of the form 𝕂^{X} = ∏_{X} 𝕂 with a finite set X this functor recovers X as X ≅ Spec(𝕂^{X}). Thus the dream of geometry is satisfied in this particular example.

The above example shows that we may hope to gain some information on the space (set) X by knowing its algebra of functions 𝕂^{X} and applying the functor Spec to it. For finite sets and certain algebras the functors 𝕂- and Spec actually define a category duality. We are going to expand this duality to larger categories.

We shall carry some geometric structure into the sets X and will study the connection between these geometric spaces and their algebras of functions. For this purpose we will describe sets of points by their coordinates. Examples are the circle or the parabola. More generally the geometric spaces we are going to consider are so called affine schemes described by polynomial equations. We will see that such geometric spaces are completely described by their algebras of functions. Here the Yoneda Lemma will play a central rôle.

We will, however, take a different approach to functions algebras and geometric spaces, than one does in algebraic geometry. We use the functorial approach, which lends itself to an easier access to the principles of noncommutative geometry. We will define geometric spaces as certain functors from the category of commutative algebras to the category of sets. These sets will have a strong geometrical meaning. The functors will associate with each algebra A the set of points of a “geometric variety”, where the points have coordinates in the algebra A.

Definition 1.1.2. The functor 𝔸 = 𝔸^{1} : 𝕂-cAlg ⟶ Set (the underlying functor) that associates with each commutative 𝕂-algebra A its space (set) of points (elements) A is called the affine line.

Lemma 1.1.3. The functor “affine line” is a representable functor.

Proof. By Lemma 2.3.5 the representing object is 𝕂[x]. Observe that it is unique up to isomorphism. □

Definition 1.1.4. The functor 𝔸^{2} : 𝕂-cAlg ⟶ Set that associates with each commutative algebra A the space (set) of points (elements) of the plane A^{2} is called the affine plane.

Lemma 1.1.5. The functor “affine plane” is a representable functor.

Proof. Similar to Lemma 2.3.9 the representing object is 𝕂[x_{1}, x_{2}]. This algebra is unique up to isomorphism. □

Let p_{1}(x_{1}, …, x_{n}), …, p_{m}(x_{1}, …, x_{n}) ∈ 𝕂[x_{1}, …, x_{n}] be a family of polynomials. We want to consider the (geometric) variety of zeros of these polynomials. Observe that 𝕂 may not contain sufficiently many zeros for these polynomials. Thus we are going to admit zeros in extension fields of 𝕂 or more generally in arbitrary commutative 𝕂-algebras.

In the following rather simple buildup of commutative algebraic geometry, the reader should carefully verify in which statements and proofs the commutativity is really needed. Most of the following will be verbally generalized to not necessarily commutative algebras.

Definition 1.1.6. Given a set of polynomials {p_{1}, …, p_{m}} ⊆ 𝕂[x_{1}, …, x_{n}]. The functor X that associates with each commutative algebra A the set X (A) of zeros of the polynomials (p_{i}) in A^{n} is called an affine algebraic variety or an affine scheme (in 𝔸^{n}) with defining polynomials p_{1}, …, p_{m}. The elements in X (A) are called the A-points of X.

Theorem 1.1.7. The affine scheme X in 𝔸^{n} with defining polynomials p_{1}, …, p_{m} is a representable functor with representing algebra

O(X) := 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}),

called the affine algebra of the functor X.

Proof. First we show that the affine scheme X : 𝕂-cAlg ⟶ Set with the defining polynomials p_{1}, …, p_{m} is a functor. Let f : A ⟶ B be a homomorphism of commutative algebras. The induced map f^{n} : A^{n} ⟶ B^{n} defined by application of f on the components restricts to X (A) ⊆ A^{n} as X (f) : X (A) ⟶ X (B). This map is well-defined for let (a_{1}, …, a_{n}) ∈ X (A) be a zero for all polynomials p_{1}, …, p_{m} then p_{i}(f(a_{1}), …, f(a_{n})) = f(p_{i}(a_{1}, …, a_{n})) = f(0) = 0 for all i hence f^{n}(a_{1}, …, a_{n}) = (f(a_{1}), …, f(a_{n})) ∈ B^{n} is a zero for all polynomials. Thus X (f) : X (A) ⟶ X (B) is well-defined. Functoriality of X is clear now.

Now we show that X is representable by O(X) = 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}). Observe that (p_{1}, …, p_{m}) denotes the (two-sided) ideal in 𝕂[x_{1}, …, x_{n}] generated by the polynomials p_{1}, …, p_{m}. We know that each n-tupel (a_{1}, …, a_{n}) ∈ A^{n} uniquely determines an algebra homomorphism f : 𝕂[x_{1}, …, x_{n}] ⟶ A by f(x_{1}) = a_{1}, …, f(x_{n}) = a_{n}. (The polynomial ring 𝕂[x_{1}, …, x_{n}] in 𝕂-cAlg is free over the set {x_{1}, …, x_{n}}, or 𝕂[x_{1}, …, x_{n}] together with the embedding ι : {x_{1}, …, x_{n}} ⟶ 𝕂[x_{1}, …, x_{n}] is a couniversal solution of the problem given by the underlying functor 𝔸 : 𝕂-cAlg ⟶ Set and the set {x_{1}, …, x_{n}} ∈ Set .) This homomorphism of algebras maps polynomials p(x_{1}, …, x_{n}) into f(p) = p(a_{1}, …, a_{n}). Hence (a_{1}, …, a_{n}) is a common zero of the polynomials p_{1}, …, p_{m} if and only if f(p_{i}) = p_{i}(a_{1}, …, a_{n}) = 0, i.e. p_{1}, …, p_{m} are in the kernel of f. This happens if and only if f vanishes on the ideal (p_{1}, …, p_{m}) or in other word can be factorized through the residue class map

ν : 𝕂[x_{1}, …, x_{n}] ⟶ 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m})

This induces a bijection

Mor_{𝕂-cAlg}(𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}), A) ∋ f ↦ (f(x_{1}), …, f(x_{n})) ∈ X (A).

Now it is easy to see that this bijection is a natural isomorphism (in A). □

If no polynomials are given for the above construction, then the functor under this construction is the affine space 𝔸^{n} of dimension n. By giving polynomials the functor X becomes a subfunctor of 𝔸^{n}, because it defines subsets X (A) ⊆ 𝔸^{n}(A) = A^{n}. Both functors are representable functors. The embedding is induced by the homomorphism of algebras ν : 𝕂[x_{1}, …, x_{n}] ⟶ 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}).

Problem 1.1.1. (1) Determine the affine algebra of the functor “unit circle”

S^{1} in 𝔸^{2}.

(2) Determine the affine algebra of the functor “unit sphere” S^{n−1} in 𝔸^{n}.

(3) Determine the affine algebra of the functor “torus” T and find an “embed-

ding” of T into 𝔸^{3}.

Actually T can be considered as product §^{1} × §^{1}(B) = §^{1}(B) × §^{1}(B). Take the first copy of §^{1}(B) as the circle with radius 2, then we have §_{1}^{1}(B) × S_{2}^{1}(B) = {(u, v, x, y)|u^{2} + v^{2} = 4; x^{2} + y^{2} = 1}. The embedding is

(u, v, x, y) ↦ (u, v, 0) + 1/2x(u, v, 0) + (0, 0, y)

hence the algebra map is

𝕂[x, y, z] ⟶ 𝕂[u, v]/(u^{2} + v^{2} − 4) ⊗ 𝕂[x, y]/(x^{2} + y^{2} − 1)

with

x ↦ u + 1/2xu,
y ↦ v + 1/2xv,
z ↦ y.

(4) Let U denote the plane curve xy = 1. Then U is not isomorphic to the affine line. (Hint: An isomorphism 𝕂[x, x^{−1}] ⟶ 𝕂[y] sends x to a polynomial p(y) which must be invertible. Consider the highest coefficient of p(y) and show that p(y) ∈ 𝕂. But that means that the map cannot be bijective.) U is also called the unit functor. Can you explain, why?

(5) Let X denote the plane curve y = x^{2}. Then X is isomorphic to the affine line.

(6) Let 𝕂 = ℂ be the field of complex numbers. Show that the unit functor U : 𝕂-cAlg ⟶ Set in Problem (3) is naturally isomorphic to the unit circle S^{1}. (Hint: There is an algebra isomorphism between the representing algebras 𝕂[e, e^{−1}] and 𝕂[c, s]/(c^{2} + s^{2} − 1).)

(7) ^{∗} Let 𝕂 be an algebraically closed field. Let p be an irreducible square polynomial in 𝕂[x, y]. Let Z be the conic section defined by p with the affine algebra 𝕂[x, y]/(p). Show that Z is naturally isomorphic either to X or to U from parts (4) resp. (5).

Remark 1.1.8. Affine algebras of affine schemes are finitely generated commutative algebras and any such algebra is an affine algebra of some affine scheme, since A ≅ 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}) (Hilbert basis theorem).

The polynomials p_{1}, …, p_{m} are not uniquely determined by the affine algebra of an affine scheme. Not even the ideal generated by the polynomials in the polynomial ring 𝕂[x_{1}, …, x_{n}] is uniquely determined. Also the number of variables x_{1}, …, x_{n} is not uniquely determined.

The 𝕂-points (α_{1}, …, α_{n}) ∈ X (𝕂) of an affine scheme X (with coefficients in the base field 𝕂) are called rational points. They do not suffice to completely describe the affine scheme.

Let for example 𝕂 = ℝ the set of rational numbers. If X and Y are affine schemes with affine algebras O(X) := 𝕂[x, y]/(x^{2} +y^{2} +1) and O(Y) := 𝕂[x]/(x^{2} +1) then both schemes have no rational points. The scheme Y, however, has exactly two complex points (with coefficients in the field ℂ of complex numbers) and the scheme X has infinitely many complex points, hence X (ℂ) ≅ Y̸(ℂ). This does not result from the embeddings into different spaces 𝔸^{2} resp. 𝔸^{1}. In fact we also have O(Y) = 𝕂[x]/(x^{2}+1) ≅ 𝕂[x, y]/(x^{2}+1, y), so Y can be considered as an affine scheme in 𝔸^{2}.

Since each affine scheme X is isomorphic to the functor Mor_{𝕂-cAlg}(O(X), -) we will henceforth identify these two functors, thus removing annoying isomorphisms.

Definition 1.1.9. Let 𝕂-Aff denote the category of all commutative finitely generated (or affine cf. 1.1.8) 𝕂-algebras. An affine algebraic variety is a representable functor 𝕂-Aff (A, -) : 𝕂-Aff ⟶ Set . The affine algebraic varieties together with the natural transformations form the category of affine algebraic varieties 𝕂-Var over 𝕂. The functor that associates with each affine algebra A its affine algebraic variety represented by A is denoted by Spec : 𝕂-Aff ⟶ 𝕂-Var , Spec(A) = 𝕂-Aff (A, -).

By the Yoneda Lemma the functor

Spec : 𝕂-Aff ⟶ 𝕂-Var

is an antiequivalence (or duality) of categories with inverse functor

O : 𝕂-Var ⟶ 𝕂-Aff .

An affine algebraic variety is completely described by its affine algebra O(X). Thus the dream of geometry is realized.

Arbitrary (not necessarily finitely generated) commutative algebras also define representable functors (defined on the category of all commutative algebras). Thus we also have “infinite dimensional” varieties which we will call geometric spaces or affine varieties. We denote their category by Geom (𝕂) and get a commutative diagram

[figure: 𝕂-cAlg; Geom (𝕂); ≅^{o}; 𝕂-Aff; 𝕂-Var; Spec]

We call the representable functors X : 𝕂-cAlg ⟶ Set geometric spaces or affine varieties, and the representable functors X : 𝕂-Aff ⟶ Set affine schemes or affine algebraic varieties. This is another realization of the dream of geometry.

The geometric spaces can be viewed as sets of zeros in arbitrary commutative 𝕂-algebras B of arbitrarily many polynomials with arbitrarily many variables. The function algebra of X will be called the affine algebra of X in both cases.

Example 1.1.10. A somewhat less trivial example is the state space of a circular pendulum (of length 1). The location is in L = {(a, b) ∈ A^{2}|a^{2} + b^{2} = 1}, the momentum is in M = {p ∈ A} which is a straight line. So the whole geometric space for the pendulum is (L × M)(A) = {(a, b, p)|a, b, p ∈ A; a^{2} + b^{2} = 1}. This geometric space is represented by 𝕂[x, y, z]/(x^{2} + y^{2} − 1) since

(L×M)(A) = {(a, b, p)|a, b, p ∈ A; a^{2} +b^{2} = 1} ≅ 𝕂-cAlg (𝕂[x, y, z]/(x^{2} +y^{2} −1), A).

The two antiequivalences of categories above give rise to the question for the function algebra. If a representable functor X = 𝕂-cAlg (A, -) is viewed as geometric sets of zeros of certain polynomials, i.e. as spaces with coordinates in arbitrary commutative algebras B, (plus functorial behavior), then it is not clear why the representing algebra A should be anything like an algebra of functions on these geometric sets. It is not even clear where these functions should assume their values. Only if we can show that A can be viewed as a reasonable algebra of functions, we should talk about a realization of the dream of geometry. But this will be done in the following theorem. We will consider functions as maps (coordinate functions) from the geometric set X (B) to the set of coordinates B, maps that are natural in B. Such coordinate functions are just natural transformations from X to the underlying functor 𝔸.

Theorem 1.1.11. Let X be a geometric space with the affine algebra A = O(X). Then A ≅ Nat(X , 𝔸) as 𝕂-algebras, where 𝔸 : 𝕂-cAlg ⟶ Set is the underlying functor or affine line. The isomorphism A ≅ Nat(X , 𝔸) induces a natural transformation A × X (B) ⟶ B (natural in B).

Proof. First we define an isomorphism between the sets A and Nat(X , 𝔸). Because of X = Mor_{𝕂-cAlg}(A, -) =: 𝕂-cAlg (A, -) and 𝔸 = Mor_{𝕂-cAlg}(𝕂[x], -) =: 𝕂-cAlg (𝕂[x], -) the Yoneda Lemma gives us

Nat(X , 𝔸) = Nat(𝕂-cAlg (A, -), 𝕂-cAlg (𝕂[x], -)) ≅ 𝕂-cAlg (𝕂[x], A) = 𝔸(A) ≅ A

on the set level. Let ϕ : A ⟶ Nat(X , 𝔸) denote the given isomorphism. ϕ is defined by ϕ(a)(B)(p)(x) := p(a). By the Yoneda Lemma its inverse is given by ϕ^{−1}(α := α((A)(1)(x).

Nat(X , 𝔸) carries an algebra structure given by the algebra structure of the coefficients. For a coefficient algebra B, a B-point p : A ⟶ B in X (B) = 𝕂-Alg (A, B), and α, β ∈ Nat(X , 𝔸) we have α(B)(p) ∈ 𝔸(B) = B. Hence (α + β)(B)(p) := (α(B) + β(B))(p) = α(B)(p) + β(B)(p) and (α · β)(B)(p) := (α(B) · β(B))(p) = α(B)(p) · β(B)(p) make Nat(X , 𝔸) an algebra.

Let a be an arbitrary element in A. By the isomorphism given above this element induces an algebra homomorphism g_{a} : 𝕂[x] ⟶ A mapping x onto a. This algebra homomorphism induces the natural transformation ϕ(a) : X ⟶ 𝔸. On the B-level it is just the composition with g_{a}, i.e. ϕ(a)(B)(p) = (𝕂[x] ⟶^{g_{a}} A ⟶^{p} B). Since such a homomorphism is completely described by the image of x we get ϕ(a)(B)(p)(x) = p(a). To compare the algebra structures of A and Nat(X , 𝔸) let a, a′ ∈ A. We have ϕ(a)(B)(p)(x) = p(a) and ϕ(a′)(B)(p)(x) = p(a′), hence ϕ(a + a′)(B)(p)(x) = p(a + a′) = p(a) + p(a′) = ϕ(a)(B)(p)(x) + ϕ(a′)(B)(p)(x) = (ϕ(a)(B)(p)+ϕ(a′)(B)(p))(x) = (ϕ(a)(B)+ϕ(a′)(B))(p)(x) = (ϕ(a)+ϕ(a′))(B)(p)(x). Analogously we get ϕ(aa′)(B)(p)(x) = p(aa′) = p(a)p(a′) = (ϕ(a) · ϕ(a′))(B)(p)(x), and thus ϕ(a + a′) = ϕ(a) + ϕ(a′) and ϕ(aa′) = ϕ(a) · ϕ(a′). Hence addition and multiplication in Nat(X , 𝔸) are defined by the addition and the multiplication of the values p(a) + p(a′) resp. p(a)p(a′).

We describe the action ψ(B) : A × X (B) ⟶ B of A on X (B). Let p : A ⟶ B be a B-point in 𝕂-cAlg (A, B) = X (B). For each a ∈ A the image ϕ(a) : X ⟶ 𝔸 is a natural transformation hence we have maps ψ(B) : A × X (B) ⟶ B such that ψ(B)(a, p) = p(a). Finally each homomorphism of algebras f : B ⟶ B′ induces a commutative diagram

[figure: A × X (B′); B′; ψ(B′); A × X (B); B; ψ(B); A × X (f); f]

Thus ψ(B) : A × X (B) ⟶ B is a natural transformation. □

Remark 1.1.12. Observe that the isomorphism A ≅ Nat(X , 𝔸) induces a natural transformation A × X (B) ⟶ B (natural in B). In particular the affine algebra A can be viewed as the set of functions from the set of B-points X (B) into the “base” ring B (functions which are natural in B). In this sense the algebra A may be considered as function algebra of the geometric space X. Thus we will call A the function algebra of X.

One can show that the algebra A is universal with respect to the property, that for each commutative algebra D and each natural transformation ρ : D × X (-) ⟶ - there is a unique homomorphism of algebras f : D ⟶ A, such that the triangle

[figure: A × X (B); B; ψ(B); ρ(B); D × X (B); f×1_{X (B)}]

commutes. We will show this result later on for noncommutative algebras. The universal property implies that the function algebra A of an geometric space X is unique up to isomorphism.

Let X be an geometric space with function algebra A = O(X). If p : A ⟶ 𝕂 is a rational point of X, i.e. a homomorphism of algebras, then Im(p) = 𝕂 hence Ker(p) is a maximal ideal of A of codimension 1. Conversely let 𝔪 be a maximal ideal of A of codimension 1 then this defines a rational point p : A ⟶ A/𝔪 ≅ 𝕂. If 𝕂 is algebraicly closed and 𝔪 an arbitrary maximal ideal of A, then A/𝔪 is a finitely generated 𝕂-algebra and a field extension of 𝕂, hence it coincides with 𝕂. Thus the codimension of 𝔪 is 1. The set of maximal ideals of A is called the maximal spectrum Spec_{m}(A). This is the approach of algebraic geometry to recover the geometric space of (rational) points from the function algebra A. We will not follow this approach since it does not easily extend to noncommutative geometry.

Problem 1.1.2. 1. Let X be an geometric space with affine algebra A. Show that the algebra A is universal with respect to the property, that for each commutative algebra D and each natural transformation ρ : D × X (-) ⟶ - there is a unique homomorphism of algebras f : D ⟶ A, such that the triangle

[figure: A × X (B); B.; ψ(B); ρ(B); D × X (B); f×1_{X (B)}]

2. Let X be an affine scheme with affine algebra

A = 𝕂[x_{1}, …, x_{n}]/(p_{1}, …, p_{m}).

Define “coordinate functions” q_{i} : X (B) ⟶ B which describe the coordinates of B-points and identify these coordinate functions with elements of A.

Now we will study morphisms between geometric spaces.

Theorem 1.1.13. Let X ⊆ 𝔸^{r} and Y ⊆ 𝔸^{s} be affine algebraic varieties and let ϕ : X ⟶ Y be a natural transformation. Then there are polynomials

p_{1}(x_{1}, …, x_{r}), …, p_{s}(x_{1}, …, x_{r}) ∈ 𝕂[x_{1}, …, x_{r}],

such that

ϕ(A)(a_{1}, …, a_{r}) = (p_{1}(a_{1}, …, a_{r}), …, p_{s}(a_{1}, …, a_{r})),

for all A ∈ 𝕂-Aff and all (a_{1}, …, a_{r}) ∈ X (A), i.e. the morphisms between affine algebraic varieties are of polynomial type.

Proof. Let O(X) = 𝕂[x_{1}, …, x_{r}]/I and O(Y) = 𝕂[y_{1}, …, y_{s}]/J. For A ∈ 𝕂-Alg and (a_{1}, …, a_{r}) ∈ X (A) let f : 𝕂[x_{1}, …, x_{r}]/I ⟶ A with f(x_{i}) = a_{i} be the homomorphism obtained from X (A) ≅ 𝕂-Alg (𝕂[x_{1}, …, x_{r}]/I, A). The natural transformation ϕ is given by composition with a homomorphism g : 𝕂[y_{1}, …, y_{s}]/J ⟶ 𝕂[x_{1}, …, x_{r}]/I hence we get

ϕ(A) : 𝕂-cAlg (𝕂[x_{1}, …, x_{r}]/I, A) ∋ f ↦ fg ∈ 𝕂-cAlg (𝕂[y_{1}, …, y_{s}]/J, A).

Since g is described by g(y_{i}) = p_{i}(x_{1}, …, x_{r}) ∈ 𝕂[x_{1}, …, x_{r}] we get

ϕ(A)(a_{1}, …, a_{s}) = (fg(y_{1}), …, fg(y_{s}))
= (f(p_{1}(x_{1}, …, x_{r})), …, f(p_{s}(x_{1}, …, x_{r})))
= (p_{1}(a_{1}, …, a_{r}), …, p_{s}(a_{1}, …, a_{r})).
□

An analogous statement holds for geometric spaces.

Example 1.1.14. The isomorphism between the affine line (1.1.2) and the parabola is given by the isomorphism f : 𝕂[x, y]/(y − x^{2}) ⟶ 𝕂[z], f(x) = z, f(y) = z^{2} that has the inverse function f^{−1}(z) = x. On the affine schemes 𝔸, the affine line, and ℙ, the parabola, the induced map is f : 𝔸(A) ∋ a ↦ (a, a^{2}) ∈ ℙ(A) resp. f^{−1} : ℙ(A) ∋ (a, b) ↦ a ∈ 𝔸(A).

2. Quantum Spaces and Noncommutative Geometry

Now we come to noncommutative geometric spaces and their function algebras. Many of the basic principles of commutative algebraic geometry as introduced in 1.1 carry over to noncommutative geometry. Our main aim, however, is to study the symmetries (automorphisms) of noncommutative spaces which lead to the notion of a quantum group.

Since the construction of noncommutative geometric spaces has deep applications in theoretical physics we will also call these spaces quantum spaces.

Definition 1.2.1. Let A be a (not necessarily commutative) 𝕂-algebra. Then the functor X := 𝕂-Alg (A, -) : 𝕂-Alg ⟶ Set represented by A is called (affine) noncommutative (geometric) space or quantum space. The elements of 𝕂-Alg (A, B) are called B-points of X. A morphism of noncommutative spaces f : X ⟶ Y is a natural transformation.

This definition implies immediately

Corollary 1.2.2. The noncommutative spaces form a category QS that is dual to the category of 𝕂-algebras.

Remark 1.2.3. Thus one often calls the dual category 𝕂-Alg^{op} category of noncommutative spaces.

If A is a finitely generated algebra then it may be considered as a residue class algebra A ≅ 𝕂⟨x_{1}, …, x_{n}⟩/I of a polynomial algebra in noncommuting variables (cf. [Advanced Algebra] 2.2). If I = (p_{1}(x_{1}, …, x_{n}), …, p_{m}(x_{1}, …, x_{n})) is the two-sided ideal generated by the polynomials p_{1}, …, p_{m} then the sets 𝕂-Alg (A, B) can be considered as sets of zeros of these polynomials in B^{n}. In fact, we have 𝕂-Alg (𝕂⟨x_{1}, …, x_{n}⟩, B) ≅ Map({x_{1}, …, x_{n}}, B) = B^{n}. Thus 𝕂-Alg (A, B) can be considered as the set of those homomorphisms of algebras from 𝕂⟨x_{1}, …, x_{n}⟩ to B that vanish on the ideal I or as the set of zeros of these polynomials in B^{n}.

Similar to Theorem 1.1.13 one shows also in the noncommutative case that morphisms between noncommutative spaces are described by polynomials.

The Theorem 1.1.11 on the operation of the affine algebra A = O(X) on X as function algebra can be carried over to the noncommutative case as well: the natural transformation ψ(B) : A × X (B) ⟶ B (natural in B) is given by ψ(B)(a, p) := p(a) and comes from the isomorphism A ≅ Nat(X , 𝔸).

Now we come to a claim on the function algebra A that we did not prove in the commutative case, but that holds in the commutative as well as in the noncommutative situation.

Lemma 1.2.4. Let D be a set and ϕ : D × X (-) ⟶ 𝔸(-) be a natural transformation. Then there exists a unique map f : D ⟶ A such that the diagram

[figure: A × X (B); B; ψ(B); D × X (B); f×1; ϕ(B)]

commutes.

Proof. Let ϕ : D × X ⟶ 𝔸 be given. We first define a map f′ : D ⟶ Nat(X , 𝔸) by f′(d)(B)(p) := ϕ(B)(d, p).

We claim that f′(d) : X ⟶ 𝔸 is a natural transformation. Observe that the diagram

[figure: D × X (B′); 𝔸(B′) = B′; ϕ(B′); D × X (B); 𝔸(B) = B; ϕ(B); D × X (g); g]

commutes for any g : B ⟶ B′, since ϕ is a natural transformation. Thus the diagram

[figure: X (B′); 𝔸(B′) = B′; f′(d)(B′); X (B); 𝔸(B) = B; f′(d)(B); X (g); g]

commutes since

(g ∘ f′(d)(B))(p) = (g ∘ ϕ(B))(d, p)
= ϕ(B′) ∘ (1 × X (g))(d, p)
= ϕ(B′)(d, X (g)(p))
= f′(d)(B′)(X (g)(p)).

Hence f′(d) ∈ Nat(X, 𝔸) and f′ : D ⟶ Nat(X , 𝔸).

Now we define f : D ⟶ A as D ⟶^{f′} Nat(X , 𝔸) ≅ A. By using the isomorphism from 1.1.11 we get f(d) = f′(d)(A)(1). (Actually we get f(d) = f′(d)(A)(1)(x) but we identify 𝔸(B) and B by 𝔸(B) ∋ p ↦ p(x) ∈ B.)

Then we get

ψ(B)(f × 1)(d, p) = ψ(B)(f(d), p)

= ψ(B)(f′(d)(A)(1)(x), p) (by definition of f)

= p ∘ f′(d)(A)(1) (since we may omit x) = p ∘ ϕ(A)(d, 1) (by definition of f′) = ϕ(B)(D × X (p))(d, 1) (since ϕ is a natural transformation) = ϕ(B)(d, p).

Hence the diagram in the Lemma commutes.

To show the uniqueness of f let g : D ⟶ A be a homomorphism such that ψ(B)(g × 1) = ϕ(B). Then we have

f(d) = f′(d)(A)(1) = ϕ(A)(d, 1) = ψ(A)(g×1)(d, 1) = ψ(A)(f(d), 1) = 1∘g(d) = g(d)

hence f = g. □

Problem 1.2.3. Definition: Let X be an geometric space with affine algebra A. Let D be an algebra. A natural transformation ρ : D × X ⟶ 𝔸 is called an algebra action if ρ(B)(-, p) : D ⟶ 𝔸(B) = B is an algebra homomorphism for all B and all p ∈ X (B).

Give proofs for:

Lemma: The natural transformation ψ : A × X ⟶ 𝔸 is an algebra action.

Theorem: Let D be an algebra and ρ : D × X (-) ⟶ 𝔸(-) be an algebra action. Then there exists a unique algebra homomorphism f : D ⟶ A such that the diagram

[figure: A × X (B); B; ψ(B); ρ(B); D × X (B); f×1]

commutes.

Definition 1.2.5. The noncommutative space 𝔸_{q}^{2|0} with the function algebra

O(𝔸_{q}^{2|0}) := 𝕂⟨x, y⟩/(xy − q^{−1}yx)

with q ∈ 𝕂 \ {0} is called the (deformed) quantum plane. The noncommutative space 𝔸_{q}^{0|2} with the function algebra

O(𝔸_{q}^{0|2}) := 𝕂⟨ξ, η⟩/(ξ^{2}, η^{2}, ξη + qηξ)

is called the dual (deformed) quantum plane. We have

𝔸_{q}^{2|0}(A) = \{(\begin{matrix}x \\ y \end{matrix})|x, y ∈ A; xy = q^{−1}yx\}

and

𝔸_{q}^{0|2}(A) = {(ξ, η)|ξ, η ∈ A; ξ^{2} = 0, η^{2} = 0, ξη = −qηξ } .

Definition 1.2.6. Let X be a noncommutative space with function algebra A and let X_{c} be the restriction of the functor X : 𝕂-Alg ⟶ Set to the category of commutative algebras: X_{c} : 𝕂-cAlg ⟶ Set . Then we call X_{c} the commutative part of the noncommutative space X.

Lemma 1.2.7. The commutative part X_{c} of a noncommutative space X is an affine variety.

Proof. The underlying functor 𝔸 : 𝕂-cAlg ⟶ 𝕂-Alg has a left adjoint functor 𝕂-Alg ∋ A ↦ A/[A, A] ∈ 𝕂-cAlg where [A, A] denotes the two-sided ideal of A generated by the elements ab − ba. In fact for each homomorphism of algebras f : A ⟶ B with a commutative algebra B there is a factorization through A/[A, A] since f vanishes on the elements ab − ba.

Hence if A = O(X) is the function algebra of X then A/[A, A] is the representing algebra for X_{c}. □

Remark 1.2.8. For any commutative algebra (of coefficients) B the spaces X and X_{c} have the same B-points: X (B) = X_{c}(B). The two spaces differ only for noncommutative algebras of coefficients. In particular for commutative fields B as algebras of coefficients the quantum plane 𝔸_{q}^{2|0} has only B-points on the two axes since the function algebra 𝕂⟨x, y⟩/(xy − q^{−1}yx, xy − yx) ≅ K[x, y]/(xy) defines only B-points (b_{1}, b_{2}) where at least one of the coefficients is zero.

Problem 1.2.4. Let S_{3} be the symmetric group and A := 𝕂[S_{3}] be the group algebra on S_{3}. Describe the points of X (B) = 𝕂-Alg (A, B) as a subspace of 𝔸^{2}(B). What is the commutative part X_{c}(B) of X and what is the affine algebra of X_{c}?

To understand how Hopf algebras fit into the context of noncommutative spaces we have to better understand the tensor product in 𝕂-Alg .

Definition 1.2.9. Let A = O(X) and A′ = O(Y) be the function algebras of the noncommutative spaces X resp. Y. Two B-points p : A ⟶ B in X (B) and p′ : A′ ⟶ B in Y(B) are called commuting points if we have for all a ∈ A and all a′ ∈ A′

p(a)p′(a′) = p′(a′)p(a),

i.e. if the images of the two homomorphisms p and p′ commute.

Remark 1.2.10. To show that the points p and p′ commute, it is sufficient to check that the images of the algebra generators p(x_{1}), …, p(x_{m}) commute with the images of the algebra generators p′(y_{1}), …, p′(y_{n}) under the multiplication. This means that we have

b_{i}b′_{j} = b′_{j}b_{i}

for the B-points (b_{1}, …, b_{m}) ∈ X (B) and (b′_{1}, …, b′_{n}) ∈ Y(B).

Definition 1.2.11. The functor

(X ⊥ Y)(B) := {(p, p′) ∈ X (B) × Y(B)|p, p′ commute}

is called the orthogonal product of the noncommutative spaces X and Y.

Remark 1.2.12. Together with X and Y the orthogonal product X ⊥ Y is again a functor, since homomorphisms f : B ⟶ B′ are compatible with the multiplication and thus preserve commuting points. Hence X ⊥ Y is a subfunctor of X × Y.

Lemma 1.2.13. If X and Y are noncommutative spaces, then X ⊥ Y is a noncommutative space with function algebra O(X ⊥ Y) = O(X) ⊗ O(Y).

If X and Y have finitely generated function algebras then the function algebra of X ⊥ Y is also finitely generated.

Proof. Let A := O(X) and A′ := O(Y). Let (p, p′) ∈ (X ⊥ Y)(B) be a pair of commuting points. Then there is a unique homomorphism of algebras h : A ⊗ A′ ⟶ B such that the following diagram commutes

[figure: A; A ⊗ A′; ι; p; B.; h; A′; ι′; p′]

Define h(a ⊗ a′) := p(a)p′(a′) and check the necessary properties. Observe that for an arbitrary homomorphism of algebras h : A ⊗ A′ ⟶ B the images of elements of the form a ⊗ 1 and 1 ⊗ a′ commute since these elements already commute in A ⊗ A′. Thus we have

(X ⊥ Y)(B) ≅ 𝕂-Alg (A ⊗ A′, B).

If the algebra A is generated by the elements a_{1}, …, a_{m} and the algebra A′ is generated by the elements a′_{1}, …, a′_{n} then the algebra A ⊗ A′ is generated by the elements a_{i} ⊗ 1 and 1 ⊗ a′_{j}. □

Proposition 1.2.14. The orthogonal product of noncommutative spaces is associative, i.e. for noncommutative spaces X, Y, and Z we have

(X ⊥ Y) ⊥ Z ≅ X ⊥ (Y ⊥ Z).

Proof. Let B be a coefficient algebra and let p_{x} ∈ X (B), p_{y} ∈ Y(B), and p_{z} ∈ Z(B) be points such that ((p_{x}, p_{y}), p_{z}) is a pair of commuting points in ((X ⊥ Y) ⊥ Z)(B). In particular (p_{x}, p_{y}) is also a pair of commuting points. Thus we have for all a ∈ A := O(X), a′ ∈ A′ := O(Y), and a′′ ∈ A′′ := O(Z)

p_{x}(a)p_{y}(a′)p_{z}(a′′) = (p_{x}, p_{y})(a ⊗ a′)p_{z}(a′′) = p_{z}(a′′)(p_{x}, p_{y})(a ⊗ a′) = p_{z}(a′′)p_{x}(a)p_{y}(a′)

and

p_{x}(a)p_{y}(a′) = p_{y}(a′)p_{x}(a).

If we choose a = 1 then we get p_{y}(a′)p_{z}(a′′) = p_{z}(a′′)p_{y}(a′). For arbitrary a, a′, a′′ we then get

p_{x}(a)p_{y}(a′)p_{z}(a′′) = p_{z}(a′′)p_{x}(a)p_{y}(a′) = p_{z}(a′′)p_{y}(a′)p_{x}(a) = p_{y}(a′)p_{z}(a′′)p_{x}(a)

hence (p_{y}, p_{z}) and (p_{x}, (p_{y}, p_{z})) are also pairs of commuting points. □

Problem 1.2.5. Show that the orthogonal product of quantum spaces X ⊥ Y is a tensor product for the category QS (in the sense of monoidal categories – if you know already what that is).

3. Quantum Monoids and their Actions on Quantum Spaces

We use the orthogonal product introduced in the previous section as “product” to define the notion of a monoid (some may call it an algebra w.r.t. the orthogonal product). Observe that on the geometric level the orthogonal product consists only of commuting points. So whenever we define a morphism on the geometric side with domain an orthogonal product of quantum spaces f : X ⊥ Y ⟶ Z then we only have to define what happens to commuting pairs of points. That makes it much easier to define such morphisms for noncommutative coordinate algebras.

We are going to define monoids in this sense and study their actions on quantum spaces.

Let E be the functor represented by 𝕂. It maps each algebra H to the one-element set {ι : 𝕂 ⟶ H}.

Definition 1.3.1. Let M be a noncommutative space and let

m : M ⊥ M ⟶ M and e : E ⟶ M

be morphisms in QS such that the diagrams

[figure: M ⊥ M; M; m; M ⊥ M ⊥ M; M ⊥ M; m ⊥ 1; 1 ⊥ m; m]

and

[figure: E ⊥ M ≅ M ≅ M ⊥ E; M ⊥ M; id⊥η; η⊥id; ∇; M ⊥ M; M; ∇; 1_{M}]

commute. Then (M, m, e) is called a quantum monoid.

Proposition 1.3.2. Let M be a noncommutative space with function algebra H. Then H is a bialgebra if and only if M is a quantum monoid.

Proof. Since the functors M ⊥ M, M ⊥ E and E ⊥ M are represented by H ⊗ H resp. H ⊗ 𝕂 ≅ H resp. 𝕂 ⊗ H ≅ H the Yoneda Lemma defines a bijection between the morphisms m : M ⊥ M ⟶ M and the algebra homomorphisms Δ : H ⟶ H ⊗ H and similarly a bijection between the morphisms e : E ⟶ M and the algebra homomorphisms ε : H ⟶ 𝕂. Again by the Yoneda Lemma the bialgebra diagrams in 𝕂-Alg commute if and only if the corresponding diagrams for a quantum monoid commute. □

Observe that a similar result cannot be formulated for Hopf algebras H since neither the antipode S nor the multiplication ∇ : H ⊗ H ⟶ H are algebra homomorphisms. In contrast to affine algebraic groups (2.3.2) Hopf algebras in the category 𝕂-Alg^{op} ≅ QR are not groups. Nevertheless, one defines

Definition 1.3.3. A functor defined on the category of 𝕂-algebras and represented by a Hopf algebra H is called a quantum group.

Definition 1.3.4. Let X be a noncommutative space and let M be a quantum monoid. A morphism (a natural transformation) of quantum spaces ρ : M ⊥ X ⟶ X is called an operation of M on X if the diagrams

[figure: M ⊥ X; X; ρ; M ⊥ M ⊥ X; M ⊥ X; m ⊥ 1; 1 ⊥ ρ; ρ]

and

[figure: X ≅ E ⊥ X; M ⊥ X; η⊥id; X; ρ; id_{X}]

commute. We call X a noncommutative M-space.

Proposition 1.3.5. Let X be a noncommutative space with function algebra A = O(X). Let M be a quantum monoid with function algebra B = O(M). Let ρ : M ⊥ X ⟶ X be a morphism in QS and let f : A ⟶ B ⊗ A be the associated homomorphism of algebras. Then the following are equivalent

1. (X , M, ρ) is an operation of the quantum monoid M on the noncommutative space X;

2. (A, H, f) define an H-comodule algebra.

Proof. The homomorphisms of algebras Δ ⊗ 1_{A}, 1_{B} ⊗ f, ϵ ⊗ 1_{A} etc. represent the morphisms of quantum spaces m ⊥ id, id ⊥ ρ, η ⊥ id etc. Hence the required diagrams are transferred by the Yoneda Lemma. □

Example 1.3.6. 1. The quantum monoid of “quantum matrices”:

We consider the algebra

M_{q}(2) := 𝕂⟨a, b, c, d⟩/I = 𝕂 ⟨\begin{matrix}a & b \\ c & d \end{matrix}⟩ /I

where the two-sided ideal I is generated by the elements

ab − q^{−1}ba, ac − q^{−1}ca, bd − q^{−1}db, cd − q^{−1}dc, ad − da − (q^{−1} − q)bc, bc − cb.

The quantum space M_{q}(2) associated with the algebra M_{q}(2) is given by

\begin{matrix}M_{q}(2)(A) & = 𝕂-Alg (M_{q}(2), A) \\ = \{(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) |a′, b′, c′, d′ ∈ A; a′b′ = q^{−1}b′a′, …, b′c′ = c′b′\} \end{matrix}

where each homomorphism of algebras f : M_{q}(2) ⟶ A is described by the quadruple (a′, b′, c′, d′) of images of the algebra generators a, b, c, d. The images must satisfy the same relations that generate the two-sided ideal I hence

\begin{matrix}a′b′ = q^{−1}b′a′, a′c′ = q^{−1}c′a′, b′d′ = q^{−1}d′b′, c′d′ = q^{−1}d′c′, \\ b′c′ = c′b′, a′d′ − q^{−1}b′c′ = d′a′ − qc′b′. \end{matrix}

We write these quadruples as 2 × 2-matrices and call them quantum matrices. The unusual commutation relations are chosen so that the following examples work.

The quantum space of quantum matrices turns out to be a quantum monoid. We give both the algebraic (with function algebras) and the geometric (with quantum spaces) approach to define the multiplication.

a) The algebraic approach:

The algebra M_{q}(2) is a bialgebra with the diagonal

Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) ,

i.e. by Δ(a) = a ⊗ a + b ⊗ c, Δ(b) = a ⊗ b + b ⊗ d, Δ(c) = c ⊗ a + d ⊗ c and Δ(d) = c ⊗ b + d ⊗ d, and with the counit

ε (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ,

i.e. ε(a) = 1, ε(b) = 0, ε(c) = 0, and ε(d) = 1. We have to prove that Δ and ε are homomorphisms of algebras and that the coalgebra laws are satisfied. To obtain a homomorphism of algebras Δ : M_{q}(2) ⟶ M_{q}(2) ⊗ M_{q}(2) we define Δ : 𝕂⟨a, b, c, d⟩ ⟶ M_{q}(2)⊗M_{q}(2) on the free algebra (the polynomial ring in noncommuting variables) 𝕂⟨a, b, c, d⟩ generated by the set {a, b, c, d} and show that it vanishes on the ideal I or more simply on the generators of the ideal. Then it factors through a unique homomorphism of algebras Δ : M_{q}(2) ⟶ M_{q}(2) ⊗ M_{q}(2). We check this only for one generator of the ideal I:

Δ(ab − q^{−1}ba) = Δ(a)Δ(b) − q^{−1}Δ(b)Δ(a) = = (a ⊗ a + b ⊗ c)(a ⊗ b + b ⊗ d) − q^{−1}(a ⊗ b + b ⊗ d)(a ⊗ a + b ⊗ c) = aa ⊗ ab + ab ⊗ ad + ba ⊗ cb + bb ⊗ cd − q^{−1}(aa ⊗ ba + ab ⊗ bc + ba ⊗ da + bb ⊗ dc) = aa ⊗ (ab − q^{−1}ba) + ab ⊗ (ad − q^{−1}bc) + ba ⊗ (cb − q^{−1}da) + bb ⊗ (cd − q^{−1}dc) = ba ⊗ (q^{−1}ad − q^{−2}bc + cb − q^{−1}da) ≡ 0 mod (I).

The reader should check the other identities.

The coassociativity follows from

\begin{matrix}(Δ ⊗ 1)Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = Δ (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) = ((\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix})) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) = \\ = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ ((\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix})) = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = (1 ⊗ Δ)Δ (\begin{matrix}a & b \\ c & d \end{matrix}) . \end{matrix}

The reader should check the properties of the counit.

b) The geometric approach:

M_{q}(2) has a rather remarkable (and actually well known) comultiplication that is better understood by using the induced multiplication of commuting points. Given

two commuting quantum matrices (\begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix}) and (\begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix}) in M_{q}(2)(A). Then their

matrix product

(\begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix}) (\begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix}) = (\begin{matrix}a_{1}a_{2} + b_{1}c_{2} & a_{1}b_{2} + b_{1}d_{2} \\ c_{1}a_{2} + d_{1}c_{2} & c_{1}b_{2} + d_{1}d_{2} \end{matrix})

is again a quantum matrix. To prove this we only check one of the relations

(a_{1}a_{2} + b_{1}c_{2})(a_{1}b_{2} + b_{1}d_{2}) = a_{1}a_{2}a_{1}b_{2} + a_{1}a_{2}b_{1}d_{2} + b_{1}c_{2}a_{1}b_{2} + b_{1}c_{2}b_{1}d_{2}
= a_{1}a_{1}a_{2}b_{2} + a_{1}b_{1}a_{2}d_{2} + b_{1}a_{1}c_{2}b_{2} + b_{1}b_{1}c_{2}d_{2}

= q^{−1}a_{1}a_{1}b_{2}a_{2} + q^{−1}b_{1}a_{1}(d_{2}a_{2} + (q^{−1} − q)b_{2}c_{2}) + b_{1}a_{1}b_{2}c_{2} + q^{−1}b_{1}b_{1}d_{2}c_{2}

= q^{−1}(a_{1}a_{1}b_{2}a_{2} + a_{1}b_{1}b_{2}c_{2} + b_{1}a_{1}d_{2}a_{2} + b_{1}b_{1}d_{2}c_{2})
= q^{−1}(a_{1}b_{2}a_{1}a_{2} + a_{1}b_{2}b_{1}c_{2} + b_{1}d_{2}a_{1}a_{2} + b_{1}d_{2}b_{1}c_{2})
= q^{−1}(a_{1}b_{2} + b_{1}d_{2})(a_{1}a_{2} + b_{1}c_{2})

We have used that the two points are commuting points. This multiplication obviously is a natural transformation M_{q}(2) ⊥ M_{q}(2)(A) ⟶ M_{q}(2)(A) (natural in A). It is

associative and has unit (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}). For the associativity observe that by 1.2.14

(((\begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix}) , (\begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix})), (\begin{matrix}a_{3} & b_{3} \\ c_{3} & d_{3} \end{matrix}))

is a pair of commuting points if and only if

((\begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix}) , ((\begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix}) , (\begin{matrix}a_{3} & b_{3} \\ c_{3} & d_{3} \end{matrix})))

is a pair of commuting points.

Since (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) for all quantum matrices

(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ M_{q}(2)(B) we see that M_{q}(2) is a quantum monoid.

It remains to show that the multiplication of M_{q}(2) and the comultiplication of M_{q}(2) correspond to each other by the Yoneda Lemma. The identity morphism of M_{q}(2) ⊗ M_{q}(2) is given by the pair of commuting points

(ι_{1}, ι_{2}) ∈ M_{q}(2) ⊥ M_{q}(2)(M_{q}(2) ⊗ M_{q}(2)) = 𝕂-Alg (M_{q}(2) ⊗ M_{q}(2), M_{q}(2) ⊗ M_{q}(2)).

Since ι_{1} = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ 1 = (\begin{matrix}a ⊗ 1 & b ⊗ 1 \\ c ⊗ 1 & d ⊗ 1 \end{matrix}) and ι_{2} = 1 ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}1 ⊗ a & 1 ⊗ b \\ 1 ⊗ c & 1 ⊗ d \end{matrix})

we have id = (ι_{1}, ι_{2}) = ((\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ 1, 1 ⊗ (\begin{matrix}a & b \\ c & d \end{matrix})). The Yoneda Lemma defines the

diagonal as the image of the identity under 𝕂-Alg (M_{q}(2) ⊗ M_{q}(2), M_{q}(2) ⊗ M_{q}(2))

⟶ 𝕂-Alg (M_{q}(2), M_{q}(2) ⊗ M_{q}(2)) by the multiplication. So Δ((\begin{matrix}a & b \\ c & d \end{matrix})) = Δ =

ι_{1} ∗ ι_{2} = ((\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ 1) ∗ (1 ⊗ (\begin{matrix}a & b \\ c & d \end{matrix})) = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}).

Thus M_{q}(2) defines a quantum monoid M_{q}(2) with

M_{q}(2)(B) = \{(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix})|a′, b′, c′, d′ ∈ B; a′b′ = q^{−1}b′a′, …, b′c′ = c′b′ \} .

This is the deformed version of M_{2}^{×} the multiplicative monoid of the 2 × 2-matrices of commutative algebras.

2. Let A_{q}^{2|0} = 𝕂⟨x, y⟩/(xy − q^{−1}yx) be the function algebra of the quantum plane 𝔸_{q}^{2|0}. By the definition 1.2.5 we have

𝔸_{q}^{2|0}(A′) = \{(\begin{matrix}x \\ y \end{matrix})|x, y ∈ A′; xy = q^{−1}yx\} .

The set

M_{q}(2)(A′) = \{(\begin{matrix}u & x \\ y & z \end{matrix})|u, x, y, z ∈ A′; ux = q^{−1}xu, …, xy = yx\}

operates on this quantum plane by matrix multiplication

M_{q}(2)(A′) ⊥ 𝔸_{q}^{2|0}(A′) ∋ ((\begin{matrix}a & b \\ c & d \end{matrix}) , (\begin{matrix}x \\ y \end{matrix})) ↦ (\begin{matrix}a & b \\ c & d \end{matrix}) · (\begin{matrix}x \\ y \end{matrix}) ∈ 𝔸_{q}^{2|0}(A′).

Again one should check that the required equations are preserved. Since we have a matrix multiplication we get an operation as in the preceding proposition. In particular A_{q}^{2|0} is a M_{q}(2)-comodule algebra.

As in example 1. we get the comultiplication as δ((\begin{matrix}x \\ y \end{matrix})) = δ = ((\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ 1) ∗

(1 ⊗ (\begin{matrix}x \\ y \end{matrix})) = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}x \\ y \end{matrix}) .

3. Let A_{q}^{0|2} = 𝕂⟨ξ, η⟩/(ξ^{2}, η^{2}, ξη+qηξ) be the function algebra of the dual quantum plane 𝔸_{q}^{0|2}. By the definition 1.2.5 we have

𝔸_{q}^{0|2}(A′) = {(a′ b′)|a′, b′ ∈ A′; a′^{2} = 0, b′^{2} = 0, a′b′ = −qb′a′ } .

The quantum monoid M_{q}(2) also operates on the dual quantum plane by matrix multiplication

𝔸_{q}^{0|2}(A′) ⊥ M_{q}(2)(A′) ∋ ((ξ η) , (\begin{matrix}a & b \\ c & d \end{matrix})) ↦ (ξ η) · (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ 𝔸_{q}^{0|2}(A′).

This gives another example of a M_{q}(2)-comodule algebra A_{q}^{0|2} ⟶ A_{q}^{0|2} ⊗ M_{q}(2) with

δ((ξ η)) = δ = ((ξ η) ⊗ 1) ∗ (1 ⊗ (\begin{matrix}a & b \\ c & d \end{matrix})) = (ξ η) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) .

What is now the reason for the remarkable relations on M_{q}(2)? It is based on a fact that we will show later namely that M_{q}(2) is the universal quantum monoid acting on the quantum plane 𝔸_{q}^{2|0} from the left and on the dual quantum plane 𝔸_{q}^{0|2} from the right. This however happens in the category of quantum planes represented by quadratic algebras. Here we will show a simpler theorem for finite dimensional algebras.

Problem 1.3.6. (1) Consider the following subset H of the set of complex 2 × 2-matrices:

H := \{( \begin{matrix}x & −y \\ ȳ & x̄ \end{matrix} ) ∈ M_{ℂ}(2 × 2)|x, y ∈ ℂ\}

We call H the set of Hamiltonian quaternions. For

h = ( \begin{matrix}x & −y \\ ȳ & x̄ \end{matrix} )

we define:

h̄ := ( \begin{matrix}x̄ & y \\ −ȳ & x \end{matrix} )

Show:

(a) hh̄ = (|x|^{2} + |y|^{2})E (E the unit matrix),

(b) H is a real subalgebra of the complex algebra of 2 × 2-matrices.

(c) H is a division algebra, i. e. each element different from zero has an inverse under the multiplication.

(d) Let

I := ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) J := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) K := ( \begin{matrix}0 & −i \\ −i & 0 \end{matrix} )

Then E, I, J, K is an R-basis of H an we have the following multiplication table:

I^{2} = J^{2} = K^{2} = −1

IJ = −JI = K JK = −KJ = I KI = −IK = J.

(2) Compute the H-points A^{2|0}(H) of the quantum plane. Definition 1.3.7. (1) Let X be a quantum space. A quantum space M(X)

together with a morphism of quantum spaces μ : M(X) ⊥ X ⟶ X is called a quantum space acting universally on X (or simply a universal quantum space for X) if for every quantum space Y and every morphism of quantum spaces f : Y ⊥ X ⟶ X there is a unique morphism of quantum spaces g : Y ⟶ M(X) such that the following diagram commutes

[figure: M(X) ⊥ X; X .; μ; f; Y ⊥ X; g⊥1_{X}]

(2) Let A be a 𝕂-algebra. A 𝕂-algebra M(A) together with a homomorphism of algebras δ : A ⟶ M(A) ⊗ A is called an algebra coacting universally on A (or simply a universal algebra for A) if for every 𝕂-algebra B and every homomorphism of 𝕂-algebras f : A ⟶ B ⊗ A there exists a unique homomorphism of algebras g : M(A) ⟶ B such that the following diagram commutes

[figure: A; M(A) ⊗ A; δ; f; B ⊗ A; g⊗1_{A}]

By the universal properties the universal algebra M(A) for A and the universal quantum space M(X) for X are unique up to isomorphism.

Proposition 1.3.8. (1) Let A be a 𝕂-algebra with universal algebra M(A) and δ : A ⟶ M(A) ⊗ A. Then M(A) is a bialgebra and A is an M(A)- comodule algebra by δ.

(2) If B is a bialgebra and if f : A ⟶ B ⊗ A defines the structure of a Bcomodule algebra on A then there is a unique homomorphism g : M(A) ⟶ B of bialgebras such that the following diagram commutes

[figure: A; M(A) ⊗ A; δ; f; B ⊗ A; g ⊗ 1_{A}]

The corresponding statement for quantum spaces and quantum monoids is the following.

Proposition 1.3.9. (1) Let X be a quantum space with universal quantum space M(X) and μ : M(X) ⊥ A ⟶ A. Then M(X) is a quantum monoid and X is an M(X)-space by μ.

(2) If Y is another quantum monoid and if f : Y ⊥ X ⟶ X defines the structure of a Y-space on X then there is a unique morphism of quantum monoids g : Y ⟶ M(X) such that the following diagram commutes

[figure: M(X) ⊥ X; X .; μ; f; Y ⊥ X; g⊥1_{X}]

Proof. We give the proof for the algebra version of the proposition. Consider the following commutative diagram

[figure: M(A) ⊗ A; M(A) ⊗ M(A) ⊗ A; 1_{M(A)}⊗δ; A; M(A) ⊗ A; δ; δ; Δ⊗1_{A}]

where the morphism of algebras Δ is defined by the universal property of M(A) with respect to the algebra morphism (1_{M(A)} ⊗ δ)δ. Furthermore there is a unique morphism of algebras ϵ : M(A) ⟶ 𝕂 such that

[figure: A; M(A) ⊗ A; δ; 1_{A}; A ≅ 𝕂 ⊗ A; ϵ⊗1_{A}]

commutes.

The coalgebra axioms arise from the following commutative diagrams and

[figure: A; M(A) ⊗ A; δ; δ; Δ⊗1_{A}; M(A) ⊗ A; M(A) ⊗ M(A) ⊗ A; 1_{M(A)}⊗δ; Δ⊗1_{A}; 1_{M(A)}⊗δ; Δ⊗1_{M(A)}⊗1_{A}; 1_{M(A)}⊗Δ⊗1_{A}; M(A) ⊗ M(A) ⊗ A; M(A) ⊗ M(A) ⊗ M(A) ⊗ A; 1_{M(A)}⊗1_{M(A)}⊗δ]

[figure: A; M(A) ⊗ A; δ; δ; Δ⊗1_{A}; 1_{M(A)}⊗1_{A}; M(A) ⊗ A; M(A) ⊗ M(A) ⊗ A; 1_{M(A)}⊗δ; 1_{M(A)}⊗1_{A}; M(A) ⊗ A ≅ M(A) ⊗ 𝕂 ⊗ A; 1_{M(A)}⊗ϵ⊗1_{A}]

and

[figure: A; M(A) ⊗ A; δ; 1_{A}; δ; Δ⊗1_{A}; M(A) ⊗ A; M(A) ⊗ M(A) ⊗ A; 1_{M(A)}⊗δ; ϵ⊗1_{A}; ϵ⊗1_{M(A)}⊗1_{A}; A; M(A) ⊗ A ≅ 𝕂 ⊗ M(A) ⊗ A.; δ]

In fact these diagrams imply by the uniqueness of the induced homomorphisms of algebras (Δ ⊗ 1_{M(A)})Δ = (1_{M(A)} ⊗ Δ)Δ, (1_{M(A)} ⊗ ϵ)Δ = 1_{M(A)} and ϵ ⊗ (1_{M(A)})Δ = 1_{M(A)}. Finally A is an M(A)-comodule algebra by the definition of Δ and ϵ.

Now assume that a structure of a B-comodule algebra on A is given by a bialgebra B and f : A ⟶ B ⊗ A. Then there is a unique homomorphism of algebras g : M(A) ⟶ B such that the diagram

[figure: A; M(A) ⊗ A; δ; f; B ⊗ A; g⊗1_{A}]

commutes. Then the following diagram

[figure: A; M(A) ⊗ A; δ; M(A) ⊗ A; M(A) ⊗ M(A) ⊗ A; Δ⊗1_{A}; 1_{M(A)}⊗δ; f; g⊗1_{A}; g⊗g⊗1_{A}; B ⊗ A; B ⊗ B ⊗ A; Δ_{B}⊗1_{A}; 1_{B}⊗f]

implies ((g ⊗ g)Δ ⊗ 1_{A})δ = (g ⊗ g ⊗ 1_{A})(Δ ⊗ 1_{A})δ = (g ⊗ g ⊗ 1_{A})(1_{M(A)} ⊗ δ)δ = (g ⊗ (g ⊗ 1_{A})δ)δ = (1_{B} ⊗ (g ⊗ 1_{A})δ)(g ⊗ 1_{A})δ = (1_{B} ⊗ f)f = (Δ_{B} ⊗ 1_{A})f = (Δ_{B} ⊗ 1_{A})(g ⊗ 1_{A})δ = (Δ_{B}g ⊗ 1_{A})δ hence (g ⊗ g)Δ = Δ_{B}g. Furthermore the diagram

[figure: A; M(A) ⊗ A; δ; B ⊗ A; f; g⊗1_{A}; ϵ⊗1_{A}; A ≅ 𝕂 ⊗ A; 1_{A}; ϵ_{B}⊗1_{A}]

implies ϵ_{B}g = ϵ. Thus g is a homomorphism of bialgebras. □

Since universal algebras for algebras A tend to become very big they do not exist in general. But a theorem of Tambara’s says that they exist for finite dimensional algebras (over a field 𝕂).

Definition 1.3.10. If X is a quantum space with finite dimensional function algebra then we call X a finite quantum space.

The following theorem is the quantum space version and equivalent to a theorem of Tambara.

Theorem 1.3.11. Let X be a finite quantum space. Then there exists a (universal) quantum space M(X) with morphism of quantum spaces μ : M(X) ⊥ X ⟶ X.

The algebra version of this theorem is

Theorem 1.3.12. (Tambara) Let A be a finite dimensional 𝕂-algebra. Then there exists a (universal) 𝕂-algebra M(A) with homomorphism of algebras δ : A ⟶ M(A) ⊗ A.

Proof. We are going to construct the 𝕂-algebra M(A) quite explicitly. First we observe that A^{∗} = Hom_{𝕂}(A, 𝕂) is a coalgebra (cf. [Advanced Algebra] Problem 2.25) with the structural morphism Δ : A^{∗} ⟶ (A ⊗ A)^{∗} ≅ A^{∗} ⊗ A^{∗}. Denote the dual basis by ∑_{i=1}^{n} a_{i} ⊗ ā^{i} ∈ A ⊗ A^{∗}. Now let T (A ⊗ A^{∗}) be the tensor algebra of the vector space A ⊗ A^{∗}. Consider elements of the tensor algebra

xy ⊗ ζ ∈ A ⊗ A^{∗},
x ⊗ y ⊗ Δ(ζ) ∈ A ⊗ A ⊗ A^{∗} ⊗ A^{∗} ≅ A ⊗ A^{∗} ⊗ A ⊗ A^{∗},
1 ⊗ ζ ∈ A ⊗ A^{∗},
ζ(1) ∈ 𝕂.

The following elements

(1)  xy ⊗ ζ − x ⊗ y ⊗ Δ(ζ)

and

(2)  1 ⊗ ζ − ζ(1)

generate a two-sided ideal I ⊆ T (A ⊗ A^{∗}). Now we define

M(A) := T (A ⊗ A^{∗})/I

and the cooperation δ : A ∋ a ⟶ ∑_{i=1}^{n}(a ⊗ ā^{i}) ⊗ a_{i} ∈ T (A ⊗ A^{∗})/I ⊗ A. This is a well-defined linear map.

To show that this map is a homomorphism of algebras we first describe the multiplication of A by a_{i}a_{j} = ∑_{k} α_{ij}^{k}a_{k}. Then the comultiplication of A^{∗} is given by Δ(ā^{k}) = ∑_{ij} α_{ij}^{k}ā^{i} ⊗ ā^{j} since (Δ(ā^{k}), a_{l} ⊗ a_{m}) = (ā^{k}, a_{l}a_{m}) = ∑_{r} α_{lm}^{r}(ā^{k}, a_{r}) = α_{lm}^{k} = ∑_{ij} α_{ij}^{k}(ā^{i}, a_{l})(ā^{j}, a_{m}) = (∑_{ij} α_{ij}^{k}ā^{i} ⊗ ā^{j}, a_{l} ⊗ a_{m}). Now write 1 = ∑ β^{k}a_{k}. Then we get ϵ(ā^{i}) = β^{i} since ϵ(ā^{i}) = (ā^{i}, 1) = ∑_{j} β^{j}(ā^{i}, a_{j}) = β^{i}. So we have δ(a)δ(b) = (∑_{i=1}^{n}(a ⊗ ā^{i}) ⊗ a_{i}) · (∑_{j=1}^{n}(b ⊗ ā^{j}) ⊗ a_{j}) = ∑_{ij}(a ⊗ b ⊗ ā^{i} ⊗ ā^{j}) ⊗ a_{i}a_{j} = ∑_{ijk} α_{ij}^{k}(a ⊗ b ⊗ ā^{i} ⊗ ā^{j}) ⊗ a_{k} = ∑_{k}(a ⊗ b ⊗ Δ(ā^{k})) ⊗ a_{k} = ∑_{k}(ab ⊗ ā^{k}) ⊗ a_{k} = δ(ab). Furthermore we have δ(1) = ∑_{i}(1 ⊗ ā^{i}) ⊗ a_{i} = ∑_{i} ā^{i}(1) ⊗ a_{i} = 1 ⊗ ∑_{i} ā^{i}(1)a_{i} = 1 ⊗ 1. Hence δ is a homomorphism of algebras.

Now we have to show that there is a unique g for each f. First of all f : A ⟶ B⊗A induces uniquely determined linear maps f_{i} : A ⟶ B with f(a) = ∑_{i} f_{i}(a) ⊗ a_{i} since the a_{i} form a basis. Since f is a homomorphism of algebras we get from ∑_{k} f_{k}(a) ⊗ a_{k} = f(ab) = f(a)f(b) = ∑_{ij}(f_{i}(a) ⊗ a_{i})(f_{j}(b) ⊗ a_{j}) = ∑_{ij} f_{i}(a)f_{j}(b) ⊗ a_{i}a_{j} = ∑_{ijk} α_{ij}^{k}f_{i}(a)f_{j}(b) ⊗ a_{k} by comparison of coefficients

f_{k}(ab) = ∑_{ij} α_{ij}^{k}f_{i}(a)f_{j}(b).

Furthermore we define g(a ⊗ ā) := (1 ⊗ ā)f(a) ∈ B. Then we get in particular g(a ⊗ ā^{i}) = (1 ⊗ ā^{i})(∑_{j} f_{j}(a) ⊗ a_{j}) = f_{i}(a). We can extend the map g to a homomorphism of algebras g : T (A ⊗ A^{∗}) ⟶ B. Applied to the generators of the ideal we get g(ab ⊗ ā^{k} − a ⊗ b ⊗ Δ(ā^{k})) = (1 ⊗ ā^{k}) ∑_{l} f_{l}(ab) ⊗ a_{l} − ∑_{rsij} α_{ij}^{k}(1 ⊗ ā^{i})(f_{r}(a) ⊗ a_{r}) · (1 ⊗ ā^{j})(f_{s}(b) ⊗ a_{s}) = f_{k}(ab) − ∑_{ij} α_{ij}^{k}f_{i}(a)f_{j}(b) = 0. We have furthermore g(1 ⊗ ζ − ζ(1)) = (1 ⊗ ζ)f(1) − ζ(1) = (1 ⊗ ζ)(1 ⊗ 1) − ζ(1) = 1ζ(1) − ζ(1) = 0. Thus the homomorphism of algebras g vanishes on the ideal I so it may be factored through M(A) = T (A)/I. Denote this factorization also by g. Then the diagram commutes since (g ⊗ 1_{A})δ(a) = (g ⊗ 1_{A})(∑_{i}(a ⊗ ā^{i}) ⊗ a_{i}) = ∑_{i}(1 ⊗ ā^{i})f(a) ⊗ a_{i} = ∑_{ij} f_{j}(a)(ā^{i}, a_{j}) ⊗ a_{i} = ∑_{i} f_{i}(a) ⊗ a_{i} = f(a).

We still have to show that g is uniquely determined. Assume that we also have (h ⊗ 1_{A})δ = f then ∑_{i} h(a ⊗ ā^{i}) ⊗ a_{i} = (h ⊗ 1_{A})δ(a) = f(a) = ∑_{i} f_{i}(a) ⊗ a_{i} hence h(a ⊗ ā^{i}) = f_{i}(a) = g(a ⊗ ā^{i}), i.e. g = h. □

Definition 1.3.13. Let A be a 𝕂-algebra. The universal algebra M(A) for A that is a bialgebra is also called the coendomorphism bialgebra of A.

Problem 1.3.7. (1) Determine explicitly the dual coalgebra A^{∗} of the algebra A := 𝕂⟨x⟩/(x^{2}). (Hint: Find a basis for A.)

(2) Determine and describe the coendomorphism bialgebra of A from problem 1.1. (Hint: Determine first a set of algebra generators of M(A). Then describe the relations.)

(3) Determine explicitly the dual coalgebra A^{∗} of A := 𝕂⟨x⟩/(x^{3}).

(4) Determine and describe the coendomorphism bialgebra of A from problem 1.3.

(5) (*) Determine explicitly the dual coalgebra A^{∗} of A := 𝕂⟨x, y⟩/I where the ideal I is generated as a two-sided ideal by the polynomials

xy − q^{−1}yx, x^{2}, y^{2}.

(6) (*) Determine the coendomorphism bialgebra of A from problem 1.5.

(7) Let A be a finite dimensional 𝕂-algebra with universal bialgebra A ⟶ B⊗A.

Show

i) that A^{op} ⟶ B^{op} ⊗ A^{op} is universal (where A^{op} has the multiplication ∇τ : A ⊗ A ⟶ A ⊗ A ⟶ A);

ii) that A ≅ A^{op} implies B ≅ B^{op} (as bialgebras); iii) that for commutative algebras A the algebra B satisfies B ≅ B^{op} but

that B need not be commutative. iv) Find an isomorphism B ≅ B^{op} for the bialgebra B = 𝕂⟨a, b⟩/(a^{2}, ab +

ba). (compare Problem 1.7 2).

(8) Consider the algebra K[ϵ]/(ϵ^{2}) the so called algebra of dual numbers over a field K. Consider the algebra B with (noncommuting) generators a, b, c, d and relations:

ac = acac ad = acad + adac bc = acbc = bcac
bd = acbd + adbc = bcad + bdac bcbc = bcbd + bdbc = 0
ac = 1 ad = 0

Show that B together with the cooperation

δ : A ⟶ B ⊗ A

with δ(ϵ) = bc ⊗ 1 + bd ⊗ ϵ is the coendomorphism bialgebra of A. Sketch of solution: 1/2. The bialgebra has the form B = 𝕂⟨a, b⟩/(a^{2}, ab+ba) with Δ(a) = a⊗1+b⊗a,

Δ(b) = b ⊗ b and ε(a) = 0, ε(b) = 1. The coaction is δ(x) = a ⊗ 1 + b ⊗ x.

3/4. A has the basis 1, x, x^{2}. The dual coalgebra has the dual basis e, ζ, ζ_{2} with Δ(e) = e ⊗ e, Δ(ζ) = ζ ⊗ e + e ⊗ ζ and Δ(ζ_{2}) = ζ_{2} ⊗ e + ζ ⊗ ζ + e ⊗ ζ_{2}.

The universal bialgebra B = T (A ⊗ A^{∗})/I satisfies δ(x) = x ⊗ e ⊗ 1 + x ⊗ ζ ⊗ x + x ⊗ ζ_{2} ⊗ x^{2} = a ⊗ 1 + b ⊗ x + c ⊗ x^{2}. Thus it is generated by the elements a = x ⊗ e, b = x ⊗ ζ and c = x ⊗ ζ_{2}. The multiplication table and the relations arise from

1 ⊗ e = 1,
1 ⊗ ζ = 1 ⊗ ζ_{2} = 0,
x^{2} ⊗ e = (x ⊗ e)(x ⊗ e),
x^{2} ⊗ ζ = (x ⊗ ζ)(x ⊗ e) + (x ⊗ e)(x ⊗ ζ),
x^{2} ⊗ ζ_{2} = (x ⊗ ζ_{2})(x ⊗ e) + (x ⊗ ζ)(x ⊗ ζ) + (x ⊗ e)(x ⊗ ζ_{2}),
0 = x^{3} ⊗ e = (x^{2} ⊗ e)(x ⊗ e),
0 = x^{3} ⊗ ζ = (x^{2} ⊗ ζ)(x ⊗ e) + (x^{2} ⊗ e)(x ⊗ ζ),
0 = x^{3} ⊗ ζ_{2} = (x^{2} ⊗ ζ_{2})(x ⊗ e) + (x^{2} ⊗ ζ)(x ⊗ ζ) + (x^{2} ⊗ e)(x ⊗ ζ_{2})

We use the abbreviation {u, v} := u^{2}v + uvu + vu^{2} and have

a^{3} = 0,
{a, b} = 0,
{a, c} + {b, a} = 0.

The condition (1 ⊗ δ)δ = (Δ ⊗ 1)δ implies

Δ(a) = a ⊗ 1 + b ⊗ a + c ⊗ a^{2},
Δ(b) = b ⊗ b + c ⊗ (ba + ab),
Δ(c) = b ⊗ c + c ⊗ b^{2} + c ⊗ (ca + ac),
ϵ(a) = 0,
ϵ(b) = 1,
ϵ(c) = 0.

5/6. A has the basis 1, x, y, xy. The dual basis of A^{∗} is denoted by e, ξ, η, θ. The diagonal is

Δ(e) = e ⊗ e,
Δ(ξ) = ξ ⊗ e + e ⊗ ξ,
Δ(η) = η ⊗ e + e ⊗ η,
Δ(θ) = θ ⊗ e + e ⊗ θ + ξ ⊗ η + qη ⊗ ξ.

Thus the coendomorphism bialgebra has the algebra generators a ⊗ ζ with a ∈ {1, x, y, xy} and ζ ∈ {e, ξ, η, θ}. The generators of the relations (of I) are given by the equations 1.1 and 1.2. They imply that 1 ⊗ e is the unit element, that 1 ⊗ ξ = 1 ⊗ η = 1 ⊗ θ = 0 and that

ab ⊗ e = (a ⊗ e)(b ⊗ e),
ab ⊗ ξ = (a ⊗ ξ)(b ⊗ e) + (a ⊗ 1)(b ⊗ ξ),
ab ⊗ η = (a ⊗ η)(b ⊗ e) + (a ⊗ 1)(b ⊗ η),
ab ⊗ θ = (a ⊗ θ)(b ⊗ e) + (a ⊗ 1)(b ⊗ θ) + (a ⊗ ξ)(b ⊗ η) + q(a ⊗ η)(b ⊗ ξ).

Furthermore for ab we have to take into account the relations in A.

We define

\begin{matrix}a := x ⊗ e, & b := x ⊗ ξ, & c := x ⊗ η, & d := x ⊗ θ, \\ e := y ⊗ e, & f := y ⊗ ξ, & g := y ⊗ η, & h := x ⊗ θ, \end{matrix}

30 1. COMMUTATIVE AND NONCOMMUTATIVE ALGEBRAIC GEOMETRY

and get δ(x) = a ⊗ 1 + b ⊗ x + c ⊗ y + d ⊗ xy and δ(y) = e ⊗ 1 + f ⊗ x + g ⊗ y + h ⊗ xy. Hence B is generated by a, …, h as an algebra. The relations are

a^{2} = e^{2} = 0, ab + ba = ac + ca = ef + fe = eg + ge = 0, ad + da + bc + qcb = eh + he + fg + qgf = 0, ae = qea, af + be = q(fa + eb), ag + ce = q(ga + ec), ah − qha + de − qed + bg − q^{2}gb + qcf − qfc = 0.

The diagonal is Δ(a) = a ⊗ 1 + b ⊗ a + c ⊗ e + d ⊗ ae, Δ(b) = b ⊗ b + c ⊗ f + d ⊗ (af + be), Δ(c) = b ⊗ c + c ⊗ g + d ⊗ (ag + ce), Δ(d) = b ⊗ d + c ⊗ h + d ⊗ (ah + de + bg + q^{−1}cf) etc.

CHAPTER 2 Hopf Algebras, Algebraic, Formal, and Quantum Groups

Introduction

In the first chapter we have encountered quantum monoids and studied their role as monoids operating on quantum spaces. The “elements” of quantum monoids operating on quantum spaces should be understood as endomorphisms of the quantum spaces. In the construction of the multiplication for universal quantum monoids of quantum spaces we have seen that this multiplication is essentially the “composition” of endomorphisms.

We are, however, primarily interested in automorphisms and we know that automorphisms should form a group under composition. This chapter is devoted to finding group structures on quantum monoids, i.e. to define and study quantum groups.

This is easy in the commutative situation, i.e. if the representing algebra of a quantum monoid is commutative. Then we can define a morphism that sends elements of the quantum group to their inverses. This will lead us to the notion of affine algebraic groups.

In the noncommutative situation, however, it will turn out that such an inversion morphism (of quantum spaces) does not exist. It will have to be replaced by a more complicated construction. Thus quantum groups will not be groups in the sense of category theory. Still we will be able to perform one of the most important and most basic constructions in group theory, the formation of the group of invertible elements of a monoid. In the case of a quantum monoid acting universally on a quantum space this will lead to the good definition of a quantum automorphism group of the quantum space.

In order to have the appropriate tools for introducing quantum groups we first introduce Hopf algebras which will be the representing algebras of quantum groups. Furthermore we need the notion of a monoid and of a group in a category. We will see, however, that quantum groups are in general not groups in the category of quantum spaces.

We first study the simple cases of affine algebraic groups and of formal groups. They will have Hopf algebras as representing objects and will indeed be groups in reasonable categories. Then we come to quantum groups, and construct quantum automorphism groups of quantum spaces.

At the end of the chapter you should

• know what a Hopf algebra is, • know what a group in a category is,

• know the definition and examples of affine algebraic groups and formal groups,

• know the definition and examples of quantum groups and be able to construct quantum automorphism groups for small quantum spaces,

• understand why a Hopf algebra is a reasonable representing algebra for a quantum group.

1. Hopf Algebras

The difference between a monoid and a group lies in the existence of an additional map S : G ∋ g ↦ g^{−1} ∈ G for a group G that allows forming inverses. This map satisfies the equation S(g)g = 1 or in a diagrammatic form

[figure: G; {1}; ε; G; 1; Δ; G × G; G × G; S×id; mult]

We want to carry this property over to bialgebras B instead of monoids. An “inverse map” shall be a morphism S : B ⟶ B with a similar property. This will be called a Hopf algebra.

Definition 2.1.1. A left Hopf algebra H is a bialgebra H together with a left antipode S : H ⟶ H, i.e. a 𝕂-module homomorphism S such that the following diagram commutes:

[figure: H; 𝕂; ε; H; η; Δ; H ⊗ H; H ⊗ H; S⊗id; ∇]

Symmetrically we define a right Hopf algebra H. A Hopf algebra is a left and right Hopf algebra. The map S is called a (left, right, two-sided) antipode.

Using the Sweedler notation (2.20) the commutative diagram above can also be expressed by the equation

∑ S(a_{(1)})a_{(2)} = ηε(a)

for all a ∈ H. Observe that we do not require that S : H ⟶ H is an algebra homomorphism.

Problem 2.1.8. (1) Let H be a bialgebra and S ∈ Hom(H, H). Then S is an antipode for H (and H is a Hopf algebra) iff S is a two sided inverse for id in the algebra (Hom(H, H), ∗, ηε) (see 2.21). In particular S is uniquely determined.

(2) Let H be a Hopf algebra. Then S is an antihomomorphism of algebras and coalgebras i.e. S “inverts the order of the multiplication and the comultiplication”.

(3) Let H and K be Hopf algebras and let f : H ⟶ K be a homomorphism of bialgebras. Then fS_{H} = S_{K}f, i.e. f is compatible with the antipode.

Definition 2.1.2. Because of Problem (22) every homomorphism of bialgebras between Hopf algebras is compatible with the antipodes. So we define a homomorphism of Hopf algebras to be a homomorphism of bialgebras. The category of Hopf algebras will be denoted by 𝕂-Hopf .

Proposition 2.1.3. Let H be a bialgebra with an algebra generating set X. Let S : H ⟶ H^{op} be an algebra homomorphism such that ∑ S(x_{(1)})x_{(2)} = ηε(x) for all x ∈ X. Then S is a left antipode of H.

Proof. Assume a, b ∈ H such that ∑ S(a_{(1)})a_{(2)} = ηε(a) and ∑ S(b_{(1)})b_{(2)} = ηε(b). Then

\begin{aligned}∑ S((ab)_{(1)})(ab)_{(2)} & = ∑ S(a_{(1)}b_{(1)})a_{(2)}b_{(2)} = ∑ S(b_{(1)})S(a_{(1)})a_{(2)}b_{(2)} \\ & = ∑ S(b_{(1)})ηε(a)b_{(2)} = ηε(a)ηε(b) = ηε(ab). \end{aligned}

Since every element of H is a finite sum of finite products of elements in X, for which the equality holds, this equality extends to all of H by induction. □

Example 2.1.4. (1) Let V be a vector space and T (V) the tensor algebra over V. We have seen in Problem 2.2 that T (V) is a bialgebra and that V generates T (V) as an algebra. Define S : V ⟶ T (V)^{op} by S(v) := −v for all v ∈ V. By the universal property of the tensor algebra this map extends to an algebra homomorphism S : T (V) ⟶ T (V)^{op}. Since Δ(v) = v ⊗1+1⊗v we have ∑ S(v_{(1)})v_{(2)} = ∇(S ⊗1)Δ(v) = −v + v = 0 = ηε(v) for all v ∈ V, hence T (V) is a Hopf algebra by the preceding proposition.

(2) Let V be a vector space and S(V) the symmetric algebra over V (that is commutative). We have seen in Problem 2.3 that S(V) is a bialgebra and that V generates S(V) as an algebra. Define S : V ⟶ S(V) by S(v) := −v for all v ∈ V. S extends to an algebra homomorphism S : S(V) ⟶ S(V). Since Δ(v) = v ⊗ 1 + 1 ⊗ v we have ∑ S(v_{(1)})v_{(2)} = ∇(S ⊗ 1)Δ(v) = −v + v = 0 = ηε(v) for all v ∈ V, hence S(V) is a Hopf algebra by the preceding proposition.

Example 2.1.5. (Group Algebras) For each algebra A we can form the group of units U(A) := {a ∈ A|∃a^{−1} ∈ A} with the multiplication of A as composition of the group. Then U is a covariant functor U : 𝕂-Alg ⟶ Gr . This functor leads to the following universal problem.

Let G be a group. An algebra 𝕂G together with a group homomorphism ι : G ⟶ U(𝕂G) is called a (the) group algebra of G, if for every algebra A and for every group homomorphism f : G ⟶ U(A) there exists a unique homomorphism of algebras g : 𝕂G ⟶ A such that the following diagram commutes

[figure: G; U(𝕂G); ι; f; U(A).; g]

The group algebra 𝕂G is (if it exists) unique up to isomorphism. It is generated as an algebra by the image of G. The map ι : G ⟶ U(𝕂G) ⊆ 𝕂G is injective and the image of G in 𝕂G is a basis.

The group algebra can be constructed as the free vector space 𝕂G with basis G and the algebra structure of 𝕂G is given by 𝕂G ⊗ 𝕂G ∋ g ⊗ h ↦ gh ∈ 𝕂G and the unit η : 𝕂 ∋ α ↦ αe ∈ 𝕂G.

The group algebra 𝕂G is a Hopf algebra. The comultiplication is given by the diagram

[figure: G; 𝕂G; ι; f; 𝕂G ⊗ 𝕂G; Δ]

with f(g) := g ⊗ g which defines a group homomorphism f : G ⟶ U(𝕂G ⊗ 𝕂G). The counit is given by

[figure: G; 𝕂G; ι; f; 𝕂; ε]

where f(g) = 1 for all g ∈ G. One shows easily by using the universal property, that Δ is coassociative and has counit ε. Define an algebra homomorphism S : 𝕂G ⟶ (𝕂G)^{op} by

[figure: G; 𝕂G; ι; f; (𝕂G)^{op}; S]

with f(g) := g^{−1} which is a group homomorphism f : G ⟶ U((𝕂G)^{op}). Then one shows with Proposition 1.3 that 𝕂G is a Hopf algebra.

The example 𝕂G of a Hopf algebra gives rise to the definition of particular elements in arbitrary Hopf algebras, that share certain properties with elements of a group. We will use and study these elements later on in in chapter 4.

Definition 2.1.6. Let H be a Hopf algebra. An element g ∈ H, g ≠ 0 is called a grouplike element if

Δ(g) = g ⊗ g.

Observe that ε(g) = 1 for each grouplike element g in a Hopf algebra H. In fact we have g = ∇(ε ⊗ 1)Δ(g) = ε(g)g ≠ 0 hence ε(g) = 1. If the base ring is not a field then one adds this property to the definition of a grouplike element.

Problem 2.1.9. (1) Let 𝕂 be a field. Show that an element x ∈ 𝕂G satisfies

Δ(x) = x ⊗ x and ε(x) = 1 if and only if x = g ∈ G.

(2) Show that the grouplike elements of a Hopf algebra form a group under multiplication of the Hopf algebra.

Example 2.1.7. (Universal Enveloping Algebras) A Lie algebra consists of a vector space 𝔤 together with a (linear) multiplication 𝔤 ⊗ 𝔤 ∋ x ⊗ y ↦ [x, y] ∈ 𝔤 such that the following laws hold:

[x, x] = 0,
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (Jacobi identity).

A homomorphism of Lie algebras f : 𝔤 ⟶ 𝔥 is a linear map f such that f([x, y]) = [f(x), f(y)]. Thus Lie algebras form a category 𝕂-Lie .

An important example is the Lie algebra associated with an associative algebra (with unit). If A is an algebra then the vector space A with the Lie multiplication

[x, y] := xy − yx

is a Lie algebra denoted by A^{L}. This construction of a Lie algebra defines a covariant functor -^{L} : 𝕂-Alg ⟶ 𝕂-Lie . This functor leads to the following universal problem.

Let 𝔤 be a Lie algebra. An algebra U(𝔤) together with a Lie algebra homomorphism ι : 𝔤 ⟶ U(𝔤)^{L} is called a (the) universal enveloping algebra of 𝔤, if for every algebra A and for every Lie algebra homomorphism f : 𝔤 ⟶ A^{L} there exists a unique homomorphism of algebras g : U(𝔤) ⟶ A such that the following diagram commutes

[figure: 𝔤; U(𝔤)^{L}; ι; f; A^{L}.; g]

The universal enveloping algebra U(𝔤) is (if it exists) unique up to isomorphism. It is generated as an algebra by the image of 𝔤.

The universal enveloping algebra can be constructed as U(𝔤) = T (𝔤)/(x ⊗ y − y ⊗ x − [x, y]) where T (𝔤) = 𝕂 ⊕ 𝔤 ⊕ 𝔤 ⊗ 𝔤 … is the tensor algebra. The map ι : 𝔤 ⟶ U(𝔤)^{L} is injective.

The universal enveloping algebra U(𝔤) is a Hopf algebra. The comultiplication is given by the diagram

[figure: 𝔤; U(𝔤); ι; f; U(𝔤) ⊗ U(𝔤); Δ]

with f(x) := x ⊗ 1 + 1 ⊗ x which defines a Lie algebra homomorphism f : 𝔤 ⟶ (U(𝔤) ⊗ U(𝔤))^{L}. The counit is given by

[figure: 𝔤; U(𝔤); ι; f; 𝕂; ε]

with f(x) = 0 for all x ∈ 𝔤. One shows easily by using the universal property, that Δ is coassociative and has counit ε. Define an algebra homomorphism S : U(𝔤) ⟶ (U(𝔤))^{op} by

[figure: 𝔤; U(𝔤); ι; f; (U(𝔤))^{op}; S]

with f(x) := −x which is a Lie algebra homomorphism f : 𝔤 ⟶ (U(𝔤)^{op})^{L}. Then one shows with Proposition 1.3 that U(𝔤) is a Hopf algebra.

(Observe, that the meaning of U in this example and the previous example (group of units, universal enveloping algebra) is totally different, in the first case U can be applied to an algebra and gives a group, in the second case U can be applied to a Lie algebra and gives an algebra.)

The preceding example of a Hopf algebra gives rise to the definition of particular elements in arbitrary Hopf algebras, that share certain properties with elements of a Lie algebra. We will use and study these elements later on in chapter 4.

Definition 2.1.8. Let H be a Hopf algebra. An element x ∈ H is called a primitive element if

Δ(x) = x ⊗ 1 + 1 ⊗ x.

Let g ∈ H be a grouplike element. An element x ∈ H is called a skew primitive or g-primitive element if

Δ(x) = x ⊗ 1 + g ⊗ x.

Problem 2.1.10. Show that the set of primitive elements P (H) = {x ∈ H|Δ(x) = x ⊗ 1 + 1 ⊗ x} of a Hopf algebra H is a Lie subalgebra of H^{L}.

Proposition 2.1.9. Let H be a Hopf algebra with antipode S. The following are equivalent:

(1) S^{2} =id.

(2) ∑ S(a_{(2)})a_{(1)} = ηε(a) for all a ∈ H.

(3) ∑ a_{(2)}S(a_{(1)}) = ηε(a) for all a ∈ H.

Proof. Let S^{2} =id. Then

\begin{aligned}∑ S(a_{(2)})a_{(1)} & = S^{2}(∑ S(a_{(2)})a_{(1)}) = S(∑ S(a_{(1)})S^{2}(a_{(2)})) \\ & = S(∑ S(a_{(1)})a_{(2)}) = S(ηε(a)) = ηε(a) \end{aligned}

by using Problem (21).

Conversely assume that (2) holds. Then

\begin{aligned}S ∗ S^{2}(a) & = ∑ S(a_{(1)}S^{2}(a_{(2)}) = S(∑ S(a_{(2)})a_{(1)} \\ & = S(ηε(a)) = ηε(a). \end{aligned}

Thus S^{2} and id are inverses of S in the convolution algebra Hom(H, H), hence S^{2} = id. Analogously one shows that (1) and (3) are equivalent. □

Corollary 2.1.10. If H is a commutative Hopf algebra or a cocommutative Hopf algebra with antipode S, then S^{2} =id.

Remark 2.1.11. Kaplansky: Ten conjectures on Hopf algebras In a set of lecture notes on bialgebras based on a course given at Chicago university

in 1973, made public in 1975, Kaplansky formulated a set of conjectures on Hopf algebras that have been the aim of intensive research.

(1) If C is a Hopf subalgebra of the Hopf algebra B then B is a free left C-module.

(Yes, if H is finite dimensional [Nichols-Zoeller]; No for infinite dimensional Hopf algebras [Oberst-Schneider]; B : C is not necessarily faithfully flat [Schauenburg])

(2) Call a coalgebra C admissible if it admits an algebra structure making it a Hopf algebra. The conjecture states that C is admissible if and only if every finite subset of C lies in a finite-dimensional admissible subcoalgebra.

(Remarks.

(a) Both implications seem hard.

(b) There is a corresponding conjecture where “Hopf algebra” is replaced by “bialgebra”.

(c) There is a dual conjecture for locally finite algebras.)

(No results known.)

(3) A Hopf algebra of characteristic 0 has no non-zero central nilpotent elements.

(First counter example given by [Schmidt-Samoa]. If H is unimodular and not semisimple, e.g. a Drinfel’d double of a not semisimple finite dimensional Hopf algebra, then the integral Λ satisfies Λ ≠ 0, Λ^{2} = ε(Λ)Λ = 0 since D(H) is not semisimple, and aΛ = ε(a)Λ = Λε(a) = Λa since D(H) is unimodular [Sommerhäuser].)

(4) (Nichols). Let x be an element in a Hopf algebra H with antipode S. Assume that for any a in H we have

∑ b_{i}xS(c_{i}) = ε(a)x

where Δa = ∑ b_{i} ⊗ c_{i}. Conjecture: x is in the center of H. (ax = ∑ a_{(1)}xε(a_{(2)}) = ∑ a_{(1)}xS(a_{(2)})a_{(3)}) = ∑ ε(a_{(1)})xa_{(2)} = xa.) In the remaining six conjectures H is a finite-dimensional Hopf algebra

over an algebraically closed field.

(5) If H is semisimple on either side (i.e. either H or the dual H^{∗} is semisimple

as an algebra) the square of the antipode is the identity.

(Yes if char(𝕂) = 0 [Larson-Radford], yes if char(𝕂) is large [Sommerhäuser])

(6) The size of the matrices occurring in any full matrix constituent of H divides

the dimension of H.

(Yes is Hopf algebra is defined over ℤ [Larson]; in general not known; work by [Montgomery-Witherspoon], [Zhu], [Gelaki])

(7) If H is semisimple on both sides the characteristic does not divide the di-

mension.

(Larson-Radford)

(8) If the dimension of H is prime then H is commutative and cocommutative. (Yes in characteristic 0 [Zhu: 1994]) Remark. Kac, Larson, and Sweedler have partial results on 5 – 8. (Was also proved by [Kac]) In the two final conjectures assume that the characteristic does not divide

the dimension of H.

(9) The dimension of the radical is the same on both sides.

(Counterexample by [Nichols]; counterexample in Frobenius-Lusztig kernel of U_{q}(sl(2)) [Schneider])

(10) There are only a finite number (up to isomorphism) of Hopf algebras of a

given dimension.

(Yes for semisimple, cosemisimple Hopf algebras: Stefan 1997) (Counterexamples: [Andruskiewitsch, Schneider], [Beattie, others] 1997)

2. Monoids and Groups in a Category

Before we use Hopf algebras to describe quantum groups and some of the better known groups, such as affine algebraic groups and formal groups, we introduce the concept of a general group (and of a monoid) in an arbitrary category. Usually this concept is defined with respect to a categorical product in the given category. But in some categories there are in general no products. Still, one can define the concept of a group in a very simple fashion. We will start with that definition and then show that it coincides with the usual notion of a group in a category in case that category has finite products.

Definition 2.2.1. Let C be an arbitrary category. Let G ∈ C be an object. We use the notation G(X) := Mor_{C}(X, G) for all X ∈ C, G(f) := Mor_{C}(f, G) for all morphisms f : X ⟶ Y in C, and f(X) := Mor_{C}(X, f) for all morphisms f : G ⟶ G′.

G together with a natural transformation m : G(-) × G(-) ⟶ G(-) is called a group (monoid) in the category C, if the sets G(X) together with the multiplication m(X) : G(X) × G(X) ⟶ G(X) are groups (monoids) for all X ∈ C.

Let (G, m) and (G′, m′) be groups in C. A morphism f : G ⟶ G′ in C is called a homomorphism of groups in C, if the diagrams

[figure: G(X) × G(X); G(X); m(X); G′(X) × G′(X); G′(X); m′(X); f(X)×f(X); f(X)]

commute for all X ∈ C.

Let (G, m) and (G′, m′) be monoids in C. A morphism f : G ⟶ G′ in C is called a homomorphism of monoids in C, if the diagrams

[figure: G(X) × G(X); G(X); m(X); G′(X) × G′(X); G′(X); m′(X); f(X)×f(X); f(X)]

and

[figure: {∗}; u; u′; G(X); G′(X); f(X)]

commute for all X ∈ C.

Problem 2.2.11. (1) Let the set Z together with the multiplication m : Z × Z ⟶ Z be a monoid. Show that the unit element e ∈ Z is uniquely determined.

Let (Z, m) be a group. Show that also the inverse i : Z ⟶ Z is uniquely determined.

Show that unit element and inverses of groups are preserved by maps that are compatible with the multiplication.

(2) Find an example of monoids Y and Z and a map f : Y ⟶ Z with f(y_{1}y_{2}) = f(y_{1})f(y_{2}) for all y_{1}, y_{2} ∈ Y, but f(e_{Y}) ≠ e_{Z}.

(3) Let (G, m) be a group in C and i_{X} : G(X) ⟶ G(X) be the inverse for all

X ∈ C. Show that i is a natural transformation.

Show that the Yoneda Lemma provides a morphism S : G ⟶ G such that i_{X} = Mor_{C}(X, S) = S(X) for all X ∈ C.

Formulate and prove properties of S of the type S ∗ id = . . .. Proposition 2.2.2. Let C be a category with finite (categorical) products. An

object G in C carries the structure m : G(-) × G(-) ⟶ G(-) of a group in C if and only if there are morphisms m : G × G ⟶ G, u : E ⟶ G, and S : G ⟶ G such that the diagrams

[figure: G × G × G; G × G; m×1; G × G; G; m; 1×m; m; E × G ≅ G ≅ G × E; G × G; 1×u; G × G; G; m; u×1; m; 1]

[figure: G; E; G; u; G × G; G × G; 1×S; S×1; Δ; m]

commute where Δ is the morphism defined in [Advanced Algebra] 4.2. The multiplications are related by m_{X} = Mor_{C}(X, m) = m(X).

An analogous statement holds for monoids.

Proof. The Yoneda Lemma defines a bijection between the set of morphisms f : X ⟶ Y and the set of natural transformations f(-) : X(-) ⟶ Y (-) by f(Z) = Mor_{C}(Z, f). In particular we have m_{X} = Mor_{C}(X, m) = m(X). The diagram

[figure: G(-) × G(-) × G(-); G(-) × G(-); m-×1; G(-) × G(-); G(-); m-; 1×m-; m-]

commutes if and only if Mor_{C}(-, m(m × 1)) = Mor_{C}(-, m)(Mor_{C}(-, m) × 1) = m-(m- × 1) = m-(1 × m-) = Mor_{C}(-, m)(1 × Mor_{C}(-, m)) = Mor_{C}(-, m(1 × m)) if and only if m(m × 1) = m(1 × m) if and only if the diagram

[figure: G × G × G; G × G; m×1; G × G; G; m; 1×m; m]

commutes. In a similar way one shows the equivalence of the other diagram(s). □

Problem 2.2.12. Let C be a category with finite products. Show that a morphism f : G ⟶ G′ in C is a homomorphism of groups if and only if

[figure: G × G; G; m; G′ × G′; G′; m′; f×f; f]

commutes.

Definition 2.2.3. A cogroup (comonoid) G in C is a group (monoid) in C^{op}, i.e. an object G ∈ Ob C = Ob C^{op} together with a natural transformation m(X) : G(X) × G(X) ⟶ G(X) where G(X) = Mor_{C^{op}}(X, G) = Mor_{C}(G, X), such that (G(X), m(X)) is a group (monoid) for each X ∈ C.

Remark 2.2.4. Let C be a category with finite (categorical) coproducts. An object G in C carries the structure m : G(-) × G(-) ⟶ G(-) of a cogroup in C if and only if there are morphisms Δ : G ⟶ G ⨿ G, ε : G ⟶ I, and S : G ⟶ G such that the diagrams

[figure: G ⨿ G; G ⨿ G ⨿ G; Δ⨿1; G; G ⨿ G; Δ; 1⨿Δ; Δ; G ⨿ G; I ⨿ G ≅ G ≅ G ⨿ I; ε⨿1; G; G ⨿ G; Δ; 1⨿ε; Δ; 1]

[figure: G; I; G; ε; G ⨿ G; G ⨿ G; 1⨿S; S⨿1; Δ; ∇]

commute where ∇ is dual to the morphism Δ defined in [Advanced Algebra] 4.2. The multiplications are related by Δ_{X} = Mor_{C}(Δ, X) = Δ(X).

Let C be a category with finite coproducts and let G and G′ be cogroups in C. Then a homomorphism of groups f : G′ ⟶ G is a morphism f : G ⟶ G′ in C such that the diagram

[figure: G; G × G; Δ; G′; G′ × G′; Δ′; f×f; f]

commutes. An analogous result holds for comonoids.

Remark 2.2.5. Obviously similar observations and statements can be made for other algebraic structures in a category C. So one can introduce vector spaces and covector spaces, monoids and comonoids, rings and corings in a category C. The structures can be described by morphisms in C if C is a category with finite (co-) products.

Problem 2.2.13. Determine the structure of a covector space on a vector space V from the fact that Hom(V, W) is a vector space for all vector spaces W.

Proposition 2.2.6. Let G ∈ C be a group with multiplication a ∗ b, unit e, and inverse a^{−1} in G(X). Then the morphisms m : G × G ⟶ G, u : E ⟶ G, and S : G ⟶ G are given by

m = p_{1} ∗ p_{2},  u = e_{E},  S = id_{G}^{−1}.

Proof. By the Yoneda Lemma [Advanced Algebra] 5.7 these morphisms can be constructed from the natural transformation as follows. Under Mor_{C}(G×G, G×G) =

G × G(G × G) ≅ G(G × G) × G(G × G) ⟶^{∗} G(G × G) = Mor_{C}(G × G, G) the

identity id_{G×G} = (p_{1}, p_{2}) is mapped to m = p_{1} ∗ p_{2}. Under Mor_{C}(E, E) = E(E) ⟶ G(E) = Mor_{C}(E, G) the identity of E is mapped to the neutral element u = e_{E}. Under Mor_{C}(G, G) = G(G) ⟶ G(G) = Mor_{C}(G, G) the identity is mapped to its ∗-inverse S = id_{G}^{−1}. □

Corollary 2.2.7. Let G ∈ C be a cogroup with multiplication a ∗ b, unit e, and inverse a^{−1} in G(X). Then the morphisms Δ : G ⟶ G ⨿ G, ε : G ⟶ I, and S : G ⟶ G are given by

Δ = ι_{1} ∗ ι_{2},  ε = e_{I},  S = id_{G}^{−1}.

3. Affine Algebraic Groups

We apply the preceding considerations to the categories 𝕂-cAlg and 𝕂-cCoalg . Consider 𝕂-cAlg , the category of commutative 𝕂-algebras. Let A, B ∈ 𝕂-cAlg .

Then A ⊗ B is again a commutative 𝕂-algebra with componentwise multiplication. In fact this holds also for non-commutative 𝕂-algebras ([Advanced Algebra] 2.3), but in 𝕂-cAlg we have

Proposition 2.3.1. The tensor product in 𝕂-cAlg is the (categorical) coproduct.

Proof. Let f ∈ 𝕂-cAlg (A, Z), g ∈ 𝕂-cAlg (B, Z). The map [f, g] : A ⊗ B ⟶ Z defined by [f, g](a ⊗ b) := f(a)g(b) is the unique algebra homomorphism such that [f, g](a ⊗ 1) = f(a) and [f, g](1 ⊗ b) = g(b) or such that the diagram

[figure: A; A ⊗ B; ι_{A}; B; ι_{B}; f; g; Z; [f,g]]

commutes, where ι_{A}(a) = a ⊗ 1 and ι_{B}(b) = 1 ⊗ b are algebra homomorphisms. □

So the category 𝕂-cAlg has finite coproducts and also an initial object 𝕂.

A more general property of the tensor product of arbitrary algebras was already considered in 1.2.13.

Observe that the following diagram commutes

[figure: A; A ⊗ A; ι_{1}; A; ι_{2}; 1_{A}; 1_{A}; A; ∇]

where ∇ is the multiplication of the algebra and by the diagram the codiagonal of the coproduct.

Definition 2.3.2. An affine algebraic group is a group in the category of commutative geometric spaces.

By the duality between the categories of commutative geometric spaces and commutative algebras, an affine algebraic group is represented by a cogroup in the category of 𝕂-cAlg of commutative algebras.

For an arbitrary affine algebraic group H we get by Corollary 2.2.7

Δ = ι_{1} ∗ ι_{2} ∈ 𝕂-cAlg (H, H ⊗ H),
ε = e ∈ 𝕂-cAlg (H, 𝕂), and S = (id)^{−1} ∈ 𝕂-cAlg (H, H).

These maps and Corollary 2.2.7 lead to

Proposition 2.3.3. Let H ∈ 𝕂-cAlg . H is a representing object for a functor 𝕂-cAlg ⟶ Gr if and only if H is a Hopf algebra.

Proof. Both statements are equivalent to the existence of morphisms in 𝕂-cAlg

Δ : H ⟶ H ⊗ H ε : H ⟶ 𝕂 S : H ⟶ H

such that the following diagrams commute

[figure: H; H ⊗ H; Δ; Δ; (coassociativity); Δ⊗1; H ⊗ H; H ⊗ H ⊗ H; 1⊗Δ] [figure: H; H ⊗ H; Δ; H ⊗ H; 𝕂 ⊗ H ≅ H ≅ H ⊗ 𝕂; ε⊗1; (counit); Δ; 1⊗ε; 1]

[figure: (coinverse); H; 𝕂; ϵ; H; η; Δ; H ⊗ H; H ⊗ H; S⊗id; id ⊗S; ∇]

□

This Proposition says two things. First of all each commutative Hopf algebra H defines a functor 𝕂-cAlg (H, -) : 𝕂-cAlg ⟶ Set that factors through the category of groups or simply a functor 𝕂-cAlg (H, -) : 𝕂-cAlg ⟶ Gr . Secondly each representable functor 𝕂-cAlg (H, -) : 𝕂-cAlg ⟶ Set that factors through the category of groups is represented by a commutative Hopf algebra.

Corollary 2.3.4. An algebra H ∈ 𝕂-cAlg represents an affine algebraic group if and only if H is a commutative Hopf algebra.

The category of commutative Hopf algebras is dual to the category of affine algebraic groups.

In the following lemmas we consider functors represented by commutative algebras. They define functors on the category 𝕂-cAlg as well as more generally on 𝕂-Alg . We first study the functors and the representing algebras. Then we use them to construct commutative Hopf algebras.

Lemma 2.3.5. The functor 𝔾_{a} : 𝕂-Alg ⟶ Ab defined by 𝔾_{a}(A) := A^{+}, the underlying additive group of the algebra A, is a representable functor represented by the algebra 𝕂[x] the polynomial ring in one variable x.

Proof. 𝔾_{a} is an underlying functor that forgets the multiplicative structure of the algebra and only preserves the additive group of the algebra. We have to determine natural isomorphisms (natural in A) 𝔾_{a}(A) ≅ 𝕂-Alg (𝕂[x], A). Each element a ∈ A^{+} is mapped to the homomorphism of algebras a_{∗} : 𝕂[x] ∋ p(x) ↦ p(a) ∈ A. This is a homomorphism of algebras since a_{∗}(p(x) + q(x)) = p(a) + q(a) = a_{∗}(p(x)) + a_{∗}(q(x)) and a_{∗}(p(x)q(x)) = p(a)q(a) = a_{∗}(p(x))a_{∗}(q(x)). Another reason to see this is that 𝕂[x] is the free (commutative) 𝕂-algebra over {x} i.e. since each map {x} ⟶ A can be uniquely extended to a homomorphism of algebras 𝕂[x] ⟶ A. The map A ∋ a ↦ a_{∗} ∈ 𝕂-Alg (𝕂[x], A) is bijective with the inverse map 𝕂-Alg (𝕂[x], A) ∋ f ↦ f(x) ∈ A. Finally this map is natural in A since

[figure: B; 𝕂-Alg (𝕂[x], B); -_{∗}; A; 𝕂-Alg (𝕂[x], A); -_{∗}; g; K-Alg(K[x],g)]

commutes for all g ∈ 𝕂-Alg (A, B). □

Remark 2.3.6. Since A^{+} has the structure of an additive group the sets of homomorphisms of algebras 𝕂-Alg (𝕂[x], A) are also additive groups.

Lemma 2.3.7. The functor 𝔾_{m} = U : 𝕂-Alg ⟶ Gr defined by 𝔾_{m}(A) := U(A), the underlying multiplicative group of units of the algebra A, is a representable functor represented by the algebra 𝕂[x, x^{−1}] = 𝕂[x, y]/(xy−1) the ring of Laurent polynomials in one variable x.

Proof. We have to determine natural isomorphisms (natural in A) 𝔾_{m}(A) ≅ 𝕂-Alg (𝕂[x, x^{−1}], A). Each element a ∈ 𝔾_{m}(A) is mapped to the homomorphism of algebras a_{∗} := (𝕂[x, x^{−1}] ∋ x ↦ a ∈ A). This defines a unique homomorphism of algebras since each homomorphism of algebras f from 𝕂[x, x^{−1}] = 𝕂[x, y]/(xy − 1) to A is completely determined by the images of x and of y but in addition the images have to satisfy f(x)f(y) = 1, i.e. f(x) must be invertible and f(y) must be the inverse to f(x). This mapping is bijective. Furthermore it is natural in A since

[figure: B; 𝕂-Alg (𝕂[x, x^{−1}], B); -_{∗}; A; 𝕂-Alg (𝕂[x, x^{−1}], A); -_{∗}; g; K-Alg(K[x,x^{−1}],g)]

for all g ∈ 𝕂-Alg (A, B) commute. □

Remark 2.3.8. Since U(A) has the structure of a (multiplicative) group the sets 𝕂-Alg (𝕂[x, x^{−1}], A) are also groups.

Lemma 2.3.9. The functor 𝕄_{n} : 𝕂-Alg ⟶ 𝕂-Alg with 𝕄_{n}(A) the algebra of n × n-matrices with entries in A is representable by the algebra 𝕂⟨x_{11}, x_{12}, …, x_{nn}⟩, the non commutative polynomialring in the variables x_{ij}.

Proof. The polynomial ring 𝕂⟨x_{ij}⟩ is free over the set {x_{ij}} in the category of (non commutative) algebras, i.e. for each algebra and for each map f : {x_{ij}} ⟶ A there exists a unique homomorphism of algebras g : 𝕂⟨x_{11}, x_{12}, …, x_{nn}⟩ ⟶ A such that the diagram

[figure: {x_{ij}}; 𝕂⟨x_{ij}⟩; ι; f; A; g]

commutes. So each matrix in M_{n}(A) defines a unique a homomorphism of algebras 𝕂⟨x_{11}, x_{12}, …, x_{nn}⟩ ⟶ A and conversely. □

Example 2.3.10. 1. The affine algebraic group called additive group

𝔾_{a} : 𝕂-cAlg ⟶ Ab

with 𝔾_{a}(A) := A^{+} from Lemma 2.3.5 is represented by the Hopf algebra 𝕂[x]. We determine comultiplication, counit, and antipode.

By Corollary 2.2.7 the comultiplication is Δ = ι_{1} ∗ ι_{2} ∈ 𝕂-cAlg (𝕂[x], 𝕂[x] ⊗ 𝕂[x]) ≅ 𝔾_{a}(𝕂[x] ⊗ 𝕂[x]). Hence

Δ(x) = ι_{1}(x) + ι_{2}(x) = x ⊗ 1 + 1 ⊗ x.

The counit is ε = e_{𝕂} = 0 ∈ 𝕂-cAlg (𝕂[x], 𝕂) ≅ 𝔾_{a}(𝕂) hence

ε(x) = 0.

The antipode is S = id_{𝕂[x]}^{−1} ∈ 𝕂-cAlg (𝕂[x], 𝕂[x]) ≅ 𝔾_{a}(𝕂[x]) hence

S(x) = −x.

2. The affine algebraic group called multiplicative group

𝔾_{m} : 𝕂-cAlg ⟶ Ab

with 𝔾_{m}(A) := A^{∗} = U(A) from Lemma 2.3.7 is represented by the Hopf algebra 𝕂[x, x^{−1}] = 𝕂[x, y]/(xy − 1). We determine comultiplication, counit, and antipode.

By Corollary 2.2.7 the comultiplication is

Δ = ι_{1} ∗ ι_{2} ∈ 𝕂-cAlg (𝕂[x, x^{−1}], 𝕂[x, x^{−1}] ⊗ 𝕂[x, x^{−1}]) ≅ 𝔾_{m}(𝕂[x, x^{−1}] ⊗ 𝕂[x, x^{−1}]).

Hence

Δ(x) = ι_{1}(x) · ι_{2}(x) = x ⊗ x.

The counit is ε = e_{𝕂} = 1 ∈ 𝕂-cAlg (𝕂[x, x^{−1}], 𝕂) ≅ 𝔾_{m}(𝕂) hence

ε(x) = 1.

The antipode is S = id_{𝕂[x,x^{−1}]}^{−1} ∈ 𝕂-cAlg (𝕂[x, x^{−1}], 𝕂[x, x^{−1}]) ≅ 𝔾_{a}(𝕂[x, x^{−1}]) hence

S(x) = x^{−1}.

3. The affine algebraic group called additive matrix group

𝕄_{n}^{+} : 𝕂-cAlg ⟶ Ab ,

with 𝕄_{n}^{+}(A) the additive group of n × n-matrices with coefficients in A is represented by the commutative algebra M_{n}^{+} = 𝕂[x_{ij}|1 ≤ i, j ≤ n] (Lemma 2.3.9). This algebra must be a Hopf algebra.

The comultiplication is Δ = ι_{1} ∗ ι_{2} ∈ 𝕂-cAlg (M_{n}^{+}, M_{n}^{+} ⊗ M_{n}^{+}) ≅ 𝕄_{n}^{+}(M_{n}^{+} ⊗ M_{n}^{+}). Hence

Δ(x_{ij}) = ι_{1}(x_{ij}) + ι_{2}(x_{ij}) = x_{ij} ⊗ 1 + 1 ⊗ x_{ij}.

The counit is ε = e_{𝕂} = (0) ∈ 𝕂-cAlg (M_{n}^{+}, 𝕂) ≅ 𝕄_{n}^{+}(𝕂) hence

ε(x_{ij}) = 0.

The antipode is S = id_{M_{n}^{+}}^{−1} ∈ 𝕂-cAlg (M_{n}^{+}, M_{n}^{+}) ≅ 𝕄_{n}^{+}(M_{n}^{+}) hence

S(x_{ij}) = −x_{ij}.

4. The matrix algebra 𝕄_{n}(A) also has a noncommutative multiplication, the matrix multiplication, defining a monoid structure 𝕄_{n}^{×}(A). Thus 𝕂[x_{ij}] carries another coalgebra structure which defines a bialgebra M_{n}^{×} = 𝕂[x_{ij}]. Obviously there is no antipode.

The comultiplication is Δ = ι_{1} ∗ ι_{2} ∈ 𝕂-cAlg (M_{n}^{×}, M_{n}^{×} ⊗ M_{n}^{×}) ≅ 𝕄_{n}^{×}(M_{n}^{×} ⊗ M_{n}^{×}). Hence Δ((x_{ij})) = ι_{1}((x_{ij})) · ι_{2}((x_{ij})) = (x_{ij}) ⊗ (x_{ij}) or

Δ(x_{ik}) = ∑_{j} x_{ij} ⊗ x_{jk}.

The counit is ε = e_{𝕂} = E ∈ 𝕂-cAlg (M_{n}^{×}, 𝕂) ≅ 𝕄_{n}^{×}(𝕂) hence

ε(x_{ij}) = δ_{ij}.

5. Let 𝕂 be a field of characteristic p. The algebra H = 𝕂[x]/(x^{p}) carries the structure of a Hopf algebra with Δ(x) = x ⊗ 1 + 1 ⊗ x, ε(x) = 0, and S(x) = −x. To show that Δ is well defined we have to show Δ(x)^{p} = 0. But this is obvious by the rules for computing p-th powers in characteristic p. We have (x ⊗ 1 + 1 ⊗ x)^{p} = x^{p} ⊗ 1 + 1 ⊗ x^{p} = 0.

Thus the algebra H represents an affine algebraic group:

α_{p}(A) := 𝕂-cAlg (H, A) ≅ {a ∈ A|a^{p} = 0}.

The group multiplication is the addition of p-nilpotent elements. So we have the group of p-nilpotent elements.

6. The algebra H = 𝕂[x]/(x^{n} − 1) is a Hopf algebra with the comultiplication Δ(x) = x ⊗ x, the counit ε(x) = 1, and the antipode S(x) = x^{n−1}. These maps are well defined since we have for example Δ(x)^{n} = (x ⊗ x)^{n} = x^{n} ⊗ x^{n} = 1 ⊗ 1. Observe that this Hopf algebra is isomorphic to the group algebra 𝕂C_{n} of the cyclic group C_{n} of order n.

Thus the algebra H represents an affine algebraic group:

μ_{n}(A) := 𝕂-cAlg (H, A) ≅ {a ∈ A|a^{n} = 1},

that is the group of n-th roots of unity. The group multiplication is the ordinary multiplication of roots of unity.

7. The linear groups or matrix groups 𝔾𝕃(n)(A), 𝕊𝕃(n)(A) and other such groups are further examples of affine algebraic groups. We will discuss them in the section on quantum groups.

Problem 2.3.14. (1) The construction of the general linear group

𝔾𝕃(n)(A) = {(a_{ij}) ∈ 𝕄_{n}(A)|(a_{ij}) invertible}

defines an affine algebraic group. Describe the representing Hopf algebra.

(2) The special linear group 𝕊𝕃(n)(A) is an affine algebraic group. What is the representing Hopf algebra?

(3) The real unit circle 𝕊^{1}(ℝ) carry the structure of a group by the addition of angles. Is it possible to make 𝕊^{1} with the affine algebra 𝕂[c, s]/(s^{2} + c^{2} − 1) into an affine algebraic group? (Hint: How can you add two points (x_{1}, y_{1}) and (x_{2}, y_{2}) on the unit circle, such that you get the addition of the associated angles?)

(4) Find a group structure on the torus T .

4. Formal Groups

Consider now 𝕂-cCoalg the category of cocommutative 𝕂-coalgebras. Let C, D ∈ 𝕂-cCoalg . Then C ⊗D is again a cocommutative 𝕂-coalgebra by [Advanced Algebra] Problem 2.7.4. In fact this holds also for non-commutative 𝕂-algebras, but in 𝕂-cCoalg we have

Proposition 2.4.1. The tensor product in 𝕂-cCoalg is the (categorical) product.

Proof. Let f ∈ 𝕂-cCoalg (Z, C), g ∈ 𝕂-cCoalg (Z, D). The map (f, g) : Z ⟶ C ⊗ D defined by (f, g)(z) := ∑ f(z_{(1)}) ⊗ g(z_{(2)}) is the unique homomorphism of coalgebras such that (1 ⊗ ε_{D})(f, g)(z) = f(z) and (ε_{C} ⊗ 1)(f, g)(z) = g(z) or such that the diagram

[figure: C; C ⊗ D; p_{C}; D; p_{D}; f; g; Z; (f,g)]

commutes, where p_{C}(c ⊗ d) = (1 ⊗ ε)(c ⊗ d) = cε(d) and p_{D}(c ⊗ d) = (ε ⊗ 1)(c ⊗ d) = ε(c)d are homomorphisms of coalgebras. □

So the category 𝕂-cCoalg has finite products and also a final object 𝕂.

Definition 2.4.2. A formal group is a group in the category of 𝕂-cCoalg of cocommutative coalgebras.

A formal group G defines a contravariant representable functor from 𝕂-cCoalg to Gr , the category of groups.

Proposition 2.4.3. Let H ∈ 𝕂-cCoalg . H a represents a formal group if and only if there are given morphisms in 𝕂-cCoalg

∇ : H ⊗ H ⟶ H, u : 𝕂 ⟶ H, S : H ⟶ H

such that the following diagrams commute

[figure: H ⊗ H; H; ∇; ∇; (associativity); ∇⊗1; H ⊗ H ⊗ H; H ⊗ H; 1⊗∇]

[figure: H ⊗ H; H; ∇; 𝕂 ⊗ H ≅ H ≅ H ⊗ 𝕂; H ⊗ H; u⊗1; (unit); ∇; 1⊗u; 1]

[figure: (inverse); H; 𝕂; ε; H; η; Δ; H ⊗ H; H ⊗ H; S⊗id; id ⊗S; ∇]

Proof. For an arbitrary formal group H we get ∇ = p_{1} ∗ p_{2} ∈ 𝕂-cCoalg (H ⊗ H, H), u = e ∈ 𝕂-cCoalg (𝕂, H), and S = (id)^{−1} ∈ 𝕂-cCoalg (H, H). These maps, the Yoneda Lemma and Remark 2.2.6 lead to the proof of the proposition. □

Remark 2.4.4. In particular the representing object (H, ∇, u, Δ, ε, S) of a formal group G is a cocommutative Hopf algebra and every such Hopf algebra represents a formal group. Hence the category of formal groups is equivalent to the category of cocommutative Hopf algebras.

Corollary 2.4.5. A coalgebra H ∈ 𝕂-cCoalg represents a formal group if and only if H is a cocommutative Hopf algebra.

The category of cocommutative Hopf algebras is equivalent to the category of formal groups.

Corollary 2.4.6. The following categories are equivalent:

(1) The category of commutative, cocommutative Hopf algebras.

(2) The category of commutative formal groups.

(3) The dual of the category of commutative affine algebraic groups. Example 2.4.7. (1) Group algebras 𝕂G are formal groups.

(2) Universal enveloping algebras U(𝔤) of Lie algebras 𝔤 are formal groups.

(3) The tensor algebra T (V) and the symmetric algebra S(V) are formal groups.

(4) Let C be a cocommutative coalgebra and G be a group. Then the group 𝕂G(C) = 𝕂-cCoalg (C, 𝕂G) is isomorphic to the set of families (h_{g}^{∗}|g ∈ G) of decompositions of the unit of C^{∗} into a sum of orthogonal idempotents h_{g}^{∗} ∈ C^{∗} that are locally finite.

To see this embed 𝕂-cCoalg (C, 𝕂G) ⊆ Hom(C, 𝕂G) and embedthe set Hom(C, 𝕂G) into the set (C^{∗})^{G} of G-families of elements in the algebra C^{∗} by h ↦ (h_{g}^{∗}) with h(c) = ∑_{g∈G} h_{g}^{∗}(c)g. The linear map h is a homomorphism of coalgebras iff (h ⊗ h)Δ = Δh and εh = ε iff ∑ h(c_{(1)}) ⊗ h(c_{(2)}) = ∑ h(c)_{(1)} ⊗ h(c)_{(2)} and ε(h(c)) = ε(c) for all c ∈ C iff ∑ h_{g}^{∗}(c_{(1)})g⊗h_{l}^{∗}(c_{(2)})l = ∑ h_{g}^{∗}(c)g⊗ g and ∑ h_{g}^{∗}(c) = ε(c) iff ∑ h_{g}^{∗}(c_{(1)})h_{l}^{∗}(c_{(2)}) = δ_{gl}h_{g}^{∗}(c) and ∑ h_{g}^{∗} = ε iff h_{g}^{∗} ∗ h_{l}^{∗} = δ_{gl}h_{g}^{∗} and ∑ h_{g}^{∗} = 1_{C^{∗}}. Furthermore the families must be locally finite, i.e. for each c ∈ C only finitely many of them give non-zero values.

(5) Let C be a cocommutative coalgebra and 𝕂[x] be the Hopf algebra with Δ(x) = x ⊗ 1 + 1 ⊗ x (the symmetric algebra of the one dimensional vector space 𝕂x). We embed as before 𝕂-cCoalg (C, 𝕂[x]) ⊆ Hom(C, 𝕂[x]) = (C^{∗})^{{N_{0}}}, the set of locally finite N_{0}-families in C^{∗} by h(c) = ∑_{i=0}^{∞} h_{i}^{∗}(c)x^{i}. The map h is a homomorphism of coalgebras iff Δ(h(c)) = ∑ h_{i}^{∗}(c)(x ⊗ 1 + 1 ⊗ x)^{i} = ∑ h_{i}^{∗}(c)\binom{i}{l}x^{l} ⊗ x^{i−l} = (h ⊗ h)Δ(c) = ∑ h_{i}^{∗}(c_{(1)})h_{j}^{∗}(c_{(2)})x^{i} ⊗ x^{j} and ε(∑ h_{i}^{∗}(c)x^{i}) = ε(c) iff h_{i}^{∗} ∗ h_{j}^{∗} = \binom{i+j}{i}h_{i+j}^{∗} and h_{0}^{∗} = ε = 1_{C^{∗}}.

Now let 𝕂 be a field of characteristic zero. Let p_{i} := h_{i}^{∗}/i!. Then the conditions simplify to p_{i}p_{j} = p_{i+j} and p_{0} = 1. Hence the series for h is completely determined by the term p := p_{1} since p_{n} = p_{1}^{n}. Since the series must be locally finite we get that for each c ∈ C there must be an n ∈ N_{0} such that p^{m}(c) = 0 for all m ≥ n. Hence the element p is topologically nilpotent and

𝕂-cCoalg (C, 𝕂[x]) ≅ rad_{t}(C^{∗})

the radial of topologically nilpotent elements of C^{∗}.

It is easy to see that rad_{t}(C^{∗}) is a group under addition and that this group structure coincides with the one on 𝕂-cCoalg (C, 𝕂[x]).

Remark 2.4.8. Let H be a finite dimensional Hopf algebra. Then by [Advanced Algebra] 2.23 and 2.25 we get that H^{∗} is an algebra and a coalgebra. The commutative diagrams defining the bialgebra property and the antipode can be transferred easily, so H^{∗} is again a Hopf algebra. Hence the functor -^{∗} : vec ⟶ vec from finite dimensional vector spaces to itself induces a duality -^{∗} : 𝕂-hopf ⟶ 𝕂-hopf from the category of finite dimensional Hopf algebras to itself.

An affine algebraic group is called finite if the representing Hopf algebra is finite dimensional. A formal group is called finite if the representing Hopf algebra is finite dimensional.

Thus the category of finite affine algebraic groups is equivalent to the category of finite formal groups.

The category of finite commutative affine algebraic groups is self dual. The category of finite commutative affine algebraic groups is equivalent (and dual) to the category of finite commutative formal groups.

5. Quantum Groups

Definition 2.5.1. (Drinfel’d) A quantum group is a noncommutative noncocommutative Hopf algebra.

Remark 2.5.2. We shall consider all Hopf algebras as quantum groups. Observe, however, that the commutative Hopf algebras may be considered as affine algebraic groups and that the cocommutative Hopf algebras may be considered as formal groups. Their property as a quantum space or as a quantum monoid will play some role. But often the (possibly nonexisting) dual Hopf algebra will have the geometrical meaning. The following examples 𝕊𝕃_{q}(2) and 𝔾𝕃_{q}(2) will have a geometrical meaning.

Example 2.5.3. The smallest proper quantum group, i.e. the smallest noncommutative noncocommutative Hopf algebra, is the 4-dimensional algebra

H_{4} := 𝕂⟨g, x⟩/(g^{2} − 1, x^{2}, xg + gx)

which was first described by M. Sweedler. The coalgebra structure is given by

\begin{matrix}Δ(g) = g ⊗ g, & Δ(x) = x ⊗ 1 + g ⊗ x, \\ ε(g) = 1, & ε(x) = 0, \\ S(g) = g^{−1}(= g), & S(x) = −gx. \end{matrix}

Since it is finite dimensional its linear dual H_{4}^{∗} is also a noncommutative noncocommutative Hopf algebra. It is isomorphic as a Hopf algebra to H_{4}. In fact H_{4} is up to isomorphism the only noncommutative noncocommutative Hopf algebra of dimension 4.

Example 2.5.4. The affine algebraic group 𝕊𝕃(n) : 𝕂-cAlg ⟶ Gr defined by 𝕊𝕃(n)(A), the group of n × n-matrices with coefficients in the commutative algebra A and with determinant 1, is represented by the algebra O(𝕊𝕃(n)) = SL(n) = 𝕂[x_{ij}]/(det(x_{ij}) − 1) i.e.

𝕊𝕃(n)(A) ≅ 𝕂-cAlg (𝕂[x_{ij}]/(det(x_{ij}) − 1), A).

Since 𝕊𝕃(n)(A) has a group structure by the multiplication of matrices, the representing commutative algebra has a Hopf algebra structure with the diagonal Δ = ι_{1} ∗ ι_{2} hence

Δ(x_{ik}) = ∑ x_{ij} ⊗ x_{jk},

the counit ε(x_{ij}) = δ_{ij} and the antipode S(x_{ij}) = adj(X)_{ij} where adj(X) is the adjoint matrix of X = (x_{ij}). We leave the verification of these facts to the reader.

We consider 𝕊𝕃(n) ⊆ M_{n} = 𝔸^{n^{2}} as a subspace of the n^{2}-dimensional affine space.

Example 2.5.5. Let M_{q}(2) = 𝕂 ⟨(\begin{matrix}a & b \\ c & d \end{matrix})⟩ /I as in 1.3.6 with I the ideal gen-

erated by

ab − q^{−1}ba, ac − q^{−1}ca, bd − q^{−1}db, cd − q^{−1}dc, (ad − q^{−1}bc) − (da − qcb), bc − cb.

We first define the quantum determinant det_{q} = ad − q^{−1}bc = da − qcb in M_{q}(2). It is a central element. To show this, it suffices to show that det_{q} commutes with the generators a, b, c, d:

\begin{matrix}(ad − q^{−1}bc)a = a(da − qbc), & (ad − q^{−1}bc)b = b(ad − q^{−1}bc), \\ (ad − q^{−1}bc)c = c(ad − q^{−1}bc), & (da − qbc)d = d(ad − q^{−1}bc). \end{matrix}

We can form the quantum determinant of an arbitrary quantum matrix in A by

det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) := a′d′ − q^{−1}b′c′ = d′a′ − qc′b′ = φ(det_{q})

if φ : M_{q}(2) ⟶ A is the algebra homomorphism associated with the quantum matrix

(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}).

Given two commuting quantum 2×2-matrices (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) , (\begin{matrix}a′′ & b′′ \\ c′′ & d′′ \end{matrix}). The quantum

determinant preserves the product, since

(3)

det_{q}((\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) (\begin{matrix}a′′ & b′′ \\ c′′ & d′′ \end{matrix})) = det_{q} (\begin{matrix}a′a′′ + b′c′′ & a′b′′ + b′d′′ \\ c′a′′ + d′c′′ & c′b′′ + d′d′′ \end{matrix})

= (a′a′′ + b′c′′)(c′b′′ + d′d′′) − q^{−1}(a′b′′ + b′d′′)(c′a′′ + d′c′′) = a′c′a′′b′′ + b′c′c′′b′′ + a′d′a′′d′′ + b′d′c′′d′′

−q^{−1}(a′c′b′′a′′ + b′c′d′′a′′ + a′d′b′′c′′ + b′d′d′′c′′) = b′c′c′′b′′ + a′d′a′′d′′ − q^{−1}b′c′d′′a′′ − q^{−1}a′d′b′′c′′ = b′c′c′′b′′ + a′d′a′′d′′ − q^{−1}b′c′d′′a′′ − q^{−1}a′d′b′′c′′ −q^{−1}b′c′(a′′d′′ − d′′a′′ − q^{−1}b′′c′′ + qc′′b′′) = a′d′a′′d′′ − q^{−1}a′d′b′′c′′ − q^{−1}b′c′(a′′d′′ − q^{−1}b′′c′′) = (a′d′ − q^{−1}b′c′)(a′′d′′ − q^{−1}b′′c′′)

= det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) det_{q} (\begin{matrix}a′′ & b′′ \\ c′′ & d′′ \end{matrix}) .

In particular we have Δ(det_{q}) = det_{q} ⊗ det_{q} and ε(det_{q}) = 1. The quantum determinant is a grouplike element (see 2.1.6).

Now we define an algebra

SL_{q}(2) := M_{q}(2)/(det_{q} − 1).

The algebra SL_{q}(2) represents the functor

𝕊𝕃_{q}(2)(A) = \{(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) ∈ M_{q}(2)(A)|det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) = 1\}.

There is a surjective homomorphism of algebras M_{q}(2) ⟶ SL_{q}(2) and 𝕊𝕃_{q}(2) is a subfunctor of M_{q}(2).

Let X, Y be commuting quantum matrices satisfying det_{q}(X) = 1 = det_{q}(Y). Since det_{q}(X) det_{q}(Y) = det_{q}(XY) for commuting quantum matrices we get det_{q}(XY) = 1, hence 𝕊𝕃_{q}(2) is a quantum submonoid of M_{q}(2) and SL_{q}(2) is a bialgebra with diagonal

Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix}) ⊗ (\begin{matrix}a & b \\ c & d \end{matrix}) ,

and

ε (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) .

To show that SL_{q}(2) has an antipode we first define a homomorphism of algebras T : M_{q}(2) ⟶ M_{q}(2)^{op} by

T (\begin{matrix}a & b \\ c & d \end{matrix}) := (\begin{matrix}d & −qb \\ −q^{−1}c & a \end{matrix}) .

We check that T : 𝕂 ⟨(\begin{matrix}a & b \\ c & d \end{matrix})⟩ ⟶ M_{q}(2)^{op} vanishes on the ideal I.

T (ab − q^{−1}ba) = T (b)T (a) − q^{−1}T (a)T (b) = −qbd + q^{−1}qdb = 0.

We leave the check of the other defining relations to the reader. Furthermore T restricts to a homomorphism of algebras S : SL_{q}(2) ⟶ SL_{q}(2)^{op} since T (det_{q}) = T (ad − q^{−1}bc) = T (d)T (a) − q^{−1}T (c)T (b) = ad − q^{−1}(−q^{−1}c)(−qb) = det_{q} hence T (det_{q} −1) = det_{q} −1 = 0 in SL_{q}(2).

One verifies easily that S satisfies ∑ S(x_{(1)})x_{(2)} = ε(x) for all given generators of SL_{q}(2), hence S is a left antipode by 2.1.3. Symmetrically S is a right antipode. Thus the bialgebra SL_{q}(2) is a Hopf algebra or a quantum group.

Example 2.5.6. The affine algebraic group 𝔾𝕃(n) : 𝕂-cAlg ⟶ Gr defined by 𝔾𝕃(n)(A), the group of invertible n×n-matrices with coefficients in the commutative algebra A, is represented by the algebra O(𝔾𝕃(n)) = GL(n) = 𝕂[x_{ij}, t]/(det(x_{ij})t−1) i.e.

𝔾𝕃(n)(A) ≅ 𝕂-cAlg (𝕂[x_{ij}, t]/(det(x_{ij})t − 1), A)).

Since 𝔾𝕃(n)(A) has a group structure by the multiplication of matrices, the representing commutative algebra has a Hopf algebra structure with the diagonal Δ = ι_{1} ∗ ι_{2} hence

Δ(x_{ik}) = ∑ x_{ij} ⊗ x_{jk},

the counit ε(x_{ij}) = δ_{ij} and the antipode S(x_{ij}) = t · adj(X)_{ij} where adj(X) is the adjoint matrix of X = (x_{ij}). We leave the verification of these facts from linear algebra to the reader. The diagonal applied to t gives

Δ(t) = t ⊗ t.

Hence t(= det(X)^{−1}) is a grouplike element in GL(n). This reflects the rule det(AB) = det(A) det(B) hence det(AB)^{−1} = det(A)^{−1} det(B)^{−1}.

Example 2.5.7. Let M_{q}(2) be as in the example 2.5.5. We define

GL_{q}(2) := M_{q}(2)[t]/J

with J generated by the elements t · (ad − q^{−1}bc) − 1. The algebra GL_{q}(2) represents the functor

𝔾𝕃_{q}(2)(A) = \{(\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) ∈ M_{q}(2)(A)|det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}) invertible in A\}.

In fact there is a canonical homomorphism of algebras M_{q}(2) ⟶ GL_{q}(2). A homomorphism of algebras φ : M_{q}(2) ⟶ A has a unique continuation to GL_{q}(2) iff det_{q}(φ (\begin{matrix}a & b \\ c & d \end{matrix})) is invertible:

[figure: M_{q}(2); M_{q}(2)[t]; G_{q}(2); A]

with t ↦ det_{q} (\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix})^{−1} . Thus 𝔾𝕃_{q}(2)(A) is a subset of M_{q}(2)(A). Observe that

M_{q}(2) ⟶ GL_{q}(2) is not surjective.

Since the quantum determinant preserves products and the product of invertible elements is again invertible we get 𝔾𝕃_{q}(2) is a quantum submonoid of M_{q}(2), hence

Δ : GL_{q}(2) ⟶ GL_{q}(2)⊗GL_{q}(2) with Δ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix})⊗(\begin{matrix}a & b \\ c & d \end{matrix}) and Δ(t) = t⊗t.

We construct the antipode for GL_{q}(2). We define T : M_{q}(2)[t] ⟶ M_{q}(2)[t]^{op} by

T (\begin{matrix}a & b \\ c & d \end{matrix}) := t (\begin{matrix}d & −qb \\ −q^{−1}c & a \end{matrix})  and  T (t) := det_{q} (\begin{matrix}a & b \\ c & d \end{matrix}) = ad − q^{−1}bc.

As in 2.5.5 T defines a homomorphism of algebras. We obtain an induced homomorphism of algebras S : GL_{q}(2) ⟶ GL_{q}(2)^{op} or a GL_{q}(2)^{op}-point in GL_{q}(2) since S(t(ad − q^{−1}bc) − 1) = (S(d)S(a) − q^{−1}S(c)S(b))S(t) − S(1) = (t^{2}ad − q^{−1}t^{2}cb)(ad − q^{−1}bc) − 1 = t^{2}(ad − q^{−1}bc)^{2} − 1 = 0.

Since S satisfies ∑ S(x_{(1)})x_{(2)} = ε(x) for all given generators, S is a left antipode by 2.1.3. Symmetrically S is a right antipode. Thus the bialgebra GL_{q}(2) is a Hopf algebra or a quantum group.

Example 2.5.8. Let sl(2) be the 3-dimensional vector space generated by the matrices

X = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  Y = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) ,  H = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

Then sl(2) is a subspace of the algebra M(2) of 2×2-matrices over 𝕂. We easily verify [X, Y] = XY −Y X = H, [H, X] = HX−XH = 2X, and [H, Y] = HY −Y H = −2Y, so that sl(2) becomes a Lie subalgebra of M(2)^{L}, which is the Lie algebra of matrices of trace zero. The universal enveloping algebra U(sl(2)) is a Hopf algebra generated as an algebra by the elements X, Y, H with the relations

[X, Y] = H, [H, X] = 2X, [H, Y] = −2Y.

As a consequence of the Poincaré-Birkhoff-Witt Theorem (that we don’t prove) the Hopf algebra U(sl(2)) has the basis {X^{i}Y^{j}H^{k}|i, j, k ∈ ℕ}. Furthermore one can prove that all finite dimensional U(sl(2))-modules are semisimple.

Example 2.5.9. We define the so-called q-deformed version U_{q}(sl(2)) of U(sl(2)) for any q ∈ 𝕂, q ≠ 1, −1 and q invertible. Let U_{q}(sl(2)) be the algebra generated by the elements E, F, K, K′ with the relations

KK′ = K′K = 1,

KEK′ = q^{2}E, KF K′ = q^{−2}F,

EF − F E = \frac{K − K′}{q − q^{−1}} .

Since K′ is the inverse of K in U_{q}(sl(2)) we write K^{−1} = K′. The representation theory of this algebra is fundamentally different depending on whether q is a root of unity or not.

We show that U_{q}(sl(2)) is a Hopf algebra or quantum group. We define

Δ(E) = 1 ⊗ E + E ⊗ K, Δ(F) = K^{−1} ⊗ F + F ⊗ 1,
Δ(K) = K ⊗ K,
ε(E) = ε(F) = 0, ε(K) = 1,
S(E) = −EK^{−1}, S(F) = −KF, S(K) = K^{−1}.

.

First we show that Δ can be expanded in a unique way to an algebra homomorphism Δ : U_{q}(sl(2)) ⟶ U_{q}(sl(2)) ⊗ U_{q}(sl(2)). Write U_{q}(sl(2)) as the residue class algebra 𝕂⟨E, F, K, K^{−1}⟩/I where I is generated by

KK^{−1} − 1, K^{−1}K − 1,

KEK^{−1} − q^{2}E, KF K^{−1} − q^{−2}F,

EF − F E − \frac{K − K^{−1}}{q − q^{−1}} .

Since K^{−1} must be mapped to the inverse of Δ(K) we must have Δ(K^{−1}) = K^{−1} ⊗ K^{−1}. Now Δ can be expanded in a unique way to the free algebra Δ : K⟨E, F, K, K^{−1}⟩ ⟶ U_{q}(sl(2)) ⊗ U_{q}(sl(2)). We have Δ(KK^{−1}) = Δ(K)Δ(K^{−1}) = 1 and similarly Δ(K^{−1}K) = 1. Furthermore we have Δ(KEK^{−1}) = Δ(K)Δ(E)Δ(K^{−1}) = (K ⊗ K) (1 ⊗ E + E ⊗ K)(K^{−1} ⊗ K^{−1}) = KK^{−1} ⊗ KEK^{−1} + KEK^{−1} ⊗ K^{2}K^{−1} = q^{2}(1 ⊗ E + E ⊗ K) = q^{2}Δ(E) = Δ(q^{2}E) and similarly Δ(KF K^{−1}) = Δ(q^{−2}F). Finally we have

Δ(EF − F E) = (1 ⊗ E + E ⊗ K)(K′ ⊗ F + F ⊗ 1)
−(K′ ⊗ F + F ⊗ 1)(1 ⊗ E + E ⊗ K)

= K′ ⊗ EF + F ⊗ E + EK′ ⊗ KF + EF ⊗ K

−K′ ⊗ F E − K′E ⊗ F K − F ⊗ E − F E ⊗ K

= K′ ⊗ (EF − F E) + (EF − F E) ⊗ K

= \frac{K′ ⊗ (K − K′) + (K − K′) ⊗ K}{q − q^{−1}}

= Δ (\frac{K − K′}{q − q^{−1}} )

hence Δ vanishes on I and can be factorized through a unique algebra homomorphism

Δ : U_{q}(sl(2)) ⟶ U_{q}(sl(2)) ⊗ U_{q}(sl(2)).

In a similar way, actually much simpler, one gets an algebra homomorphism

ε : U_{q}(sl(2)) ⟶ 𝕂.

To check that Δ is coassociative it suffices to check this for the generators of the algebra. We have (Δ ⊗ 1)Δ(E) = (Δ ⊗ 1)(1 ⊗ E + E ⊗ K) = 1 ⊗ 1 ⊗ E + 1 ⊗ E ⊗ K + E ⊗ K ⊗ K = (1 ⊗ Δ)(1 ⊗ E + E ⊗ K) = (1 ⊗ Δ)Δ(E). Similarly we get (Δ ⊗ 1)Δ(F) = (1 ⊗ Δ)Δ(F). For K the claim is obvious. The counit axiom is easily checked on the generators.

Now we show that S is an antipode for U_{q}(sl(2)). First define S : 𝕂⟨E, F, K, K^{−1}⟩ ⟶ U_{q}(sl(2))^{op} by the definition of S on the generators. We have

S(KK^{−1}) = 1 = S(K^{−1}K),

S(KEK^{−1}) = −KEK^{−1}K^{−1} = −q^{2}EK^{−1} = S(q^{2}E),
S(KF K^{−1}) = −KKF K^{−1} = −q^{−2}KF = S(q^{−2}F),
S(EF − F E) = KF EK^{−1} − EK^{−1}KF = KF K^{−1}KEK − EF

= \frac{K^{−1} − K}{q − q^{−1}} = S (\frac{K − K^{−1}}{q − q^{−1}} ) .

So S defines a homomorphism of algebras S : U_{q}(sl(2)) ⟶ U_{q}(sl(2)). Since S satisfies ∑ S(x_{(1)})x_{(2)} = ε(x) for all given generators, S is a left antipode by 2.1.3. Symmetrically S is a right antipode. Thus the bialgebra U_{q}(sl(2)) is a Hopf algebra or a quantum group.

This quantum group is of central interest in theoretical physics. Its representation theory is well understood. If q is not a root of unity then the finite dimensional U_{q}(sl(2))-modules are semisimple. Many more properties can be found in [Kassel: Quantum Groups].

6. Quantum Automorphism Groups

Lemma 2.6.1. The category 𝕂-Alg of 𝕂-algebras has arbitrary coproducts.

Proof. This is a well known fact from universal algebra. In fact all equationally defined algebraic categories are complete and cocomplete. We indicate the construction of the coproduct of a family (A_{i}|i ∈ I) of 𝕂-algebras.

Define ∐_{i∈I} A_{i} := T (⊕_{i∈I}A_{i})/L where T denotes the tensor algebra and where L is the two sided ideal in T (⊕_{i∈I}A_{i}) generated by the set

J := {ιj_{k}(x_{k}y_{k}) − ι(j_{k}(x_{k}))ι(j_{k}(y_{k})), 1_{T (⨁ A_{i})} − ιj_{k}(1_{A_{k}})|x_{k}, y_{k} ∈ A_{k}, k ∈ I}.

Then one checks easily for a family of algebra homomorphisms (f_{k} : A_{k} ⟶ B|k ∈ I) that the following diagram gives the required universal property

[figure: A_{k}; ⊕ A_{i}; j_{k}; T (⊕A_{i}); ι; T (⊕A_{i})/L; ν; B; f_{k}; f; f′; f̄]
□

Corollary 2.6.2. The category of bialgebras has finite coproducts.

Proof. The coproduct ∐ B_{i} of bialgebras (B_{i}|i ∈ I) in 𝕂-Alg is an algebra. For the diagonal and the counit we obtain the following commutative diagrams

[figure: B_{k} ⊗ B_{k}; ∐ B_{i} ⊗ ∐ B_{i}; j_{k}⊗j_{k}; B_{k}; ∐ B_{i}; j_{k}; Δ_{k}; ∃_{1}Δ]

[figure: B_{k}; ∐ B_{i}; j_{k}; ε_{k}; 𝕂; ∃_{1}ε]

since in both cases ∐ B_{i} is a coproduct in 𝕂-Alg . Then it is easy to show that these homomorphisms define a bialgebra structure on ∐ B_{i} and that ∐ B_{i} satisfies the universal property for bialgebras. □

Theorem 2.6.3. Let B be a bialgebra. Then there exists a Hopf algebra H(B) and a homomorphism of bialgebras ι : B ⟶ H(B) such that for every Hopf algebra H and for every homomorphism of bialgebras f : B ⟶ H there is a unique homomorphism of Hopf algebras g : H(B) ⟶ H such that the diagram

[figure: B; H(B); ι; f; H; g]

commutes.

Proof. Define a sequence of bialgebras (B_{i}|i ∈ ℕ) by

\begin{matrix}B_{0} := B, \\ B_{i+1} := B_{i}^{opcop}, i ∈ ℕ. \end{matrix}

Let B′ be the coproduct of the family (B_{i}|i ∈ ℕ) with injections ι_{i} : B_{i} ⟶ B′. Because B′ is a coproduct of bialgebras there is a unique homomorphism of bialgebras S′ : B′ ⟶ B′^{opcop} such that the diagrams

[figure: B_{i+1}^{opcop}; B′^{opcop}; ι_{i+1}; B_{i}; B′; ι_{i}; id; S′]

commute.

Now let I be the two sided ideal in B′ generated by

{(S′ ∗ 1 − uε)(x_{i}), (1 ∗ S′ − uε)(x_{i})|x_{i} ∈ ι_{i}(B_{i}), i ∈ ℕ}.

I is a coideal, i.e. ε_{B′}(I) = 0 and Δ_{B′}(I) ⊆ I ⊗ B′ + B′ ⊗ I.

Since ε_{B′} and Δ_{B′} are homomorphisms of algebras it suffices to check this for the generating elements of I. Let x ∈ B_{i} be given. Then we have ε((1 ∗ S′)ι_{i}(x)) = ε(∇(1⊗S′)Δι_{i}(x)) = ∇_{𝕂}(ε⊗εS′)(ι_{i}⊗ι_{i})Δ_{i}(x) = (ει_{i}⊗ει_{i})Δ_{i}(x) = ε_{i}(x) = ε(uει_{i}(x)). Symmetrically we have ε((S′ ∗ 1)ι_{i}(x)) = ε(uει_{i}(x)). Furthermore we have

Δ((1 ∗ S′)ι_{i}(x))
= Δ∇(1 ⊗ S′)Δι_{i}(x)
= (∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(Δ ⊗ Δ)(1 ⊗ S′)(ι_{i} ⊗ ι_{i})Δ_{i}(x)
= (∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(Δ ⊗ τ(S′ ⊗ S′)Δ)(ι_{i} ⊗ ι_{i})Δ_{i}(x)
= ∑(∇ ⊗ ∇)(1 ⊗ τ ⊗ 1)(ι_{i}(x_{(1)}) ⊗ ι_{i}(x_{(2)}) ⊗ S′ι_{i}(x_{(4)}) ⊗ S′ι_{i}(x_{(3)}))
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(4)}) ⊗ ι_{i}(x_{(2)})S′ι_{i}(x_{(3)})
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′)ι_{i}(x_{(2)}).

Hence we have

Δ((1 ∗ S′−uε)ι_{i}(x))
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′)ι_{i}(x_{(2)}) − Δuει_{i}(x)
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ ((1 ∗ S′) − uε)ι_{i}(x_{(2)})
+ ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ uει_{i}(x_{(2)}) − Δuει_{i}(x)
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′ − uε)ι_{i}(x_{(2)})
+ ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(2)}) ⊗ 1_{B′} − uει_{i}(x) ⊗ 1_{B′}
= ∑ ι_{i}(x_{(1)})S′ι_{i}(x_{(3)}) ⊗ (1 ∗ S′ − uε)ι_{i}(x_{(2)})
+(1 ∗ S′ − uε)ι_{i}(x) ⊗ 1_{B′}
∈ B′ ⊗ I + I ⊗ B′.

Thus I is a coideal and a biideal of B′.

Now let H(B) := B′/I and let ν : B′ ⟶ H(B) be the residue class homomorphism. We show that H(B) is a bialgebra and ν is a homomorphism of bialgebras. H(B) is an algebra and ν is a homomorphism of algebras since I is a two sided ideal. Since I ⊆ Ker(ε) there is a unique factorization

[figure: B′; B′/I; ν; ε′; 𝕂; ε]

where ε : B′/I ⟶ 𝕂 is a homomorphism of algebras. Since Δ(I) ⊆ B′ ⊗ I + I ⊗ B′ ⊆ Ker(ν ⊗ ν : B′ ⊗ B′ ⟶ B′/I ⊗ B′/I) and thus I ⊆ Ker(Δ(ν ⊗ ν)) we have a unique factorization

[figure: B′ ⊗ B′; B′/I ⊗ B′/I; ν⊗ν; B′; B′/I; ν; Δ_{B′}; Δ]

by an algebra homomorphism Δ : B′/I ⟶ B′/I ⊗ B′/I. Now it is easy to verify that B′/I becomes a bialgebra and ν a bialgebra homomorphism.

We show that the map νS′ : B′ ⟶ B′/I can be factorized through B′/I in the commutative diagram

[figure: B′; B′/I; ν; νS′; B′/I; S]

This holds if I ⊆ Ker(νS′). Since Ker(ν) = I it suffices to show S′(I) ⊆ I. We have

S′((S′ ∗ 1)ι_{i}(x)) =
= ∇τ(S′^{2}ι_{i} ⊗ S′ι_{i})Δ_{i}(x)
= ∇τ(S′ ⊗ 1)(ι_{i+1} ⊗ ι_{i+1})Δ_{i}(x)
= ∇(1 ⊗ S′)(ι_{i+1} ⊗ ι_{i+1})τΔ_{i}(x)
= ∇(1 ⊗ S′)(ι_{i+1} ⊗ ι_{i+1})Δ_{i+1}(x)
= (1 ∗ S′)ι_{i+1}(x)

and

S′(uει_{i}(x)) = S′(1)ε_{i}(x) = S′(1)ε_{i+1}(x) = S′(uει_{i+1}(x))

hence we get

S′((S′ ∗ 1 − uε)ι_{i}(x)) = (1 ∗ S′ − uε)ι_{i+1}(x) ∈ I.

This shows S′(I) ⊆ I. So there is a unique homomorphism of bialgebras S : H(B) ⟶ H(B)^{opcop} such that the diagram

[figure: B′^{opcop}; H(B)^{opcop}; ν; B′; H(B); ν; S′; S]

commutes.

Now we show that H(B) is a Hopf algebra with antipode S. By Proposition 2.1.3 it suffices to test on generators of H(B) hence on images νι_{i}(x) of elements x ∈ B_{i}. We have

\begin{matrix}(1 ∗ S)νι_{i}(x) = ∇(ν ⊗ Sν)Δι_{i}(x) = ∇(ν ⊗ ν)(1 ⊗ S′)Δι_{i}(x) = \\ = ν(1 ∗ S′)ι_{i}(x) = νuει_{i}(x) = uενι_{i}(x). \end{matrix}

By Proposition 2.1.3 S is an antipode for H(B).

We prove now that H(B) together with ι := νι_{0} : B ⟶ H(B) is a free Hopf algebra over B. Let H be a Hopf algebra and let f : B ⟶ H be a homomorphism of bialgebras. We will show that there is a unique homomorphism f̄: H(B) ⟶ H such that

[figure: B; H(B); ι; f; H; f̄]

commutes.

We define a family of homomorphisms of bialgebras f_{i} : B_{i} ⟶ H by

\begin{matrix}f_{0} := f, \\ f_{i+1} := S_{H}f_{i}, i ∈ ℕ. \end{matrix}

We have in particular f_{i} = S_{H}^{i}f for all i ∈ ℕ. Thus there is a unique homomorphism of bialgebras f′ : B′ = ∐ B_{i} ⟶ H such that f′ι_{i} = f_{i} for all i ∈ ℕ.

We show that f′(I) = 0. Let x ∈ B_{i}. Then

f′((1 ∗ S′)ι_{i}(x))= f′(∇(1 ⊗ S′)(ι_{i} ⊗ ι_{i})Δ_{i}(x))
= ∑ f′ι_{i}(x_{(1)})f′S′ι_{i}(x_{(2)})
= ∑ f′ι_{i}(x_{(1)})f′ι_{i+1}(x_{(2)})
= ∑ f_{i}(x_{(1)})f_{i+1}(x_{(2)})
= ∑ f_{i}(x_{(1)})Sf_{i}(x_{(2)})
= (1 ∗ S)f_{i}(x) = uεf_{i}(x) = uε_{i}(x)
= f′(uει_{i}(x)).

This together with the symmetric statement gives f′(I) = 0. Hence there is a unique factorization through a homomorphism of algebras f̄: H(B) ⟶ H such that f′ = f̄ν.

The homomorphism f̄: H(B) ⟶ H is a homomorphism of bialgebras since the diagram

[figure: B′/I; ν; f′; f̄; f′⊗f′; B′/I ⊗ B′/I; ν⊗ν; f̄⊗f̄; H; H ⊗ H; Δ_{H}; Δ′; B′; B′ ⊗ B′; Δ]

commutes with the possible exception of the right hand square Δf̄ and (f̄⊗f̄)Δ′. But ν is surjective so also the last square commutes. Similarly we get ε_{H} f̄ = ε_{H(B)}. Thus f̄ is a homomorphism of bialgebras and hence a homomorphism of Hopf algebras. □

Remark 2.6.4. In chapter 1 we have constructed universal bialgebras M(A) with coaction δ : A ⟶ M(A) ⊗ A for certain algebras A (see 1.3.12). This induces a homomorphism of algebras

δ′ : A ⟶ H(M(A)) ⊗ A

such that A is a comodule-algebra over the Hopf algebra H(M(A)). If H is a Hopf algebra and A is an H-comodule algebra by ∂ : A ⟶ H ⊗ A then there is a unique homomorphism of bialgebras f : M(A) ⟶ H such that

[figure: A; M(A) ⊗ A; δ; ∂; H ⊗ A; f⊗1]

commutes. Since the f : M(A) ⟶ H factorizes uniquely through f̄: H(M(A)) ⟶ H we get a commutative diagram

[figure: A; H(M(A)) ⊗ A; δ′; ∂; H ⊗ A; f̄⊗1]

with a unique homomorphism of Hopf algebras f̄: H(M(A)) ⟶ H.

This proof depends only on the existence of a universal algebra M(A) for the algebra A. Hence we have

Corollary 2.6.5. Let X be a quantum space with universal quantum space (and quantum monoid) M(X). Then there is a unique (up to isomorphism) quantum group H(M(X)) acting universally on X.

This quantum group H(M(X)) can be considered as the “quantum subgroup of invertible elements” of M(X) or the quantum group of “quantum automorphisms” of X.

7. Duality of Hopf Algebras

In 2.4.8 we have seen that the dual Hopf algebra H^{∗} of a finite dimensional Hopf algebra H satisfies certain relations w.r.t. the evaluation map. The multiplication of H^{∗} is derived from the comultiplication of H and the comultiplication of H^{∗} is derived from the multiplication of H.

This kind of duality is restricted to the finite-dimensional situation. Nevertheless one wants to have a process that is close to the finite-dimensional situation. This short section is devoted to several approaches of duality for Hopf algebras.

First we use the relations of the finite-dimensional situation to give a general definition.

Definition 2.7.1. Let H and L be Hopf algebras. Let

ev : L ⊗ H ∋ a ⊗ h ↦ ⟨a, h⟩ ∈ 𝕂

be a bilinear form satisfying

(4)  ⟨a ⊗ b, ∑ h_{(1)} ⊗ h_{(2)}⟩ = ⟨ab, h⟩, ⟨1, h⟩ = ε(h)
(5)  ⟨∑ a_{(1)} ⊗ a_{(2)}, h ⊗ j⟩ = ⟨a, hj⟩, ⟨a, 1⟩ = ε(a)
(6)  ⟨a, S(h)⟩ = ⟨S(a), h⟩

Such a map is called a weak duality of Hopf algebras. The bilinear form is called left (right) nondegenerate if ⟨a, H⟩ = 0 implies a = 0 (⟨L, h⟩ = 0 implies h = 0). A duality of Hopf algebras is a weak duality that is left and right nondegenerate.

Remark 2.7.2. If H is a finite dimensional Hopf algebra then the usual evaluation ev : H^{∗} ⊗ H ⟶ 𝕂 defines a duality of Hopf algebras.

Remark 2.7.3. Assume that ev : L ⊗ H ⟶ 𝕂 defines a weak duality. By [Advanced Algebra] 1.22 we have isomorphisms Hom(L⊗H, 𝕂) ≅ Hom(L, Hom(H, 𝕂)) and Hom(L ⊗ H, 𝕂) ≅ Hom(H, Hom(L, 𝕂)). Denote the homomorphisms associated with ev : L ⊗ K ⟶ 𝕂 by φ : L ⟶ Hom(H, 𝕂) resp. ψ : H ⟶ Hom(L, 𝕂). They satisfy φ(a)(h) = ev(a ⊗ h) = ψ(h)(a).

ev : L ⊗ K ⟶ 𝕂 is left nondegenerate iff φ : L ⟶ Hom(H, 𝕂) is injective. ev : L ⊗ K ⟶ 𝕂 is right nondegenerate iff ψ : H ⟶ Hom(L, 𝕂) is injective.

Lemma 2.7.4. 1. The bilinear form ev : L ⊗ H ⟶ 𝕂 satisfies (2.4) if and only if φ : L ⟶ Hom(H, 𝕂) is a homomorphism of algebras.

2. The bilinear form ev : L ⊗ H ⟶ 𝕂 satisfies (2.5) if and only if ψ : H ⟶ Hom(L, 𝕂) is a homomorphism of algebras.

Proof. ev : L⊗H ⟶ 𝕂 satisfies the right equation of (4) iff φ(ab)(h) = ⟨ab, h⟩ = ⟨a ⊗ b, ∑ h_{(1)} ⊗ h_{(2)}⟩ = ∑⟨a, h_{(1)}⟩⟨b, h_{(2)}⟩ = ∑ φ(a)(h_{(1)})φ(b)(h_{(2)}) = (φ(a) ∗ φ(b))(h) by the definition of the algebra structure on Hom(H, 𝕂).

ev : L ⊗ H ⟶ 𝕂 satisfies the left equation of (4) iff φ(1)(h) = ⟨1, h⟩ = ε(h).

The second part of the Lemma follows by symmetry. □

Example 2.7.5. There is a weak duality between the quantum groups 𝕊𝕃_{q}(2) and U_{q}(sl(2)). (Kassel: Chapter VII.4).

Proposition 2.7.6. Let ev : L ⊗ H ⟶ 𝕂 be a weak duality of Hopf algebras. Let I := Ker(φ : L ⟶ Hom(H, 𝕂)) and J := Ker(ψ : H ⟶ Hom(L, 𝕂)). Let \overline{L} := L/I and \overline{H} := H/J. Then \overline{L} and \overline{H} are Hopf algebras and the induced bilinear form \overline{ev} : \overline{L} ⊗ \overline{H} ⟶ 𝕂 is a duality.

Proof. First observe that I and J are two sided ideals hence \overline{L} and \overline{H} are algebras. Then ev : L ⊗ H ⟶ 𝕂 can be factored through \overline{ev} : \overline{L} ⊗ \overline{H} ⟶ 𝕂 and the equations (4) and (5) are still satisfied for the residue classes.

The ideals I and J are biideals. In fact, let x ∈ I then ⟨Δ(x), a ⊗ b⟩ = ⟨x, ab⟩ = 0 hence Δ(x) ∈ Ker(φ ⊗ φ : L ⊗ L ⟶ Hom(H ⊗ H, 𝕂) = I ⊗ L + L ⊗ I (the last equality is an easy exercise in linear algebra) and ε(x) = ⟨x, 1⟩ = 0. Hence as in the proof of Theorem 2.6.3 we get that \overline{L} = L/I and \overline{H} = H/J are bialgebras. Since ⟨S(x), a⟩ = ⟨x, S(a)⟩ = 0 we have an induced homomorphism \overline{S} : \overline{L} ⟶ \overline{L}. The identities satisfied in L hold also for the residue classes in \overline{L} so that L and similarly \overline{H} become Hopf algebras. Finally we have by definition of I that ⟨\overline{x}, \overline{a}⟩ = ⟨x, a⟩ = 0 for all a ∈ H iff a ∈ I or \overline{a} = 0. Thus the bilinear form \overline{ev} : \overline{L} ⊗ \overline{H} ⟶ K defines a duality. □

Problem 2.7.15. (in Linear Algebra)

(1) For U ⊆ V define U^{⊥} := {f ∈ V^{∗}|f(U) = 0}. For Z ⊆ V^{∗} define Z^{⊥} := {v ∈ V |Z(v) = 0}. Show that the following hold:

(a) U ⊆ V ⟹ U = U^{⊥⊥};

(b) Z ⊆ V^{∗} and dim Z < ∞ ⟹ Z = Z^{⊥⊥};

(c) {U ⊆ V | dim V/U < ∞} ≅ {Z ⊆ V^{∗}| dim Z < ∞} under the maps

U ↦ U^{⊥} and Z ↦ Z^{⊥}.

(2) Let V = ⊕_{i=1}^{∞}𝕂x_{i} be an infinite-dimensional vector space. Find an element g ∈ (V ⊗ V)^{∗} that is not in V^{∗} ⊗ V^{∗} (⊆ (V ⊗ V)^{∗}).

Definition 2.7.7. Let A be an algebra. We define A^{o} := {f ∈ A^{∗}|∃ ideal _{A}I_{A} ⊆ A : dim(A/I) < ∞ and f(I) = 0}.

Lemma 2.7.8. Let A be an algebra and f ∈ A^{∗}. The following are equivalent:

(1) f ∈ A^{o};

(2) there exists I_{A} ⊆ A such that dim A/I < ∞ and f(I) = 0;

(3) A · f ⊆ _{A} Hom_{𝕂}(.A_{A}, .𝕂) is finite dimensional;

(4) A · f · A is finite dimensional; (5) ∇^{∗}(f) ∈ A^{∗} ⊗ A^{∗}.

Proof. 1. ⟹ 2. and 4. ⟹ 3. are trivial.

2. ⟹ 3. Let I_{A} ⊆ A with f(I) = 0 and dim A/I < ∞. Write A^{∗} ⊗ A ⟶ 𝕂 as ⟨g, a⟩. Then ⟨af, i⟩ = ⟨f, ia⟩ = 0 hence Af ⊂ I^{⊥} and dim Af < ∞.

3. ⟹ 2. Let dim Af < ∞. Then I_{A} := (Af)^{⊥} is an ideal of finite codimension in A and f(I) = 0 holds.

2. ⟹ 1. Let I_{A} ⊂ A with dim A/I_{A} < ∞ and f(I) = 0 be given. Then right multiplication induces φ : A ⟶ Hom_{𝕂}(A/I., A/I.) and dim End_{𝕂}(A/I) < ∞. Thus J = Ker(φ) ⊆ A is a two sided ideal of finite codimension and J ⊂ I (since φ(j)(¯1) = 0 = ¯1 · j = j̄ implies j ∈ I). Furthermore we have f(J) ⊆ f(I) = 0.

1. ⟹ 4. ⟨afb, i⟩ = ⟨f, bia⟩ = 0 implies A · f · A ⊆ _{A}I_{A}^{⊥} hence dim AfA < ∞.

3. ⟹ 5. We observe that ∇^{∗}(f) = f∇ ∈ (A ⊗ A)^{∗}. We want to show that ∇^{∗}(f) ∈ A^{∗} ⊗ A^{∗}. Let g_{1}, …, g_{n} be a basis of Af. Then there exist h_{1}, …, h_{n} ∈ A^{∗} such that bf = ∑ h_{i}(b)g_{i}. Let a, b ∈ A. Then ⟨∇^{∗}(f), a ⊗ b⟩ = ⟨f, ab⟩ = ⟨bf, a⟩ = ∑ h_{i}(b)g_{i}(a) = ⟨∑ g_{i} ⊗ h_{i}, a ⊗ b⟩ so that ∇^{∗}(f) = ∑ g_{i} ⊗ h_{i} ∈ A^{∗} ⊗ A^{∗}.

5. ⟹ 3. Let ∇^{∗}(f) = ∑ g_{i} ⊗ h_{i} ∈ A^{∗} ⊗ A^{∗}. Then bf = ∑ h_{i}(b)g_{i} for all b ∈ A as before. Thus Af is generated by the g_{1}, …, g_{n}. □

Proposition 2.7.9. Let (A, m, u) be an algebra. Then we have m^{∗}(A^{o}) ⊆ A^{o}⊗A^{o}. Furthermore (A^{o}, Δ, ε) is a coalgebra with Δ = m^{∗} and ε = u^{∗}.

Proof. Let f ∈ A^{o} and let g_{1}, …, g_{n} be a basis for Af. Then we have m^{∗}(f) = ∑ g_{i}⊗h_{i} for suitable h_{i} ∈ A^{∗} as in the proof of the previous proposition. Since g_{i} ∈ Af we get Ag_{i} ⊆ Af and dim(Ag_{i}) < ∞ and hence g_{i} ∈ A^{o}. Choose a_{1}, …, a_{n} ∈ A such that g_{i}(a_{j}) = δ_{ij}. Then (fa_{j})(a) = f(a_{j}a) = ⟨m^{∗}(f), a_{j} ⊗ a⟩ = ∑ g_{i}(a_{j})h_{i}(a) = h_{j}(a) implies fa_{j} = h_{j} ∈ fA. Observe that dim(fA) < ∞ hence dim(h_{j}A) < ∞, so that h_{j} ∈ A^{o}. This proves m^{∗}(f) ∈ A^{o} ⊗ A^{o}.

One checks easily that counit law and coassociativity hold. □

Theorem 2.7.10. (The Sweedler dual:) Let (B, m, u, Δ, ε) be a bialgebra. Then (B^{o}, Δ^{∗}, ε^{∗}, m^{∗}, u^{∗}) again is a bialgebra. If B = H is a Hopf algebra with antipode S, then S^{∗} is an antipode for B^{o} = H^{o}.

Proof. We know that (B^{∗}, Δ^{∗}, ε^{∗}) is an algebra and that (B^{o}, m^{∗}, u^{∗}) is a coalgebra. We show now that B^{o} ⊆ B^{∗} is a subalgebra. Let f, g ∈ B^{o} with dim(Bf) < ∞ and dim(Bg) < ∞. Let a ∈ B. Then we have (a(fg))(b) = (fg)(ba) = ∑ f(b_{(1)}a_{(1)})g(b_{(2)}a_{(2)}) = ∑(a_{(1)}f)(b_{(1)})(a_{(2)}g)(b_{(2)}) = ∑((a_{(1)}f)(a_{(2)}g))(b) hence a(fg) = ∑(a_{(1)}f)(a_{(2)}g) ∈ (Bf)(Bg). Since dim(Bf)(Bg) < ∞ we have dim(B(fg)) < ∞ so that fg ∈ B^{o}. Furthermore we have ε ∈ B^{o}, since Ker(ε) has codimension 1. Thus B^{o} ⊆ B^{∗} is a subalgebra. It is now easy to see that B^{o} is a bialgebra.

Now let S be the antipode of H. We show S^{∗}(H^{o}) ⊆ H^{o}. Let a ∈ H, f ∈ H^{o}. Then ⟨aS^{∗}(f), b⟩ = ⟨S^{∗}(f), ba⟩ = ⟨f, S(ba)⟩ = ⟨f, S(a)S(b)⟩ = ⟨fS(a), S(b)⟩ = ⟨S^{∗}(fS(a)), b⟩. This implies aS^{∗}(f) = S^{∗}(fS(a)) and HS^{∗}(f) = S^{∗}(fS(H)) ⊆ S^{∗}(fH). Since f ∈ H^{o} we get dim(fH) < ∞ so that dim(S^{∗}(fH)) < ∞ and dim(HS^{∗}(f)) < ∞. This shows S^{∗}(f) ∈ H^{o}. The rest of the proof is now trivial. □

Definition 2.7.11. Let G = 𝕂-cAlg (H, -) be an affine group and R ∈ 𝕂-cAlg . We define G ⊗_{𝕂} R := G|_{R-cAlg} to be the restriction to commutative R-algebras. The functor G ⊗_{𝕂} R is represented by H ⊗ R ∈ R-cAlg :

G|_{R-cAlg} (A) = 𝕂-cAlg (H, A) ≅ R-cAlg (H ⊗ R, A).

Theorem 2.7.12. (The Cartier dual:) Let H be a finite dimensional commutative cocommutative Hopf algebra. Let G = 𝕂-cAlg (H, -) be the associated affine group and let D(G) := 𝕂-cAlg (H^{∗}, -) be the dual group. Then we have

D(G) = Gr(G, G_{m})

where Gr(G, G_{m})(R) = Gr(G ⊗_{𝕂} R, G_{m} ⊗_{𝕂} R) is the set of group (-functor) homomorphisms and G_{m} is the multiplicative group.

Proof. We have Gr(G, G_{m})(R) = Gr(G⊗_{𝕂}R, G_{M}⊗_{𝕂}R) ≅ R-Hopf-Alg(𝕂[t, t^{−1}]⊗ R, H ⊗R) ≅ R-Hopf-Alg(R[t, t^{−1}], H ⊗R) ≅ {x ∈ U(H ⊗R)|Δ(x) = x⊗x, ε(x) = 1}, since Δ(x) = x ⊗ x and ε(x) = 1 imply xS(x) = ε(x) = 1.

Consider x ∈ Hom_{R}((H ⊗ R)^{∗}, R) = Hom_{R}(H^{∗} ⊗ R, R). Then Δ(x) = x ⊗ x iff x(v^{∗}w^{∗}) = ⟨x, v^{∗}w^{∗}⟩ = ⟨Δ(x), v^{∗} ⊗ w^{∗}⟩ = x(v^{∗})x(w^{∗}) and ε(x) = 1 iff ⟨x, ε⟩ = 1. Hence x ∈ R-cAlg ((H ⊗ R)^{∗}, R) ≅ 𝕂-cAlg (H^{∗}, R) = D(G)(R). □

CHAPTER 3 Representation Theory, Reconstruction and Tannaka Duality

Introduction

One of the most interesting properties of quantum groups is their representation theory. It has deep applications in theoretical physics. The mathematical side has to distinguish between the representation theory of quantum groups and the representation theory of Hopf algebras. In both cases the particular structure allows to form tensor products of representations such that the category of representations becomes a monoidal (or tensor) category.

The problem we want to study in this chapter is, how much structure of the quantum group or Hopf algebra can be found in the category of representations. We will show that a quantum monoid can be uniquely reconstructed (up to isomorphism) from its representations. The additional structure given by the antipode is itimitely connected with a certain duality of representations. We will also generalize this process of reconstruction.

On the other hand we will show that the process of reconstruction can also be used to obtain the Tambara construction of the universal quantum monoid of a noncommutative geometrical space (from chapter 1.). Thus we will get another perspective for this theorem.

At the end of the chapter you should • understand representations of Hopf algebras and of quantum groups,

• know the definition and first fundamental properties of monoidal or tensor categories,

• be familiar with the monoidal structure on the category of representations of Hopf algebras and of quantum groups,

• understand why the category of representations contains the full information about the quantum group resp. the Hopf algebra (Theorem of Tannaka-Krein),

• know the process of reconstruction and examples of bialgebras reconstructed from certain diagrams of finite dimensional vector spaces,

• understand better the Tambara construction of a universal algebra for a finite dimensional algebra.

1. Representations of Hopf Algebras

Let A be an algebra over a commutative ring 𝕂. Let A-Mod be the category of A-modules. An A-module is also called a representation of A.

Observe that the action A⊗M ⟶ M satisfying the module axioms and an algebra homomorphism A ⟶ End(M) are equivalent descriptions of an A-module structure on the 𝕂-module M.

The functor U : A-Mod ⟶ 𝕂-Mod with U(_{A}M) = M and U(f) = f is called the forgetful functor or the underlying functor.

If B is a bialgebra then a representation of B is also defined to be a B-module. It will turn out that the property of being a bialgebra leads to the possibility of building tensor products of representations in a canonical way.

Let C be a coalgebra over a commutative ring 𝕂. Let C-Comod be the category of C-comodules. A C-comodule is also called a corepresentation of C.

The functor U : C-Comod ⟶ 𝕂-Mod with U(^{C}M) = M and U(f) = f is called the forgetful functor or the underlying functor.

If B is a bialgebra then a corepresentation of B is also defined to be a B-comodule. It will turn out that the property of being a bialgebra leads to the possibility of building tensor products of corepresentations in a canonical way.

Usually representations of a ring are considered to be modules over the given ring. The role of comodules certainly arises in the context of coalgebras. But it is not quite clear what the good definition of a representation of a quantum group or its representing Hopf algebra is.

For this purpose consider representations M of an ordinary group G. Assume for the simplicity of the argument that G is finite. Representations of G are vector spaces together with a group action G × M ⟶ M. Equivalently they are vector spaces together with a group homomorphism G ⟶ Aut(M) or modules over the group algebra: 𝕂[G] ⊗ M ⟶ M. In the situation of quantum groups we consider the representing Hopf algebra H as algebra of functions on the quantum group G.

Then the algebra of functions on G is the Hopf algebra 𝕂^{G}, the dual of the group algebra 𝕂[G]. An easy exercise shows that the module structure 𝕂[G] ⊗ M ⟶ M translates to the structure of a comodule M ⟶ 𝕂^{G}⊗M and conversely. (Observe that G is finite.) So we should define representations of a quantum group as comodules over the representing Hopf algebra.

Problem 3.1.16. Let G be a finite group and 𝕂^{G} := 𝕂[G]^{∗} the dual of the group algebra. Show that 𝕂^{G} is a Hopf algebra and that each module structure 𝕂[G] ⊗ M ⟶ M translates to the structure of a comodule M ⟶ 𝕂^{G} ⊗ M and conversely. Show that this defines a monoidal equivalence of categories.

Describe the group valued functor 𝕂-cAlg (𝕂^{G}, −) in terms of sets and their group structure.

Definition 3.1.1. Let G be a quantum group with representing Hopf algebra H. A representation of G is a comodule over the representing Hopf algebra H.

From this definition we obtain immediately that we may form tensor products of representations of quantum groups since the representing algebra is a bialgebra.

We come now to the canonical construction of tensor products of (co-)represen-tations.

Lemma 3.1.2. Let B be a bialgebra. Let M, N ∈ B-Mod be two B-modules. Then M ⊗N is a B-module by the action b(m⊗n) = ∑ b_{(1)}m⊗b_{(2)}n. If f : M ⟶ M′ and g : N ⟶ N′ are homomorphisms of B-modules in B-Mod then f ⊗ g : M ⊗ N ⟶ M′ ⊗ N′ is a homomorphism of B-modules.

Proof. We have homomorphisms of 𝕂-algebras α : B ⟶ End(M) and β : B ⟶ End(N) defining the B-module structure on M and N. Thus we get a homomorphism of algebras can(α ⊗ β)Δ : B ⟶ B ⊗ B ⟶ End(M) ⊗ End(N) ⟶ End(M ⊗ N). Thus M ⊗N is a B-module. The structure is b(m⊗n) = can(α⊗β)(∑ b_{(1)})⊗b_{(2)})(m⊗n) = can(∑ α(b_{(1)}) ⊗ β(b_{(2)}))(m ⊗ n) = ∑ α(b_{(1)})(m) ⊗ β(b_{(2)})(n) = ∑ b_{(1)}m ⊗ b_{(2)}n.

Furthermore we have 1(m ⊗ n) = 1m ⊗ 1m = m ⊗ n.

If f, g are homomorphisms of B-modules, then we have (f ⊗ g)(b(m ⊗ n)) = (f ⊗ g)(∑ b_{(1)}m ⊗ b_{(2)}n) = ∑ f(b_{(1)}m) ⊗ g(b_{(2)}n) = ∑ b_{(1)}f(m) ⊗ b_{(2)}g(n) = b(f(m) ⊗ g(n)) = b(f ⊗ g)(m ⊗ n). Thus f ⊗ g is a homomorphism of B-modules. □

Corollary 3.1.3. Let B be a bialgebra. Then ⊗ : B-Mod × B-Mod ⟶ B-Mod with ⊗(M, N) = M ⊗ N and ⊗(f, g) = f ⊗ g is a functor.

Proof. The following are obvious from the ordinary properties of the tensor product over 𝕂. 1_{M} ⊗1_{N} = 1_{M⊗N} and (f ⊗g)(f′⊗g′) = ff′⊗gg′ for M, N, f, f′, g, g′ ∈ B-Mod . □

Lemma 3.1.4. Let B be a bialgebra. Let M, N ∈ B-Comod be two B-comodules. Then M⊗N is a B-comodule by the coaction δ_{M⊗N}(m⊗n) = ∑ m_{(1)}n_{(1)}⊗m_{(M)}⊗n_{(N)}.

If f : M ⟶ M′ and g : N ⟶ N′ are homomorphisms of B-comodules in B-Comod then f ⊗ g : M ⊗ N ⟶ M′ ⊗ N′ is a homomorphism of B-comodules.

Proof. The coaction on M ⊗ N may also be described by (∇_{B} ⊗ 1_{M} ⊗ 1_{N})(1_{B} ⊗ τ ⊗ 1_{N})(δ_{M} ⊗ δ_{N}) : M ⊗ N ⟶ B ⊗ M ⊗ B ⊗ N ⟶ B ⊗ B ⊗ M ⊗ N ⟶ B ⊗ M ⊗ N. Although a diagrammatic proof of the coassociativity of the coaction and the law of the counit is quite involved it allows generalization so we give it here.

Consider the next diagram.

Square (1) commutes since M and N are comodules.

Squares (2) and (3) commute since τ : M ⊗ N ⟶ N ⊗ M for 𝕂-modules M and N is a natural transformation.

Square (4) represents an interesting property of τ namely

\begin{matrix}(1 ⊗ 1 ⊗ τ)(τ_{B⊗M,B} ⊗ 1) = (1 ⊗ 1 ⊗ τ)(τ ⊗ 1 ⊗ 1)(1 ⊗ τ ⊗ 1) = \\ (τ ⊗ 1 ⊗ 1)(1 ⊗ 1 ⊗ τ)(1 ⊗ τ ⊗ 1) = (τ ⊗ 1 ⊗ 1)(1 ⊗ τ_{M,B⊗B}) \end{matrix}

that uses the fact that (1 ⊗ g)(f ⊗ 1) = (f ⊗ 1)(1 ⊗ g) holds and that τ_{B⊗M,B} = (τ ⊗ 1)(1 ⊗ τ) and τ_{M,B⊗B} = (1 ⊗ τ)(τ ⊗ 1).

Square (5) and (6) commute by the properties of the tensor product.

Square (7) commutes since B is a bialgebra.

[figure: B ⊗ M⊗; B ⊗ N; M ⊗ N; δ⊗δ; B ⊗ B⊗; M ⊗ N; 1⊗τ⊗1; B ⊗ M ⊗ N; ∇⊗1⊗1; δ⊗δ; Δ⊗1⊗Δ⊗1; Δ⊗Δ⊗1⊗1; Δ⊗1⊗1; B ⊗ M ⊗; B ⊗ B ⊗ M⊗; B ⊗ N; B ⊗ B ⊗ N; 1⊗δ⊗1⊗δ; B ⊗ B ⊗ B⊗; B ⊗ M ⊗ N; 1^{2}⊗τ_{M,B⊗B}⊗1; 1⊗τ⊗1; 1⊗τ_{B⊗M,B}⊗1⊗1; 1⊗τ⊗1⊗1⊗1; B ⊗ B ⊗; B ⊗ B ⊗ B⊗; M ⊗ N; M ⊗ B ⊗ N; 1⊗1⊗δ⊗δ; B ⊗ B ⊗ B⊗; B ⊗ M ⊗ N; 1⊗1⊗1⊗τ⊗1; ∇⊗1⊗1; ∇⊗1⊗1⊗1⊗1; ∇⊗1⊗1⊗1⊗1; B ⊗ B⊗; M ⊗ B ⊗ N; B ⊗ M ⊗ N; 1⊗δ⊗δ; B ⊗ B⊗; B ⊗ M ⊗ N; 1⊗1⊗τ⊗1; B ⊗ B ⊗ M ⊗ N; 1⊗∇⊗1⊗1; (1); (2); (3); (4); (5); (6); (7)]

The law of the counit is

[figure: B ⊗ M⊗; B ⊗ N; M ⊗ N; δ⊗δ; B ⊗ B⊗; M ⊗ N; 1⊗τ⊗1; B ⊗ M ⊗ N; ∇⊗1⊗1; 1; M ⊗ N; M ⊗ N; 1; M ⊗ N; 1; ε⊗1⊗ε⊗1; ε⊗ε⊗1⊗1; ε⊗1⊗1]

where the last square commutes since ε is a homomorphism of algebras. Now let f and g be homomorphisms of B-comodules. Then the diagram

[figure: B ⊗ M⊗; B ⊗ N; M ⊗ N; δ⊗δ; B ⊗ B⊗; M ⊗ N; 1⊗τ⊗1; B ⊗ M ⊗ N; ∇⊗1⊗1; f⊗g; 1⊗f⊗1⊗g; 1⊗1⊗f⊗g; 1⊗f⊗g; B ⊗ M′⊗; B ⊗ N′; M′ ⊗ N′; δ⊗δ; B ⊗ B⊗; M′ ⊗ N′; 1⊗τ⊗1; B ⊗ M′ ⊗ N′; ∇⊗1⊗1]

commutes. Thus f ⊗ g is a homomorphism of B-comodules. □

Corollary 3.1.5. Let B be a bialgebra. Then ⊗ : B-Comod × B-Comod ⟶ B-Comod with ⊗(M, N) = M ⊗ N and ⊗(f, g) = f ⊗ g is a functor.

Proposition 3.1.6. Let B be a bialgebra. Then the tensor product ⊗ : B-Mod × B-Mod ⟶ B-Mod satisfies the following properties:

(1) The associativity isomorphism α : (M_{1} ⊗ M_{2}) ⊗ M_{3} ⟶ M_{1} ⊗ (M_{2} ⊗ M_{3}) with α((m ⊗ n) ⊗ p) = m ⊗ (n ⊗ p) is a natural transformation from the functor ⊗ ∘ (⊗ × Id) to the functor ⊗ ∘ (Id ×⊗) in the variables M_{1}, M_{2}, and M_{3} in B-Mod .

(2) The counit isomorphisms λ : 𝕂⊗M ⟶ M with λ(κ⊗m) = κm and ρ : M ⊗𝕂 ⟶ M with ρ(m ⊗ κ) = κm are natural transformations in the variable M in B-Mod from the functor 𝕂 ⊗ - resp. - ⊗ 𝕂 to the identity functor Id.

(3) The following diagrams of natural transformations are commutative

[figure: ((M_{1} ⊗ M_{2}) ⊗ M_{3}) ⊗ M_{4}; (M_{1} ⊗ (M_{2} ⊗ M_{3})) ⊗ M_{4}; α(M_{1},M_{2},M_{3})⊗1; M_{1} ⊗ ((M_{2} ⊗ M_{3}) ⊗ M_{4}); α(M_{1},M_{2}⊗M_{3},M_{4}); α(M_{1}⊗M_{2},M_{3},M_{4}); 1⊗α(M_{2},M_{3},M_{4}); (M_{1} ⊗ M_{2}) ⊗ (M_{3} ⊗ M_{4}); M_{1} ⊗ (M_{2} ⊗ (M_{3} ⊗ M_{4})); α(M_{1},M_{2},M_{3}⊗M_{4})]

[figure: (M_{1} ⊗ 𝕂) ⊗ M_{2}; M_{1} ⊗ (𝕂 ⊗ M_{2}); α(M_{1},𝕂,M_{2}); M_{1} ⊗ M_{2}; ρ(M_{1})⊗1; 1⊗λ(M_{2})]

Proof. The homomorphisms α, λ, and ρ are already defined in the category 𝕂-Mod and satisfy the claimed properties. So we have to show, that these are homomorphisms in B-Mod and that 𝕂 is a B-module. 𝕂 is a B-module by ε ⊗ 1_{𝕂} : B ⊗ 𝕂 ⟶ 𝕂. The easy verification uses the coassociativity and the counital property of B. □

Similarly we get

Proposition 3.1.7. Let B be a bialgebra. Then the tensor product

⊗ : B-Comod × B-Comod ⟶ B-Comod

satisfies the following properties:

(1) The associativity isomorphism α : (M_{1} ⊗ M_{2}) ⊗ M_{3} ⟶ M_{1} ⊗ (M_{2} ⊗ M_{3}) with α((m ⊗ n) ⊗ p) = m ⊗ (n ⊗ p) is a natural transformation from the functor ⊗ ∘ (⊗ × Id) to the functor ⊗ ∘ (Id ×⊗) in the variables M_{1}, M_{2}, and M_{3} in B-Comod .

(2) The counit isomorphisms λ : 𝕂⊗M ⟶ M with λ(κ⊗m) = κm and ρ : M ⊗𝕂 ⟶ M with ρ(m ⊗ κ) = κm are natural transformations in the variable M in B-Comod from the functor 𝕂 ⊗ - resp. - ⊗ 𝕂 to the identity functor Id.

(3) The following diagrams of natural transformations are commutative

[figure: ((M_{1} ⊗ M_{2}) ⊗ M_{3}) ⊗ M_{4}; (M_{1} ⊗ (M_{2} ⊗ M_{3})) ⊗ M_{4}; α(M_{1},M_{2},M_{3})⊗1; M_{1} ⊗ ((M_{2} ⊗ M_{3}) ⊗ M_{4}); α(M_{1},M_{2}⊗M_{3},M_{4}); α(M_{1}⊗M_{2},M_{3},M_{4}); 1⊗α(M_{2},M_{3},M_{4}); (M_{1} ⊗ M_{2}) ⊗ (M_{3} ⊗ M_{4}); M_{1} ⊗ (M_{2} ⊗ (M_{3} ⊗ M_{4})); α(M_{1},M_{2},M_{3}⊗M_{4})]

[figure: (M_{1} ⊗ 𝕂) ⊗ M_{2}; M_{1} ⊗ (𝕂 ⊗ M_{2}); α(M_{1},𝕂,M_{2}); M_{1} ⊗ M_{2}; ρ(M_{1})⊗1; 1⊗λ(M_{2})]

Remark 3.1.8. We now get some simple properties of the underlying functors U : B-Mod ⟶ 𝕂-Mod resp. U : B-Comod ⟶ 𝕂-Mod that are easily verified.

U(M ⊗ N) = U(M) ⊗ U(N),
U(f ⊗ g) = f ⊗ g,
U(𝕂) = 𝕂,
U(α) = α, U(λ) = λ, U(ρ) = ρ.

Problem 3.1.17. We have seen that in representation theory and in corepresentation theory of quantum groups such as 𝕂G, U(𝔤), SL_{q}(2), U_{q}(sl(2)) the ordinary tensor product (in 𝕂-Mod ) of two (co-)reprensentations is in a canonical way again a (co-)reprensentation. For two H-modules M and N describe the module structure on M ⊗ N if

(1) H = 𝕂G: g(m ⊗ n) = … for g ∈ G;

(2) H = U(𝔤): g(m ⊗ n) = … for g ∈ 𝔤;

(3) H = U_{q}(sl(2)):

(a) E(m ⊗ n) = …,
(b) F (m ⊗ n) = …,
(c) K(m ⊗ n) = …
for the elements E, F, K ∈ U_{q}(sl(2)).

For two 𝕂G-modules M and N the structure is g(m ⊗ n) = gm ⊗ gn for g ∈ G. For U(𝔤)-modules it is g(m ⊗ n) = gm ⊗ n + m ⊗ gn for g ∈ 𝔤. For U_{q}(sl(2))- modules it is E(m ⊗ n) = m ⊗ En + Em ⊗ Kn, F (m ⊗ n) = K^{−1}m ⊗ F n + F m ⊗ n, K(m ⊗ n) = Km ⊗ Kn.

Remark 3.1.9. Let A and B be algebras over a commutative ring 𝕂. Let f : A ⟶ B be a homomorphism of algebras. Then we have a functor U_{f} : B-Mod ⟶ A-Mod with U_{f}(_{B}M) = _{A}M and U_{f}(g) = g where am := f(a)m for a ∈ A and m ∈ M. The functor U_{f} is also called forgetful or underlying functor.

The action of A on a B-module M can also be seen as the homomorphism A ⟶ B ⟶ End(M).

We denote the underlying functors previously discussed by

U_{A} : A-Mod ⟶ 𝕂-Mod resp. U_{B} : B-Mod ⟶ 𝕂-Mod .

Proposition 3.1.10. Let f : B ⟶ C be a homomorphism of bialgebras. Then U_{f} satisfies the following properties:

U_{f}(M ⊗ N) = U_{f}(M) ⊗ U_{f}(N), U_{f}(g ⊗ h) = g ⊗ h, U_{f}(𝕂) = 𝕂, U_{f}(α) = α, U_{f}(λ) = λ, U_{f}(ρ) = ρ, U_{B}U_{f}(M) = U_{C}(M), U_{B}U_{f}(g) = U_{C}(g).

Proof. This is clear since the underlying 𝕂-modules and the 𝕂-linear maps stay unchanged. The only thing to check is that U_{f} generates the correct B-module structure on the tensor product. For U_{f}(M ⊗ N) = M ⊗ N we have b(m ⊗ n) = f(b)(m ⊗ n) = ∑ f(b)_{(1)}m ⊗ f(b)_{(2)}n = ∑ f(b_{(1)})m ⊗ f(b_{(2)})n = ∑ b_{(1)}m ⊗ b_{(2)}n. □

Remark 3.1.11. Let C and D be coalgebras over a commutative ring 𝕂. Let f : C ⟶ D be a homomorphism of coalgebras. Then we have a functor U_{f} : C-Comod ⟶ D-Comod with U_{f}(^{C}M) = ^{D}M and U_{f}(g) = g where δ_{D} = (f ⊗1)δ_{C} : M ⟶ C⊗M ⟶ D ⊗ M. Again the functor U_{f} is called forgetful or underlying functor.

We denote the underlying functors previously discussed by

U_{C} : C-Comod ⟶ 𝕂-Mod resp. U_{D} : D-Comod ⟶ 𝕂-Mod .

Proposition 3.1.12. Let f : B ⟶ C be a homomorphism of bialgebras. Then U_{f} : C-Comod ⟶ D-Comod satisfies the following properties:

U_{f}(M ⊗ N) = U_{f}(M) ⊗ U_{f}(N), U_{f}(g ⊗ h) = g ⊗ h, U_{f}(𝕂) = 𝕂, U_{f}(α) = α, U_{f}(λ) = λ, U_{f}(ρ) = ρ, U_{C}U_{f}(M) = U_{B}(M), U_{C}U_{f}(g) = U_{B}(g).

Proof. We leave the proof to the reader. □

Proposition 3.1.13. Let H be a Hopf algebra. Let M and N be be H-modules. Then Hom(M, N), the set 𝕂-linear maps from M to N, becomes an H-module by (hf)(m) = ∑ h_{(1)}f(S(h_{(2)}m). This structure makes

Hom : H-Mod × H-Mod ⟶ H-Mod

a functor contravariant in the first variable and covariant in the second variable.

Proof. The main part to be proved is that the action H ⊗ Hom(M, N) ⟶ Hom(M, N) satisfies the associativity law. Let f ∈ Hom(M, N), h, k ∈ H, and m ∈ M. Then ((hk)f)(m) = ∑(hk)_{(1)}f(S((hk)_{(2)}) = ∑ h_{(1)}k_{(1)}f(S(k_{(2)})S(h_{(2)})m) = ∑ h_{(1)}(kf)(S(h_{(2)})m) = (h(kf))(m).

We leave the proof of the other properties, in particular the functorial properties, to the reader. □

Corollary 3.1.14. Let M be an H-module. Then the dual 𝕂-module M^{∗} = Hom(M, 𝕂) becomes an H-module by (hf)(m) = f(S(h)m).

Proof. The space 𝕂 is an H-module via ε : H ⟶ 𝕂. Hence we have (hf)(m) = ∑ h_{(1)}f(S(h_{(2)}m) = ∑ ε(h_{(1)})f(S(h_{(2)}m) = f(S(h)m). □

2. Monoidal Categories

For our further investigations we need a generalized version of the tensor product that we are going to introduce in this section. This will give us the possibility to study more general versions of the notion of algebras and representations.

Definition 3.2.1. A monoidal category (or tensor category) consists of a category C, a covariant functor ⊗ : C × C ⟶ C, called the tensor product, an object I ∈ C, called the unit, natural isomorphisms

α(A, B, C) : (A ⊗ B) ⊗ C ⟶ A ⊗ (B ⊗ C),
λ(A) : I ⊗ A ⟶ A,
ρ(A) : A ⊗ I ⟶ A,

called associativity, left unit and right unit, such that the following diagrams commute:

[figure: ((A ⊗ B) ⊗ C) ⊗ D; (A ⊗ (B ⊗ C)) ⊗ D; α(A,B,C)⊗1; A ⊗ ((B ⊗ C) ⊗ D); α(A,B⊗C,D); α(A⊗B,C,D); 1⊗α(B,C,D); (A ⊗ B) ⊗ (C ⊗ D); A ⊗ (B ⊗ (C ⊗ D)); α(A,B,C⊗D)]

[figure: (A ⊗ I) ⊗ B; A ⊗ (I ⊗ B); α(A,I,B); A ⊗ B; ρ(A)⊗1; 1⊗λ(B)]

These diagrams are called coherence diagrams or constraints.

A monoidal category is called a strict monoidal category, if the morphisms α, λ, ρ are the identity morphisms.

Remark 3.2.2. We define A_{1} ⊗ … ⊗ A_{n} := (… (A_{1} ⊗ A_{2}) ⊗ …) ⊗ A_{n}.

There is an important theorem of S. MacLane that says that all diagrams whose morphisms are constructed by using copies of α, λ, ρ, identities, inverses, tensor products and compositions of such commute. We will only prove this theorem for some very special cases (3.2.4). It implies that each monoidal category can be replaced by (is monoidally equivalent to) a strict monoidal category. That means that we may omit in diagrams the morphisms α, λ, ρ or replace them by identities. In particular there is only one automorphism of A_{1} ⊗ … ⊗ A_{n} formed by coherence morphisms namely the identity.

Remark 3.2.3. For each monoidal category C we can construct the monoidal category C^{symm} symmetric to C that coincides with C as a category and has tensor product A ⊠ B := B ⊗ A and the coherence morphisms

α(C, B, A)^{−1} : (A ⊠ B) ⊠ C ⟶ A ⊠ (B ⊠ C),
ρ(A) : I ⊠ A ⟶ A,
λ(A) : A ⊠ I ⟶ A.

Then the coherence diagrams are commutative again, so that C^{symm} is a monoidal category.

Lemma 3.2.4. Let C be a monoidal category. Then the following diagrams commute

[figure: (I ⊗ A) ⊗ B; I ⊗ (A ⊗ B); α; A ⊗ B; λ(A)⊗1_{B}; λ(A⊗B); (A ⊗ B) ⊗ I; A ⊗ (B ⊗ I); α; A ⊗ B; ρ(A⊗B); 1_{A}⊗ρ(B)]

and we have λ(I) = ρ(I).

Proof. First we observe that the identity functor Id_{C} and the functor I ⊗ - are isomorphic by the natural isomorphism λ. Thus we have I ⊗ f = I ⊗ g ⟹ f = g. In the following diagram all subdiagrams commute except for the right hand trapezoid. Since all morphisms are isomorphisms the right hand trapezoid must commute also. Hence the first diagram of the Lemma commutes.

[figure: ((I ⊗ I) ⊗ A) ⊗ B; (I ⊗ (I ⊗ A)) ⊗ B; α⊗1; I ⊗ ((I ⊗ A) ⊗ B); α; (ρ⊗1)⊗1; (1⊗λ)⊗1; 1⊗(λ⊗1); (I ⊗ A) ⊗ B; I ⊗ (A ⊗ B); α; α; 1⊗α; α; 1; I ⊗ (A ⊗ B); I ⊗ (A ⊗ B); 1; (I ⊗ I) ⊗ (A ⊗ B); I ⊗ (I ⊗ (A ⊗ B)); α; ρ⊗(1⊗1); 1⊗λ]

In a similar way one shows the commutativity of the second diagram.

Furthermore the following diagram commutes

[figure: I ⊗ (I ⊗ I); (I ⊗ I) ⊗ I; α; I ⊗ (I ⊗ I); α; I ⊗ I; 1⊗ρ; ρ; I ⊗ I; ρ⊗1; 1⊗λ; I; ρ; ρ]

Here the left hand triangle commutes by the previous property. The commutativity of the right hand diagram is given by the axiom. The lower square commutes since ρ is a natural transformation. In particular ρ(1 ⊗ ρ) = ρ(1 ⊗ λ). Since ρ is an isomorphism and I ⊗ - ≅ Id_{C} we get ρ = λ. □

Problem 3.2.18. For morphisms f : I ⟶ M and g : I ⟶ N in a monoidal category we define (f ⊗ 1 : N ⟶ M ⊗ N) := (f ⊗ 1_{I})ρ(I)^{−1} and (1 ⊗ g : M ⟶ M ⊗ N) := (1 ⊗ g)λ(I)^{−1}. Show that the diagram

[figure: N; M ⊗ N; f⊗1; I; M; f; g; 1⊗g]

commutes.

We continue with a series of examples of monoidal categories.

Example 3.2.5. (1) Let R be an arbitrary ring. The category R-Mod -R of R-R-bimodules with the tensor product M ⊗_{R} N is a monoidal category. In particular the 𝕂-modules form a monoidal category. This is our most important example of a monoidal category.

(2) Let B be a bialgebra and B-Mod be the category of left B-modules. We define the structure of a B-module on the tensor product M ⊗ N = M ⊗_{𝕂} N by

B ⊗ M ⊗ N ⟶^{Δ⊗1_{M} ⊗1_{N}} B ⊗ B ⊗ M ⊗ N ⟶^{1_{B}⊗τ⊗1_{N}} B ⊗ M ⊗ B ⊗ N ⟶^{ρ_{M} ⊗ρ_{N}} M ⊗ N

as in the previous section. So B-Mod is a monoidal category by 3.1.6

(3) Let B be a bialgebra and B-Comod be the category of B-comodules. The tensor product M ⊗ N = M ⊗_{𝕂} N carries the structure of a B-comodule by

M ⊗ N ⟶^{δ_{M} ⊗δ_{N}} B ⊗ M ⊗ B ⊗ N ⟶^{1_{B}⊗τ⊗1_{N}} B ⊗ B ⊗ M ⊗ N ⟶^{∇⊗1_{M} ⊗1_{N}} B ⊗ M ⊗ N.

as in the previous section. So B-Comod is a monoidal category by 3.1.7

Definition 3.2.6. Let G be a set. The category of G-families of vector spaces M^{G} = ∏_{g∈G} Vec has families of vector spaces (V_{g}|g ∈ G) as objects and families of linear maps (f_{g} : V_{g} ⟶ W_{g}|g ∈ G) as morphisms. The composition is (f_{g}|g ∈ G) ∘ (h_{g}|g ∈ G) = (f_{g} ∘ h_{g}|g ∈ G).

Lemma 3.2.7. Let G be a monoid. Then M^{G} is a monoidal category with the tensor product

(V_{g}|g ∈ G) ⊗ (W_{g}|g ∈ G) := ( ⨁_{h,k∈G,hk=g} V_{h} ⊗ W_{k}|g ∈ G).

Proof. This is an easy exercise. □

Problem 3.2.19. Let G be a monoid. Show that M^{G} is a monoidal category. Where do unit and associativity laws of G enter the proof?

Definition 3.2.8. Let G be a set. A vector space V together with a family of subspaces (V_{g} ⊆ V |g ∈ G) is called G-graded, if V = ⊕_{g∈G}V_{g} holds.

Let (V, (V_{g}|g ∈ G)) and (W, (W_{g}|g ∈ G)) be G-graded vector spaces. A linear map f : V ⟶ W is called G-graded, if f(V_{g}) ⊆ W_{g} for all g ∈ G.

The G-graded vector spaces and G-graded linear maps form the category M^{[G]} of G-graded vector spaces.

Lemma 3.2.9. Let G be a monoid. Then M^{[G]} is a monoidal category with the tensor product V ⊗ W, where the subspaces (V ⊗ W)_{g} are defined by

(V ⊗ W)_{g} := ∑_{h,k∈G,hk=g} V_{h} ⊗ W_{k}.

Proof. This is an easy exercise. □

Problem 3.2.20. Let G be a monoid. Show that M^{[G]} is a monoidal category. Definition 3.2.10. (1) A chain complex of 𝕂-modules

M = (… ⟶^{∂_{3}} M_{2} ⟶^{∂_{2}} M_{1} ⟶^{∂_{1}} M_{0})

consists of a family of 𝕂-modules M_{i} and a family of homomorphisms ∂_{n} : M_{n} ⟶ M_{n−1} with ∂_{n−1}∂_{n} = 0. This chain complex is indexed by the monoid ℕ_{0}. One may also consider more general chain complexes indexed by an arbitrary cyclic monoid. Chain complexes indexed by ℕ_{0} × ℕ_{0} are called double complexes. So much more general chain complexes may be considered. We restrict ourselves to chain complexes over ℕ_{0}.

Let M and N be chain complexes. A homomorphism of chain complexes f : M ⟶ N consists of a family of homomorphisms of 𝕂-modules f_{n} : M_{n} ⟶ N_{n} such that f_{n}∂_{n+1} = ∂_{n+1}f_{n+1} for all n ∈ ℕ_{0}.

The chain complexes with these homomorphisms form the category of chain complexes 𝕂-Comp .

If M and N are chain complexes then we form a new chain complex M ⊗ N with (M ⊗ N)_{n} := ⊕_{i=0}^{n}M_{i} ⊗ N_{n−i} and ∂ : (M ⊗ N)_{n} ⟶ (M ⊗ N)_{n−1} given by ∂(m_{i} ⊗ n_{n−i}) := (−1)^{i}∂_{M}(m_{i}) ⊗ n_{n−i} + m_{i} ⊗ ∂(n_{n−i}). This is often called the total complex associated with the double complex of the tensor product of M and N. Then it is easily checked that 𝕂-Comp is a monoidal category with this tensor product.

(2) A cochain complex has the form

M = (M_{0} ⟶^{∂_{0}} M_{1} ⟶^{∂_{1}} M_{2} ⟶^{∂_{2}} …)

with ∂_{i+1}∂_{i} = 0. The cochain complexes form a monoidal category of cochain complexes 𝕂-Cocomp .

Problem 3.2.21. Show that the cochain complexes form a monoidal category 𝕂-Cocomp .

Definition 3.2.11. Let (C, ⊗) and (D, ⊗) be monoidal categories. A functor

F : C ⟶ D

together with a natural transformation

ξ(M, N) : F(M) ⊗ F(N) ⟶ F(M ⊗ N)

and a morphism

ξ_{0} : I_{D} ⟶ F(I_{C})

is called weakly monoidal if the following diagrams commute

[figure: (F(M) ⊗ F(N)) ⊗ F(P); F(M ⊗ N) ⊗ F(P); ξ⊗1; F((M ⊗ N) ⊗ P); ξ; α; F(α); F(M) ⊗ (F(N) ⊗ F(P)); F(M) ⊗ F(N ⊗ P); 1⊗ξ; F(M ⊗ (N ⊗ P)); ξ]

[figure: I ⊗ F(M); F(I) ⊗ F(M); ξ_{0}⊗1; F(I ⊗ M); ξ; F(M); λ; F(λ)]

[figure: F(M) ⊗ I; F(M) ⊗ F(I); 1⊗ξ_{0}; F(M ⊗ I); ξ; F(M).; ρ; F(ρ)]

If, in addition, the morphisms ξ and ξ_{0} are isomorphisms then the functor is called a monoidal functor. The functor is called a strict monoidal functor if ξ and ξ_{0} are the identity morphisms.

A natural transformation ζ : F ⟶ F′ between weakly monoidal functors is called a monoidal natural transformation if the diagrams

[figure: F′(M) ⊗ F′(N); F′(M ⊗ N); ξ′; F(M) ⊗ F(N); F(M ⊗ N); ξ; ζ⊗ζ; ζ; F(I); I; ξ_{0}; F′(I); ξ′_{0}; ζ]

commute.

Let C and D be monoidal categories. A monoidal functor F : C ⟶ D is called a monoidal equivalence if there exists a monoidal functor G : D ⟶ C and monoidal natural isomorphisms φ : GF ≅ Id_{C} and ψ : FG ≅ Id_{D}.

Proposition 3.2.12. Let G be a monoid.

(1) The monoidal category M^{G} of G-families of vector spaces is monoidally equivalent to the monoidal category M^{[G]} of G-graded vector spaces.

(2) The monoidal category M^{[G]} of G-graded vector spaces is monoidally equivalent to the monoidal category of 𝕂G-comodules Comod -𝕂G.

Proof. We use the following constructions.

(1) For a G-family (V_{g}|g ∈ G) we construct a G-graded vector space V̂ := ⊕_{g∈G}V_{g} (exterior direct sum) with the subspaces V̂_{g} := Im(V_{g}) in the direct sum. Conversely if (V, (V_{g}|g ∈ G)) is a G-graded vector space then (V_{g}|g ∈ G) is a G-family of vector spaces. Similar constructions are used for morphisms. It is easy to see that the categories M^{G} and M^{[G]} are equivalent monoidal categories.

(2) For a G-graded vector space (V, (V_{g}|g ∈ G)) we construct the 𝕂G-comodule V with the structure map δ : V ⟶ V ⊗𝕂G, δ(v) := v⊗g for all (homogeneous elements) v ∈ V_{g} and for all g ∈ G. Conversely let (V, δ : V ⟶ V ⊗ 𝕂G) be a 𝕂G-comodule. Then one constructs the vector space V with den graded (homogenous) components V_{g} := {v ∈ V |δ(v) = v ⊗ g}. It is easy to verify, that this is an equivalence of categories.

Since 𝕂G is a bialgebra, the category of 𝕂G-comodules is a monoidal category by Proposition 3.1.7. One then checks that under the equivalence between M^{[G]} and Comod -𝕂G tensor products are mapped into corresponding tensor products so that we have a monoidal equivalence. □

Problem 3.2.22. Give a detailed proof of Proposition 3.2.12.

Example 3.2.13. The following is a bialgebra B = 𝕂⟨x, y⟩/I, where I is generated by x^{2}, xy + yx. The diagonal is Δ(y) = y ⊗ y, Δ(x) = x ⊗ y + 1 ⊗ x and the counit is ϵ(y) = 1, ϵ(x) = 0.

Proposition 3.2.14. The monoidal category Comp -𝕂 of chain complexes over 𝕂 is monoidally equivalent to the category of B-comodules Comod -B with B as in the preceding example.

Proof. We use the following construction. A chain complex M is mapped to the B-comodule M = ⊕_{i∈ℕ}M_{i} with the structure map δ : M ⟶ M ⊗ B, δ(m) := ∑ m ⊗ y^{i} + ∂_{i}(m) ⊗ xy^{i−1} for all m ∈ M_{i} and for all i ∈ ℕ resp. δ(m) := m ⊗ 1 for m ∈ M_{0}. Conversely if M, δ : M ⟶ M ⊗ B is a B-comodule, then one associates with it the vector spaces M_{i} := {m ∈ M|∃m′ ∈ M[δ(m) = m ⊗ y^{i} + m′ ⊗ xy^{i−1}} and the linear maps ∂_{i} : M_{i} ⟶ M_{i−1} with ∂_{i}(m) := m′ for δ(m) = m ⊗ y^{i} + m′ ⊗ xy^{i−1}. It is checked easily that this is an equivalence of categories. By Problem (41) this is a monoidal equivalence. □

Problem 3.2.23. (1) Give a detailed proof that Comp -𝕂 and Comod -B with B as in the preceding Proposition 3.2.14 are monoidally equivalent categories.

(You may use the following arguments: Let M ∈ Comod -B. Define

M_{i} := {m ∈ M|∃m′ ∈ M : δ(m) = m ⊗ y^{i} + m′ ⊗ xy^{i−1}}.

Let m ∈ M. Since y^{i}, xy^{i} form a basis of B we have δ(m) = ∑_{i} m_{i} ⊗ y^{i} + ∑_{i} m′_{i} ⊗ xy^{i}. Apply (δ ⊗ 1)δ = (1 ⊗ Δ)δ to this equation and compare coefficients to get

δ(m_{i}) = m_{i} ⊗ y^{i} + m′_{i−1} ⊗ xy^{i−1}, δ(m′_{i}) = m′_{i} ⊗ y^{i}

for all i ∈ ℕ_{0} (with m′_{−1} = 0). Consequently for each m_{i} ∈ M_{i} there is exactly one ∂(m_{i}) = m′_{i−1} ∈ M such that

δ(m_{i}) = m_{i} ⊗ y^{i} + ∂(m_{i}) ⊗ xy^{i−1}.

Since δ(m′_{i−1}) = m′_{i−1} ⊗ y^{i−1} for all i ∈ ℕ we see that ∂(m_{i}) ∈ M_{i−1}. So we have defined ∂ : M_{i} ⟶ M_{i−1}. Furthermore we see from this equation that ∂^{2}(m_{i}) = 0 for all i ∈ ℕ. Hence we have obtained a chain complex from (M, δ).

If we apply (1 ⊗ ϵ)δ(m) = m then we get m = ∑ m_{i} with m_{i} ∈ M_{i} hence M = ⊕_{i∈ℕ}M_{i}. This together with the inverse construction leads to the required equivalence.)

(2) Show that the category 𝕂-Cocomp of cochain complexes is monoidally equivalent to B-Comod , where B is chosen as in Example 3.2.13.

(3) Show that the bialgebra B from Example 3.2.13 is a Hopf algebra. We can generalize the notions of an algebra or of a coalgebra in the context of a

monoidal category. We define

Definition 3.2.15. Let C be a monoidal category. An algebra or a monoid in C consists of an object A together with a multiplication ∇ : A ⊗ A ⟶ A that is associative

[figure: A ⊗ A; A; ∇; A ⊗ A ⊗ A; A ⊗ A; id ⊗∇; ∇⊗1; ∇]

or more precisely

[figure: (A ⊗ A) ⊗ A; A ⊗ (A ⊗ A); α; A ⊗ A; id ⊗∇; A ⊗ A; A; ∇; ∇⊗id; ∇]

and has a unit η : I ⟶ A such that the following diagram commutes

[figure: I ⊗ A ≅ A ≅ A ⊗ I; A ⊗ A; id ⊗η; η⊗id; ∇; A ⊗ A; A.; ∇; id]

Let A and B be algebras in C. A morphism of algebras f : A ⟶ B is a morphism in C such that

[figure: A; B; f; A ⊗ A; B ⊗ B; f⊗f; ∇_{A}; ∇_{B}; and; I; η_{A}; η_{B}; A; B; f]

commute.

Remark 3.2.16. It is obvious that the composition of two morphisms of algebras is again a morphism of algebras. The identity also is a morphism of algebras. Thus we obtain the category Alg (C) of algebras in C.

Definition 3.2.17. Let C be a monoidal category. A coalgebra or a comonoid in C consists of an object C together with a comultiplication Δ : A ⟶ A ⊗ A that is coassociative or more precisely

[figure: C ⊗ C; C ⊗ C ⊗ C; id ⊗Δ; C; C ⊗ C; Δ; Δ; Δ⊗id]

[figure: (C ⊗ C) ⊗ C; C ⊗ (C ⊗ C); α; C ⊗ C; Δ⊗id; C; C ⊗ C; Δ; id ⊗Δ; Δ]

and has a counit ε : C ⟶ I such that the following diagram commutes

[figure: C; C ⊗ C; Δ; Δ; id ⊗ϵ; C ⊗ C; I ⊗ C ≅ C ≅ C ⊗ I.; ϵ⊗id; id]

Let C and D be coalgebras in C. A morphism of coalgebras f : C ⟶ D is a morphism in C such that

[figure: C ⊗ C; D ⊗ D; f⊗f; C; D; f; Δ_{C}; Δ_{D}; and; I; ε_{C}; ε_{D}; C; D; f]

commute.

Remark 3.2.18. It is obvious that the composition of two morphisms of coalgebras is again a morphism of coalgebras. The identity also is a morphism of coalgebras. Thus we obtain the category Coalg (C) of coalgebras in C.

Remark 3.2.19. Observe that the notions of bialgebra, Hopf algebra, and comodule algebra cannot be generalized to an arbitrary monoidal category since we need to have an algebra structure on the tensor product of two algebras and this requires us to interchange the middle tensor factors. These interchanges or flips are known under the name symmetry, quasisymmetry or braiding and will be discussed later on.

3. Dual Objects

At the end of the first section in Corollary 3.1.14 we saw that the dual of an Hmodule can be constructed. We did not show the corresponding result for comodules. In fact such a construction for comodules needs some finiteness conditions. With this restriction the notion of a dual object can be introduced in an arbitrary monoidal category.

Definition 3.3.1. Let (C, ⊗) be a monoidal category M ∈ C be an object. An object M^{∗} ∈ C together with a morphism ev : M^{∗} ⊗ M ⟶ I is called a left dual for M if there exists a morphism db : I ⟶ M ⊗ M^{∗} in C such that

(M ⟶^{db ⊗1} M ⊗ M^{∗} ⊗ M ⟶^{1⊗ev} M) = 1_{M}

(M^{∗} ⟶^{1⊗db} M^{∗} ⊗ M ⊗ M^{∗} ⟶^{ev ⊗1} M^{∗}) = 1_{M^{∗}}.

A monoidal category is called left rigid if each object M ∈ C has a left dual.

Symmetrically we define: an object^{∗}M ∈ C together with a morphism ev : M⊗^{∗}M ⟶ I is called a right dual for M if there exists a morphism db : I ⟶ ^{∗}M ⊗ M in C such that

(M ⟶^{1⊗db} M ⊗^{∗}M ⊗ M ⟶^{ev ⊗1} M) = 1_{M}

(^{∗}M ⟶^{db ⊗1} ^{∗}M ⊗ M ⊗^{∗}M ⟶^{1⊗ev} ^{∗}M) = 1_{^{∗}M}.

A monoidal category is called right rigid if each object M ∈ C has a left dual. The morphisms ev and db are called the evaluation respectively the dual basis. Remark 3.3.2. If (M^{∗}, ev) is a left dual for M then obviously (M, ev) is a right

dual for M^{∗} and conversely. One uses the same morphism db : I ⟶ M ⊗ M^{∗}.

Lemma 3.3.3. Let (M^{∗}, ev) be a left dual for M. Then there is a natural isomorphism

Mor_{C}(- ⊗ M, -) ≅ Mor_{C}(-, - ⊗ M^{∗}),

i. e. the functor - ⊗ M : C ⟶ C is left adjoint to the functor - ⊗ M^{∗} : C ⟶ C.

Proof. We give the unit and the counit of the pair of adjoint functors. We define Φ(A) := 1_{A} ⊗ db : A ⟶ A ⊗ M ⊗ M^{∗} and Ψ(B) := 1_{B} ⊗ ev : B ⊗ M^{∗} ⊗ M ⟶ B. These are obviously natural transformations. We have commutative diagrams

[figure: (A ⊗ M; A ⊗ M ⊗ M^{∗} ⊗ M; FΦ(A)=; 1_{A}⊗db ⊗1_{M}; A ⊗ M) = 1_{A⊗M}; ΨF(A)=; 1_{A}⊗1_{M} ⊗ev]

and

[figure: (B ⊗ M^{∗}; B ⊗ M^{∗} ⊗ M ⊗ M^{∗}; ΦG(B)=; 1_{B}⊗1_{M^{∗}}⊗db; B ⊗ M^{∗}) = 1_{B⊗M^{∗}}; GΨ(B)=; 1_{B}⊗ev ⊗1_{M^{∗}}]

thus the Lemma has been proved by [Advanced Algebra] Corollary 5.17. □

The converse holds as well. If - ⊗ M is left adjoint to - ⊗ M^{∗} then the unit Φ gives a morphism db := Φ(I) : I ⟶ M ⊗ M^{∗} and the counit Ψ gives a morphism ev := Ψ(I) : M^{∗} ⊗ M ⟶ I satisfying the required properties. Thus we have

Corollary 3.3.4. If - ⊗ M : C ⟶ C is left adjoint to - ⊗ M^{∗} : C ⟶ C then M^{∗} is a left dual for M.

Corollary 3.3.5. (M^{∗}, ev) is a left dual for M if and only if there is a natural isomorphism

Mor_{C}(M^{∗} ⊗ -, -) ≅ Mor_{C}(-, M ⊗ -),

i. e. the functor M^{∗} ⊗ - : C ⟶ C is left adjoint to the functor M ⊗ - : C ⟶ C. The natural isomorphism if given by

(f : M^{∗} ⊗ N ⟶ P) ↦ ((1_{M} ⊗ f)(db ⊗1_{N}) : N ⟶ M ⊗ M^{∗} ⊗ N ⟶ M ⊗ P)

and

(g : N ⟶ M ⊗ P) ↦ ((ev ⊗1_{P})(1_{M^{∗}} ⊗ g) : M^{∗} ⊗ N ⟶ M^{∗} ⊗ M ⊗ P ⟶ P).

Proof. We have a natural isomorphism

Mor_{C}(M^{∗} ⊗ -, -) ≅ Mor_{C}(-, M ⊗ -),

iff (M, ev) is a right dual for M^{∗} (as a symmetric statement to Lemma 3.3.3) iff (M^{∗}, ev) is a left dual for M. □

Corollary 3.3.6. If M has a left dual then this is unique up to isomorphism.

Proof. Let (M^{∗}, ev) and (M^{!}, ev^{!}) be left duals for M. Then the functors -⊗M^{∗} and -⊗M^{!} are isomorphic by [Advanced Algebra] Lemma 5.13. In particular we have M^{∗} ≅ I ⊗ M^{∗} ≅ I ⊗ M^{!} ≅ M^{!}. If we consider the construction of the isomorphism then we get in particular that (ev^{!} ⊗1)(1 ⊗ db) : M^{!} ⟶ M^{!} ⊗ M ⊗ M^{∗} ⟶ M^{∗} is the given isomorphism. □

Problem 3.3.24. Let (M^{∗}, ev) be a left dual for M. Then there is a unique morphism db : I ⟶ M ⊗ M^{∗} satisfying the conditions of Definition 3.3.1. db′ = (1_{M} ⊗ 1_{M^{∗}}) db′ = (1_{M} ⊗ ev ⊗1_{M^{∗}})(1_{M} ⊗ 1_{M^{∗}} ⊗ db) db′ = (1_{M} ⊗ ev ⊗1_{M^{∗}})(db′ ⊗ db) = (1_{M} ⊗ ev ⊗1_{M^{∗}})(db′ ⊗1_{M} ⊗ 1_{M^{∗}}) db = (1_{M} ⊗ 1_{M^{∗}}) db = db

Definition 3.3.7. Let (M^{∗}, ev_{M}) and (N^{∗}, ev_{N}) be left duals for M resp. N. For each morphism f : M ⟶ N we define the transposed morphism

(f^{∗} : N^{∗} ⟶ M^{∗}) := (N^{∗} ⟶^{1⊗db_{M}} N^{∗} ⊗ M ⊗ M^{∗} ⟶^{1⊗f⊗1} N^{∗} ⊗ N ⊗ M^{∗} ⟶^{ev_{N} ⊗1} M^{∗}).

With this definition we get that the left dual is a contravariant functor, since we have

Lemma 3.3.8. Let (M^{∗}, ev_{M}), (N^{∗}, ev_{N}), and (P^{∗}, ev_{P}) be left duals for M, N and P respectively.

1. We have (1_{M})^{∗} = 1_{M^{∗}}.

2. If f : M ⟶ N and g : N ⟶ P are given then (gf)^{∗} = f^{∗}g^{∗} holds.

Proof. 1. (1_{M})^{∗} = (ev ⊗1)(1 ⊗ 1 ⊗ 1)(1 ⊗ db) = 1_{M^{∗}}.

2. The following diagram commutes

[figure: M; N ⊗ N^{∗} ⊗ M; db_{N} ⊗1; f; 1⊗1⊗f; N; N ⊗ N^{∗} ⊗ N; db_{N} ⊗1; N; 1⊗ev_{N}; g⊗1⊗1; g; P ⊗ N^{∗} ⊗ N; P; 1⊗ev_{N}]

Hence we have gf = (1 ⊗ ev_{N})(g ⊗ 1 ⊗ f)(db_{N} ⊗1). Thus the following diagram commutes

[figure: P^{∗}; P^{∗} ⊗ N ⊗ N^{∗}; 1⊗db; P^{∗} ⊗ P ⊗ N^{∗}; 1⊗g⊗1; 1⊗db; 1⊗db; 1⊗db; P^{∗} ⊗ M ⊗ M^{∗}; P^{∗} ⊗ N ⊗ N^{∗} ⊗ M ⊗ M^{∗}; 1⊗db ⊗1; P^{∗} ⊗ P ⊗ N^{∗} ⊗ M ⊗ M^{∗}; 1⊗g⊗1; 1⊗gf⊗1; P^{∗} ⊗ P ⊗ M^{∗}; P^{∗} ⊗ P ⊗ N^{∗} ⊗ N ⊗ M^{∗}; 1⊗ev ⊗1; 1⊗f⊗1; ev ⊗1; ev ⊗1; ev ⊗1; M^{∗}; N^{∗} ⊗ N ⊗ M^{∗}; 1⊗ev; N^{∗} ⊗ M ⊗ M^{∗}; 1⊗f⊗1; 1⊗g⊗1⊗f⊗1; N^{∗}; ev ⊗1; 1⊗db]
□

Problem 3.3.25. (1) In the category of ℕ-graded vector spaces determine

all objects M that have a left dual.

(2) In the category of chain complexes 𝕂-Comp determine all objects M that have a left dual.

(3) In the category of cochain complexes 𝕂-Cocomp determine all objects M that have a left dual.

(4) Let (M^{∗}, ev) be a left dual for M. Show that db : I ⟶ M ⊗ M^{∗} is uniquely determined by M, M^{∗}, and ev. (Uniqueness of the dual basis.)

(5) Let (M^{∗}, ev) be a left dual for M. Show that ev : M^{∗} ⊗ M ⟶ I is uniquely determined by M, M^{∗}, and db.

Corollary 3.3.9. Let M, N have the left duals (M^{∗}, ev_{M}) and (N^{∗}, ev_{N}) and let f : M ⟶ N be a morphism in C. Then the following diagram commutes

[figure: N ⊗ N^{∗}; N ⊗ M^{∗}.; 1⊗f^{∗}; I; M ⊗ M^{∗}; db_{M}; db_{N}; f⊗1]

Proof. The following diagram commutes

[figure: M; N ⊗ N^{∗} ⊗ M; db ⊗1; N; N ⊗ N^{∗} ⊗ N; db ⊗1; f; 1⊗1⊗f; N; 1⊗ev; 1]

This implies (f ⊗ 1_{M^{∗}}) db_{M} = ((1_{N} ⊗ ev_{N})(1_{N} ⊗ 1_{N^{∗}} ⊗ f)(db_{N} ⊗1_{M}) ⊗ 1_{M^{∗}}) db_{M} = (1_{N} ⊗ev_{N} ⊗1_{M^{∗}})(1_{N} ⊗1_{N^{∗}} ⊗f ⊗1_{M^{∗}})(db_{N} ⊗1_{M} ⊗1_{M^{∗}}) db_{M} = (1_{N} ⊗ev_{N} ⊗1_{M^{∗}})(1_{N} ⊗ 1_{N^{∗}} ⊗ f ⊗ 1_{M^{∗}})(1_{N} ⊗ 1_{N^{∗}} ⊗ db_{M}) db_{N} = (1_{N} ⊗ (ev_{N} ⊗1_{M^{∗}})(1_{N^{∗}} ⊗ f ⊗ 1_{M^{∗}})(1_{N^{∗}} ⊗ db_{M})) db_{N} = (1_{N} ⊗ f^{∗}) db_{N}. □

Corollary 3.3.10. Let M, N have the left duals (M^{∗}, ev_{M}) and (N^{∗}, ev_{N}) and let f : M ⟶ N be a morphism in C. Then the following diagram commutes

[figure: N^{∗} ⊗ N; I.; ev_{N}; N^{∗} ⊗ M; M^{∗} ⊗ M; f^{∗}⊗1; 1⊗f; ev_{M}]

Proof. This statement follows immediately from the symmetry of the definition of a left dual. □

Example 3.3.11. Let M ∈ R-Mod -R be an R-R-bimodule. Then Hom_{R}(M., R.) is an R-R-bimodule by (rfs)(x) = rf(sx). Furthermore we have the morphism ev : Hom_{R}(M., R.) ⊗_{R} M ⟶ R defined by ev(f ⊗_{R} m) = f(m).

(Dual Basis Lemma:) A module M ∈ Mod -R is called finitely generated and projective if there are elements m_{1}, …, m_{n} ∈ M und m^{1}, …, m^{n} ∈ Hom_{R}(M., R.) such that

∀m ∈ M : ∑_{i=1}^{n} m_{i}m^{i}(m) = m.

Actually this is a consequence of the dual basis lemma. But this definition is equivalent to the usual definition.

Let M ∈ R-Mod -R. M as a right R-module is finitely generated and projective iff M has a left dual. The left dual is isomorphic to Hom_{R}(M., R.).

If M_{R} is finitely generated projective then we use db : R ⟶ M ⊗_{R} Hom_{R}(M., R.) with db(1) = ∑_{i=1}^{n} m_{i} ⊗_{R} m^{i}. In fact we have (1 ⊗_{R} ev)(db ⊗_{R}1)(m) = (1 ⊗_{R} ev)(∑ m_{i} ⊗_{R} m^{i} ⊗_{R} m) = ∑ m_{i}m^{i}(m) = m. We have furthermore (ev ⊗_{R}1)(1 ⊗_{R} db)(f)(m) = (ev ⊗_{R}1)(∑_{i=1}^{n} f ⊗_{R} m_{i} ⊗_{R} m^{i})(m) = ∑ f(m_{i})m^{i}(m) = f(∑ m_{i}m^{i}(m)) = f(m) for all m ∈ M hence (ev ⊗_{R}1)(1 ⊗_{R} db)(f) = f.

Conversely if M has a left dual M^{∗} then ev : M^{∗} ⊗_{R} M ⟶ R defines a homomorphism ι : M^{∗} ⟶ Hom_{R}(M., R.) in R-Mod -R by ι(m^{∗})(m) = ev(m^{∗} ⊗_{R} m). We define ∑_{i=1}^{n} m_{i} ⊗ m^{i} := db(1) ∈ M ⊗ M^{∗}, then m = (1 ⊗ ev)(db ⊗1)(m) = (1 ⊗ ev)(∑ m_{i}⊗m^{i}⊗m) = ∑ m_{i}ι(m^{i})(m) so that m_{1}, …, m_{n} ∈ M and ι(m^{1}), …, ι(m^{n}) ∈ Hom_{R}(M., R.) form a dual basis for M, i. e. M is finitely generated and projective as an R-module. Thus M^{∗} and Hom_{R}(M., R.) are isomorphic by the map ι.

Analogously Hom_{R}(.M, .R) is a right dual for M iff M is finitely generated and projective as a left R-module.

Problem 3.3.26. Find an example of an object M in a monoidal category C that has a left dual but no right dual.

Definition 3.3.12. Given objects M, N in C. An object [M, N] is called a left inner Hom of M and N if there is a natural isomorphism Mor_{C}(- ⊗ M, N) ≅ Mor_{C}(-, [M, N]), i. e. if it represents the functor Mor_{C}(- ⊗ M, N).

If there is an isomorphism Mor_{C}(P ⊗ M, N) ≅ Mor_{C}(P, [M, N]) natural in the three variable M, N, P then the category C is called monoidal and left closed.

If there is an isomorphism Mor_{C}(M ⊗ P, N) ≅ Mor_{C}(P, [M, N]) natural in the three variable M, N, P then the category C is called monoidal and right closed.

If M has a left dual M^{∗} in C then there are inner Homs [M, - ] defined by [M, N] := N ⊗ M^{∗}. In particular left rigid monoidal categories are left closed.

Example 3.3.13. (1) The category of finite dimensional vector spaces is a monoidal category where each object has a (left and right) dual. Hence it is (left and right) closed and rigid.

(2) Let Ban be the category of (complex) Banach spaces where the morphisms satisfy ∥ f(m) ∥≤∥ m ∥ i. e. the maps are bounded by 1 or contracting. Ban is a monoidal category by the complete tensor product M ˆ⊗N. In Ban exists an inner Hom functor [M, N] that consists of the set of bounded linear maps from M to N made into a Banach space by an appropriate topology. Thus Ban is a monoidal closed category.

(3) Let H be a Hopf algebra. The category H-Mod of left H-modules is a monoidal category (see Example 3.2.5 2.). Then Hom_{𝕂}(M, N) is an object in H-Mod by the multiplication

(hf)(m) := ∑ h_{(1)}f(mS(h_{(2)})

as in Proposition 3.1.13.

Hom_{𝕂}(M, N) is an inner Hom functor in the monoidal category H-Mod . The isomorphism ϕ : Hom_{𝕂}(P, Hom_{𝕂}(M, N)) ≅ Hom_{𝕂}(P ⊗ M, N) can be restricted to an isomorphism

Hom_{H}(P, Hom_{𝕂}(M, N)) ≅ Hom_{H}(P ⊗ M, N),

because ϕ(f)(h(p ⊗ m)) = ϕ(f)(∑ h_{(1)}p ⊗ h_{(2)}m) = ∑ f(h_{(1)}p)(h_{(2)}m) = ∑(h_{(1)}(f(p)))(h_{(2)}m) = ∑ h_{(1)}(f(p)(S(h_{(2)})h_{(3)}m)) = h(f(p)(m)) = h(ϕ(f) (p⊗m)) and conversely (h(f(p)))(m) = ∑ h_{(1)}(f(p)(S(h_{(2)})m)) = ∑ h_{(1)}(ϕ(f) (p⊗S(h_{(2)})m)) = ∑ ϕ(f)(h_{(1)}p⊗h_{(2)}S(h_{(3)})m) = ϕ(f)(hp⊗m) = f(hp)(m). Thus H-Mod is left closed.

If M ∈ H-Mod is a finite dimensional vector space then the dual vector space M^{∗} := Hom_{𝕂}(M, 𝕂) again is an H-module by (hf)(m) := f(S(h)m). Furthermore M^{∗} is a left dual for M with the morphisms

db : 𝕂 ∋ 1 ↦ ∑_{i} m_{i} ⊗ m^{i} ∈ M ⊗ M^{∗}

and

ev : M^{∗} ⊗ M ∋ f ⊗ m ↦ f(m) ∈ 𝕂

where m_{i} and m^{i} are a dual basis of the vector space M. Clearly we have (1 ⊗ ev)(db ⊗1) = 1_{M} and (ev ⊗1)(1 ⊗ db) = 1_{M^{∗}} since M is a finite dimensional vector space. We have to show that db and ev are H-module homomorphisms. We have (h db(1))(m) = (h(∑ m_{i} ⊗ m^{i}))(m) = (∑ h_{(1)}m_{i} ⊗ h_{(2)}m^{i})(m) = ∑(h_{(1)}m_{i})((h_{(2)}m^{i})(m)) = ∑(h_{(1)}m_{i})(m^{i}(S(h_{(2)})m)) = ∑ h_{(1)}S(h_{(2)})m = ε(h)m = ε(h)(∑ m_{i} ⊗ m^{i})(m) = ε(h) db(1)(m) = db(ε(h)1)(m) = db(h1)(m),

hence h db(1) = db(h1). Furthermore we have

\begin{matrix}h ev(f ⊗ m) = hf(m) = ∑ h_{(1)}f(S(h_{(2)})h_{(3)}m) = ∑(h_{(1)}f)(h_{(2)}m) = \\ ∑ ev(h_{(1)}f ⊗ h_{(2)}m) = ev(h(f ⊗ m)). \end{matrix}

(4) Let H be a Hopf algebra. Then the category of left H-comodules (see Example 3.2.5 3.) is a monoidal category. Let M ∈ H-Comod be a finite dimensional vector space. Let m_{i} be a basis for M and let the comultiplication of the comodule be δ(m_{i}) = ∑ h_{ij} ⊗ m_{j}. Then we have Δ(h_{ik}) = ∑ h_{ij} ⊗ h_{jk}. M^{∗} := Hom_{𝕂}(M, 𝕂) becomes a left H-comodule δ(m^{j}) := ∑ S(h_{ij}) ⊗ m^{i}. One verifies that M^{∗} is a left dual for M.

Lemma 3.3.14. Let M ∈ C be an object with left dual (M^{∗}, ev). Then

1. M ⊗ M^{∗} is an algebra with multiplication

∇ := 1_{M} ⊗ ev ⊗1_{M^{∗}} : M ⊗ M^{∗} ⊗ M ⊗ M^{∗} ⟶ M ⊗ M^{∗}

and unit

u := db : I ⟶ M ⊗ M^{∗};

2. M^{∗} ⊗ M is a coalgebra with comultiplication

Δ := 1_{M^{∗}} ⊗ db ⊗1_{M} : M^{∗} ⊗ M ⟶ M^{∗} ⊗ M ⊗ M^{∗} ⊗ M

and counit

ε := ev : M^{∗} ⊗ M ⟶ I.

Proof. 1. The associativity is given by (∇ ⊗ 1)∇ = (1_{M} ⊗ ev ⊗1_{M^{∗}} ⊗ 1_{M} ⊗ 1_{M^{∗}})(1_{M} ⊗ ev ⊗1_{M^{∗}}) = 1_{M} ⊗ ev ⊗ ev ⊗1_{M^{∗}} = (1_{M} ⊗ 1_{M^{∗}} ⊗ 1_{M} ⊗ ev ⊗1_{M^{∗}})(1_{M} ⊗ ev ⊗1_{M^{∗}}) = (1⊗∇)∇. The axiom for the left unit is ∇(u⊗1) = (1_{M} ⊗ev ⊗1_{M^{∗}})(db ⊗ 1_{M} ⊗ 1_{M^{∗}}) = 1_{M} ⊗ 1_{M^{∗}}.

2. is dual to the statement for algebras. □

Lemma 3.3.15. 1. Let A be an algebra in C and left M ∈ C be a left rigid object with left dual (M^{∗}, ev). There is a bijection between the set of morphisms f : A ⊗ M ⟶ M making M a left A-module and the set of algebra morphisms f̃: A ⟶ M ⊗M^{∗}.

2. Let C be a coalgebra in C and left M ∈ C be a left rigid object with left dual (M^{∗}, ev). There is a bijection between the set of morphisms f : M ⟶ M ⊗ C making M a right C-comodule and the set of coalgebra morphisms f̃: M^{∗} ⊗ M ⟶ C.

Proof. 1. By Lemma 3.3.14 the object M ⊗ M^{∗} is an algebra. Given f : A ⊗ M ⟶ M such that M becomes an A-module. By Lemma 3.3.3 we associate f̃ := (f ⊗ 1)(1 ⊗ db) : A ⟶ A ⊗ M ⊗ M^{∗} ⟶ M ⊗ M^{∗}. The compatibility of f̃ with the multiplication is given by the commutative diagram

[figure: A ⊗ A; A; ∇; A ⊗ A ⊗ M ⊗ M^{∗}; A ⊗ M ⊗ M^{∗}; ∇⊗1⊗1; A ⊗ M ⊗ M^{∗}; M ⊗ M^{∗}; f⊗1; A ⊗ M ⊗ M^{∗} ⊗ M ⊗ M^{∗}; A ⊗ M ⊗ M^{∗}; 1⊗1⊗ev ⊗1; M ⊗ M^{∗} ⊗ M ⊗ M^{∗}; M ⊗ M^{∗}; 1⊗ev ⊗1; f̃⊗f̃; 1⊗f⊗1; 1⊗db ⊗1⊗1; f⊗1; f̃; 1⊗1⊗db; 1⊗f̃; f̃⊗1⊗1; f⊗1⊗1; 1; 1⊗db; 1; f⊗1]

The unit axiom is given by

[figure: I; M ⊗ M^{∗}; db; A; A ⊗ M ⊗ M^{∗}; 1⊗db; M ⊗ M^{∗}; f⊗1; u; u⊗1; 1⊗1]

Conversely let g : A ⟶ M ⊗ M^{∗} be an algebra homomorphism and consider g̃ := (1 ⊗ ev)(g ⊗ 1) : A ⊗ M ⟶ M ⊗ M^{∗} ⊗ M ⟶ M. Then M becomes a left A-module since

[figure: A ⊗ A ⊗ M; A ⊗ M; ∇⊗1; A ⊗ M ⊗ M^{∗} ⊗ M; M ⊗ M^{∗} ⊗ M ⊗ M^{∗} ⊗ M; g⊗1⊗1⊗1; M ⊗ M^{∗} ⊗ M; 1⊗ev ⊗1⊗1; A ⊗ M; M; g̃; 1⊗g̃; M ⊗ M^{∗} ⊗ M; 1⊗1⊗1⊗ev; g̃; 1⊗g⊗1; 1⊗1⊗ev; g⊗g⊗1; g⊗1; 1⊗ev; 1⊗ev; g⊗1]

and

[figure: A ⊗ M; M ⊗ M^{∗} ⊗ M; g⊗1; M; 1⊗ev; M; u⊗1; db ⊗1; 1]

commute.

2. (M^{∗}, ev) is a left dual for M in the category C if and only if (M^{∗}, db) is the right dual for M in the dual category C^{op}. So if we dualize the result of part 1. we have to change sides, hence 2. □

4. Finite reconstruction

The endomorphism ring of a vector space enjoys the following universal property. It is a vector space itself and allows a homomorphism ρ : End(V) ⊗ V ⟶ V. It is universal with respect to this property, i. e. if Z is a vector space and f : Z ⊗ V ⟶ V is a homomorphism, then there is a unique homomorphism g : Z ⟶ End(V) such that

[figure: End(V) ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes.

The algebra structure of End(V) comes for free from this universal property. If we replace the vector space V by a diagram of vector spaces ω : D ⟶ Vec we

get a similar universal object End(ω). Again the universal property induces a unique algebra structure on End(ω).

Problem 3.4.27. (1) Let V be a vector space. Show that there is a universal vector space E and homomorphism ρ : E ⊗ V ⟶ V (such that for each vector space Z and each homomorphism f : Z ⊗ V ⟶ V there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes). We call E and ρ : E ⊗ V ⟶ V a vector space acting universally on V.

(2) Let E and ρ : E ⊗ V ⟶ V be a vector space acting universally on V. Show that E uniquely has the structure of an algebra such that V becomes a left E-module.

(3) Let ω : D ⟶ Vec be a diagram of vector spaces. Show that there is a universal vector space E and natural transformation ρ : E ⊗ ω ⟶ ω (such that for each vector space Z and each natural transformation f : Z ⊗ ω ⟶ ω there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ ω; ω; ρ; f; Z ⊗ ω; g⊗1]

commutes). We call E and ρ : E ⊗ ω ⟶ ω a vector space acting universally on ω.

(4) Let E and ρ : E⊗ω ⟶ ω be a vector space acting universally on ω. Show that E uniquely has the structure of an algebra such that ω becomes a diagram of left E-modules.

Similar considerations can be carried out for coactions V ⟶ V ⊗C or ω ⟶ ω ⊗C and a coalgebra structure on C. There is one restriction, however. We can only use finite dimensional vector spaces V or diagrams of finite dimensional vector spaces. This will be done further down.

We want to find a universal natural transformation δ : ω ⟶ ω ⊗ coend(ω). For this purpose we consider the isomorphisms

Mor_{C}(ω(X), ω(X) ⊗ M) ≅ Mor_{C}(ω(X)^{∗} ⊗ ω(X), M)

that are given by f ↦ (ev ⊗1)(1 ⊗ f) and as inverse g ↦ (1 ⊗ g)(db ⊗1). We first develop techniques to describe the properties of a natural transformation ϕ : ω ⟶ ω ⊗ M as properties of the associated family g(X) : ω(X)^{∗} ⊗ ω(X) ⟶ M. We will see that g : ω^{∗} ⊗ ω ⟶ M will be a cone. Then we will show that ϕ is a universal natural transformation if and only if its associated cone is universal. In the literature this is called a coend.

Throughout this section assume the following. Let D be an arbitrary diagram scheme. Let C be a cocomplete monoidal category such that the tensor product preserves colimits in both arguments. Let C_{0} be the full subcategory of those objects in C that have a left dual. Let ω : D ⟶ C be a diagram in C such that ω(X) ∈ C_{0} for all X ∈ D, i. e. ω is given by a functor ω_{0} : D ⟶ C_{0}. We call such a diagram a finite diagram in C. Finally for an object M ∈ C let ω ⊗ M : D ⟶ C be the functor with (ω ⊗ M)(X) = ω(X) ⊗ M.

Remark 3.4.1. Consider the following category D̃. For each morphism f : X ⟶ Y there is an object f̃ ∈ D̃. The object corresponding to the identity 1_{X} : X ⟶ X is denoted by X̃ ∈ D̃. For each morphism f : X ⟶ Y in D there are two morphisms f_{1} : f̃ ⟶ X̃ and f_{2} : f̃ ⟶ Ỹ in D̃. Furthermore there are the identities 1_{f} : f̃ ⟶ f̃ in D̃.

Since there are no morphisms with X̃ as domain other than (1_{X})_{i} : X̃ ⟶ X̃ and 1_{f} : f̃ ⟶ f̃ we only have to define the following compositions (1_{X})_{i} ∘ f_{j} := f_{j}. Then D̃ becomes a category and we have 1X̃ = (1_{X})_{1} = (1_{X})_{2}.

We define a diagram ω^{∗} ⊗ ω : D̃ ⟶ C as follows. If f : X ⟶ Y is given then

(ω^{∗} ⊗ ω)(f̃) := ω(Y)^{∗} ⊗ ω(X)

and

\begin{matrix}ω(f_{1}) := ω(f)^{∗} ⊗ ω(1_{X}), \\ ω(f_{2}) := ω(1_{Y})^{∗} ⊗ ω(f). \end{matrix}

The colimit of ω^{∗} ⊗ ω consists of an object coend(ω) ∈ C together with a family of morphisms ι(X, X) : ω(X)^{∗} ⊗ ω(X) ⟶ coend(ω) such that the diagrams

[figure: ω(X)^{∗} ⊗ ω(X); (ω^{∗} ⊗ ω)(f̃) = ω(Y)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; coend(ω); ι(X,X); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y)]

commute for all f : X ⟶ Y in D. Indeed, such a family ι(X̃) := ι(X, X) can be uniquely extended to a natural transformation by defining ι(f̃) := ι(X, X)(ω(f)^{∗} ⊗ ω(X)) = ι(Y, Y)(ω(Y)^{∗} ⊗ ω(f)). In addition the pair (coend(ω), ι) is universal with respect to this property.

In the literature such a universal object is called a coend of the bifunctor ω^{∗} ⊗ ω : D^{op} × D ⟶ C.

Corollary 3.4.2. The following is a coequalizer

[figure: ∐_{f∈MorD} ω(Dom(f))^{∗} ⊗ ω(Codom(f)); ∐_{X∈Ob D} ω(X)^{∗} ⊗ ω(X); p; q; coend(ω)]

Proof. This is just a reformulation of [Advanced Algebra] Remark 6.11, since the colimit may also be built from the commutative squares given above.

Observe that for the construction of the colimit not all objects of the diagram have to be used but only those of the form ω(X)^{∗} ⊗ ω(X). □

Theorem 3.4.3. (Tannaka-Krein)

Let ω : D ⟶ C_{0} ⊆ C be a finite diagram. Then there exists an object coend(ω) ∈ C and a natural transformation δ : ω ⟶ ω ⊗ coend(ω) such that for each object M ∈ C and each natural transformation φ : ω ⟶ ω ⊗ M there exists a unique morphism φ̃: coend(ω) ⟶ M such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; φ; ω ⊗ M; 1⊗φ̃]

commutes.

Proof. Let coend(ω) ∈ C together with morphisms ι(f̃) : ω(Y)^{∗} ⊗ ω(X) ⟶ coend(ω) be the colimit of the diagram ω^{∗} ⊗ ω : D̃ ⟶ C. So we get commutative diagrams

[figure: ω(X)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; ι(X,X); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y); ω(Y)^{∗} ⊗ ω(X); coend(ω); ι(f̃)]

for each f : X ⟶ Y in C.

For X ∈ C we define a morphism δ(X) : ω(X) ⟶ ω(X) ⊗ coend(ω) by (1 ⊗ ι(X, X))(db ⊗1) : ω(X) ⟶ ω(X) ⊗ ω(X)^{∗} ⊗ ω(X) ⟶ ω(X) ⊗ coend(ω). Then we get as in Corollary 3.3.5 ι(X, X) = (1 ⊗ ev)(1 ⊗ δ(X)).

We show that δ is a natural transformation. For each f : X ⟶ Y the square commutes by Corollary 3.3.9. Thus the following diagram commutes

[figure: ω(Y) ⊗ ω(Y)^{∗}; ω(Y) ⊗ ω(X)^{∗}.; 1⊗ω(f)^{∗}; I; ω(X) ⊗ ω(X)^{∗}; db_{X}; db_{Y}; ω(f)⊗1]

[figure: ω(X); ω(X) ⊗ ω(X)^{∗} ⊗ ω(X); db ⊗1; ω(X) ⊗ coend(ω); 1⊗ι(X,X); ω(Y) ⊗ ω(Y)^{∗} ⊗ ω(X); ω(Y) ⊗ ω(X)^{∗} ⊗ ω(X); 1⊗ω(f)^{∗}⊗1; ω(Y); ω(Y) ⊗ ω(Y)^{∗} ⊗ ω(Y); db ⊗1; ω(Y) ⊗ coend(ω).; 1⊗ι(Y,Y); ω(f); ω(f)⊗1; db ⊗1; 1⊗1⊗ω(f); ω(f)⊗1⊗1; 1⊗ι(X,X)]

Now let M ∈ C be an object and φ : ω ⟶ ω ⊗ M a natural transformation. Observe that

[figure: ω(Y)^{∗} ⊗ ω(Y); I; ev; ω(Y)^{∗} ⊗ ω(X); ω(X)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; 1⊗ω(f); ev]

commutes by Corollary 3.3.10. Thus also the diagram

[figure: ω(X)^{∗} ⊗ ω(X); ω(X)^{∗} ⊗ ω(X) ⊗ M; 1⊗φ(X); ω(f)^{∗}⊗1; ω(f)^{∗}⊗1⊗1; M; ev ⊗1; ω(Y)^{∗} ⊗ ω(X); ω(Y)^{∗} ⊗ ω(X) ⊗ M; 1⊗φ(X); 1⊗ω(f); 1⊗ω(f)⊗1; ev ⊗1; ω(Y)^{∗} ⊗ ω(Y); ω(Y)^{∗} ⊗ ω(Y) ⊗ M; 1⊗φ(Y)]

commutes. We define φ̃ : coend(ω) ⟶ M from the colimit property as universal factorization

[figure: ω(X)^{∗} ⊗ ω(X); ω(Y)^{∗} ⊗ ω(X); ω(f)^{∗}⊗1; ι(X,X); (ev ⊗1)(1⊗φ(X)); ω(Y)^{∗} ⊗ ω(Y); 1⊗ω(f); ι(Y,Y); coend(ω); M; φ̃; (ev ⊗1)(1⊗φ(Y))]

Hence the diagram

[figure: ω(X); ω(X) ⊗ coend(ω); δ(X); ω(X) ⊗ M; ω(X) ⊗ M; 1⊗1; φ(X); 1⊗φ̃; ω(X) ⊗ ω(X)^{∗} ⊗ ω(X); ω(X) ⊗ ω(X)^{∗} ⊗ ω(X) ⊗ M; 1⊗1⊗φ(X); db ⊗1; 1⊗ev ⊗1; 1⊗ι(X,X); db ⊗1⊗1]

commutes. The exterior portion of this diagram yields

[figure: ω(X); ω(X) ⊗ coend(ω); δ(X); ω(X) ⊗ M.; 1⊗φ̃; φ(X)]

It remains to show that φ̃ : coend(ω) ⟶ M is uniquely determined. Let φ̃_{0} : coend(ω) ⟶ M be another morphism with φ(X) = (1 ⊗ φ̃_{0})δ(X) for all X ∈ D. Then the following diagram commutes

[figure: ω(X)^{∗} ⊗ ω(X); coend(ω); ι(X,X); ω(X)^{∗} ⊗ ω(X) ⊗ coend(ω); 1⊗δ(X); 1⊗φ(X); ev ⊗1; 1⊗1⊗φ̃_{0}; φ̃_{0}; ω(X)^{∗} ⊗ ω(X) ⊗ M; M,; ev ⊗1]

hence we have φ̃_{0} = φ̃. □

Corollary 3.4.4. The functor Nat(ω, ω ⊗ M) is a representable functor in M represented by coend(ω).

Proof. The universal problem implies the isomorphism

Nat(ω, ω ⊗ M) ≅ Mor_{C}(coend(ω), M)

and the universal natural transformation δ : ω ⟶ ω ⊗ coend(ω) is mapped to the identity under this isomorphism. □

It is also possible to construct an isomorphism

Nat(ω, ω′ ⊗ M) ≅ Mor_{C}(cohom(ω′, ω), M)

for different functors ω, ω′ : D ⟶ C and thus define cohomomorphism objects. Observe that only ω′ has to take values in C_{0} since then we can build objects ω′(X)^{∗} ⊗ ω(X).

5. The coalgebra coend

Proposition 3.5.1. Let C be a monoidal category and ω : D ⟶ C be a diagram in C. Assume that there is a universal object coend(ω) and natural transformation δ : ω ⟶ ω ⊗ coend(ω).

Then there is exactly one coalgebra structure on coend(ω) such that the diagrams

[figure: ω ⊗ coend(ω); ω ⊗ coend(ω) ⊗ coend(ω); δ⊗1; ω; ω ⊗ coend(ω); δ; δ; 1⊗Δ]

and

[figure: ω; ω ⊗ coend(ω); δ; id_{ω}; ω ⊗ I; 1⊗ϵ]

commute.

Proof. Because of the universal property of coend(ω) there are structure morphisms Δ : coend(ω) ⟶ coend(ω) ⊗ coend(ω) and ϵ : coend(ω) ⟶ I. This implies the coalgebra property similar to the proof of Corollary 3.3.8. □

Observe that by this construction all objects and all morphisms of the diagram ω : D ⟶ C_{0} ⊆ C are comodules or morphisms of comodules over the coalgebra coend(ω). In fact C := coend(ω) is the universal coalgebra over which the given diagram becomes a diagram of comodules.

Corollary 3.5.2. Let (D, ω) be a diagram C with objects in C_{0}. Then all objects of the diagram are comodules over the coalgebra C := coend(ω) and all morphisms are morphisms of comodules. If D is another coalgebra and all objects of the diagram are D-comodules by φ(X) : ω(X) ⟶ ω(X) ⊗ D and all morphisms of the diagram are morphisms of D-comodules then there exists a unique morphism of coalgebras φ̃: coend(ω) ⟶ D such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; φ; ω ⊗ D; 1⊗φ̃]

commutes.

Proof. The morphisms φ(X) : ω(X) ⟶ ω(X) ⊗ D define a natural transformation since all morphisms of the diagram are morphisms of comodules. So the existence and the uniqueness of a morphism φ̃: coend(ω) ⟶ D is clear. The only thing to show is that this is a morphism of coalgebras. This follows from the universal property of C = coend(ω) and the diagram

[figure: ω; ω ⊗ C; δ; δ; 1⊗Δ; 1; 1⊗φ̃; ω ⊗ C; ω ⊗ C ⊗ C; δ⊗1; 1⊗φ̃; 1⊗φ̃⊗φ̃; ω; ω ⊗ D; φ; φ; 1⊗Δ; ω ⊗ D; ω ⊗ D ⊗ D; φ⊗1]

where the right side of the cube commutes by the universal property. Similarly we get that φ̃ preserves the counit since the following diagram commutes

[figure: ω; ω ⊗ C; δ; ω; ω ⊗ D; φ; 1; 1⊗φ̃; 1; 1⊗ε; 1⊗ε; ω ⊗ 𝕂; 1; ω ⊗ 𝕂; 1]
□

6. The bialgebra coend

Let ω : D ⟶ C and ω′ : D′ ⟶ C be diagrams in C. We call the diagram (D, ω) ⊗ (D′, ω′) := (D × D′, ω ⊗ ω′) with (ω ⊗ ω′)(X, Y) := ω(X) ⊗ ω′(Y) the tensor product of these two diagrams. The new diagram consists of all possible tensor products of all objects and all morphisms of the original diagrams.

From now on we assume that the category C is the category of vector spaces and we use the symmetry τ : V ⊗ W ⟶ W ⊗ V in Vec .

Proposition 3.6.1. Let (D, ω) and (D′, ω′) be finite diagrams in Vec . Then

coend(ω ⊗ ω′) ≅ coend(ω) ⊗ coend(ω′).

Proof. First observe the following. If two diagrams ω : D ⟶ Vec and ω′ : D′ ⟶ Vec are given then \underrightarrow{lim}_{D} \underrightarrow{lim}_{D′}(ω ⊗ ω′) ≅ \underrightarrow{lim}_{D×D′}(ω ⊗ ω′) ≅ \underrightarrow{lim}_{D}(ω) ⊗ \underrightarrow{lim}_{D′}(ω′) since the tensor product preserves colimits and colimits commute with colimits. For this consider the diagram

[figure: ω(X) ⊗ ω′(Y); ω(X) ⊗ \underrightarrow{lim}_{D′}(ω′); \underrightarrow{lim}_{D}(ω) ⊗ ω′(Y); \underrightarrow{lim}_{D}(ω) ⊗ \underrightarrow{lim}_{D′}(ω′); ≅ \underrightarrow{lim}_{D×D′}(ω ⊗ ω′).]

The maps in the diagram are the injections for the corresponding colimits. In particular we have coend(ω⊗ω′) ≅ \underrightarrow{lim}_{D×D′}((ω⊗ω′)^{∗}⊗(ω⊗ω′)) ≅ \underrightarrow{lim}_{D×D′}(ω^{∗}⊗ω⊗ω′^{∗}⊗ω′) ≅ \underrightarrow{lim}_{D}(ω^{∗} ⊗ ω) ⊗ \underrightarrow{lim}_{D′}(ω′^{∗} ⊗ ω′) ≅ coend(ω) ⊗ coend(ω′).

The (universal) morphism

(ι(X)⊗ι′(Y))(1⊗τ ⊗1) : ω(X)^{∗}⊗ω′(Y)^{∗}⊗ω(X)⊗ω′(Y) ⟶ \underrightarrow{lim}(ω^{∗}⊗ω)⊗\underrightarrow{lim}(ω′^{∗}⊗ω′)

can be identified with the universal morphism

ι(X, Y) : ω(X)^{∗} ⊗ ω′(Y)^{∗} ⊗ ω(X) ⊗ ω′(Y) ⟶ \underrightarrow{lim}((ω ⊗ ω′)^{∗} ⊗ (ω ⊗ ω′)).

Hence the induced morphisms

(1 ⊗ τ ⊗ 1)(δ ⊗ δ′) : ω(X) ⊗ ω′(Y) ⟶ ω(X) ⊗ ω′(Y) ⊗ coend(ω) ⊗ coend(ω′)

and

δ : ω(X) ⊗ ω′(Y) ⟶ ω(X) ⊗ ω′(Y) ⊗ coend(ω ⊗ ω′)

can be identified. □

Corollary 3.6.2. For all finite diagrams (D, ω) and (D′, ω′) in D there is a universal natural transformation δ : ω ⊗ ω′ ⟶ ω ⊗ ω′ ⊗ coend(ω) ⊗ coend(ω′) so that for each object M and each natural transformation φ : ω ⊗ ω′ ⟶ ω ⊗ ω′ ⊗ M there exists a unique morphism φ̃: coend(ω) ⊗ coend(ω′) ⟶ M such that

[figure: ω ⊗ ω′; ω ⊗ ω′ ⊗ coend(ω) ⊗ coend(ω′); δ; ω ⊗ ω′ ⊗ M; 1⊗1⊗φ̃; φ]

commutes.

Definition 3.6.3. Let (D, ω) be a diagram in C = Vec . Then ω is called reconstructive

• if there is an object coend(ω) in C and a universal natural transformation δ : ω ⟶ ω ⊗ coend(ω)

• and if (1⊗τ ⊗1)(δ ⊗δ) : ω ⊗ω ⟶ ω ⊗ω ⊗coend(ω)⊗coend(ω) is a univesarl natural transformation of bifunctors.

Definition 3.6.4. Let (D, ω) be a diagram in Vec . Let D be a monoidal category and ω be a monoidal functor. Then (D, ω) is called a monoidal diagram.

Let (D, ω) be a monoidal diagram Vec . Let A ∈ Vec be an algebra. A natural transformation φ : ω ⟶ ω ⊗ B is called monoidal monoidal if the diagrams

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ B ⊗ B; φ(X)⊗φ(Y); ρ; ρ⊗m; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ B; φ(X⊗Y)]

and

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; ω(I); ω(I) ⊗ B; φ(I)]

commute.

We denote the set of monoidal natural transformations by Nat^{⊗}(ω, ω ⊗ B).

Problem 3.6.28. Show that Nat^{⊗}(ω, ω ⊗ B) is a functor in B.

Theorem 3.6.5. Let (D, ω) be a reconstructive, monoidal diagram in Vec . Then coend(ω) is a bialgebra and δ : ω ⟶ ω ⊗ coend(ω) is a monoidal natural transformation.

If B is a bialgebra and ∂ : ω ⟶ ω ⊗ B is a monoidal natural transformation, then there is a unique homomorphism of bialgebras f : coend(ω) ⟶ B such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; ∂; ω ⊗ B; 1⊗f]

commutes.

Proof. The multiplication of coend(ω) arises from the following diagram

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ coend(ω) ⊗ coend(ω); δ⊗δ; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ coend(ω); δ; ≅ ω(X) ⊗ ω(Y) ⊗ coend(ω)]

For the construction of the unit we consider the diagram D_{0} = ({I}, {id}) together with ω_{0} : D_{0} ⟶ Vec , ω_{0}(I) = 𝕂, the monoidal unit object in the monoidal category of diagrams in Vec . Then (𝕂 ⟶ 𝕂 ⊗ 𝕂) = (ω_{0} ⟶ ω_{0} ⊗ coend(ω_{0})) is the universal map. The following diagram then induced the unit for coend(ω)

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; ω(I); ω(I) ⊗ coend(ω); ≅ 𝕂 ⊗ coend(ω)]

By using the universal property one checks the laws for bialgebras.

The above diagrams show in particular that the natural transformation δ : ω ⟶ ω ⊗ coend(ω) is monoidal. □

7. The quantum monoid of a quantum space

Problem 3.7.29. If A is a finite dimensional algebra and δ : A ⟶ M(A) ⊗ A the universal cooperation of the Tambara bialgebra on A from the left then τδ : A ⟶ A ⊗ M(A) (with the same multiplication on M(A)) is a universal cooperation of M(A) on A from the right. The comultiplication defined by this cooperation is τΔ : M(A) ⟶ M(A) ⊗ M(A). Thus we have to distinguish between the left and the right Tambara bialgebra on A and we have M_{r}(A) = M_{l}(A)^{cop}.

Now consider the special monoidal diagram scheme D := D[X; m, u]. To make things simpler we assume that Vec is strict monoidal. The category D[X; m, u] has the objects X ⊗ … ⊗ X = X^{⊗n} for all n ∈ ℕ (and I := X^{⊗0}) and the morphisms m : X ⊗ X ⟶ X, u : I ⟶ X and all morphisms formally constructed from m, u, id by taking tensor products and composition of morphisms.

Let A be an algebra with multiplication m_{A} : A ⊗ A ⟶ A and unit u_{A} : 𝕂 ⟶ A. Then ω_{A} : D ⟶ C defined by ω(X) = A, ω(X^{⊗n}) = A^{⊗n}, ω(m) = m_{A} and ω(u) = u_{A} is a strict monoidal functor. If A is finite dimensional then the diagram is finite. We get

Theorem 3.7.1. Let A be a finite dimensional algebra. Then the algebra M(A) coacting universally from the right on A (the right Tambara bialgebra) M(A) and coend(ω_{A}) are isomorphic as bialgebras.

Proof. We have studied the Tambara bialgebra for left coaction f : A ⟶ M(A)⊗ A but here we need the analogue for universal right coaction f : A ⟶ A ⊗ M(A) (see Problem (44).

Let B be an algebra and f : A ⟶ A ⊗ B be a homomorphism of algebras. For ω = ω_{A} we define

φ(X^{⊗n}) : ω(X^{⊗n}) = A^{⊗n} ⟶^{f^{⊗n}} A^{⊗n} ⊗ B^{⊗n} ⟶^{1⊗m_{B}^{n}} A^{⊗n} ⊗ B = ω(X^{⊗n}) ⊗ B,

where m_{B}^{n} : B^{⊗n} ⟶ B is the n-fold multiplication on B. The map φ is a natural transformation since the diagrams

[figure: A; A ⊗ B; φ(X); 𝕂; 𝕂 ⊗ B; φ(I); u; 1⊗u]

and

commute. Furthermore the following commute so that φ : ω_{A} ⟶ ω_{A} ⊗ B is a monoidal natural transformation.

[figure: A ⊗ A; A ⊗ A ⊗ B; φ(X⊗X); A ⊗ A ⊗ B ⊗ B; f⊗f; 1⊗m; m; m⊗m; m⊗1; A ⊗ B; f; 1⊗1; A; A ⊗ B; φ(X)] [figure: A^{⊗r} ⊗ A^{⊗s}; A^{⊗r} ⊗ A^{⊗s} ⊗ B ⊗ B; φ(X^{⊗r})⊗φ(X^{⊗s}); A^{⊗r} ⊗ A^{⊗s} ⊗ B^{⊗r} ⊗ B^{⊗s}; A^{⊗(r+s)} ⊗ B^{⊗(r+s)}; A^{⊗(r+s)}; A^{⊗(r+s)} ⊗ B; φ(X^{⊗(r+s)})]

Conversely let φ : ω_{A} ⟶ ω_{A} ⊗ B be a natural transformation. Let f := φ(X) : A ⟶ A ⊗ B. Then the following commute

[figure: A ⊗ A; A ⊗ A ⊗ B ⊗ B; f⊗f; =; 1⊗m; A ⊗ A; A ⊗ A ⊗ B; φ(X⊗X); m; m⊗1; A; A ⊗ B; f]

and

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; =; 𝕂; 𝕂 ⊗ B; u; u⊗1; A; A ⊗ B.; f]

Hence f : A ⟶ A ⊗ B is a homomorphism of algebras.

Thus we have defined an isomorphism

𝕂Alg (A, A ⊗ B) ≅ Nat^{⊗}(ω_{A}, ω_{A} ⊗ B)

that is natural in B. If A is finite dimensional then the left hand side is represented by the Tambara bialgebra M_{r}(A) and the right hand side by the bialgebra coend(ω_{A}). Thus both bialgebras must be isomorphic. □

Corollary 3.7.2. There is a unique isomorphism of bialgebras M_{r}(A) ≅ coend(ω_{A}) such that the diagram

[figure: A; A ⊗ M_{r}(A); A ⊗ coend(ω_{A})]

commutes

Proof. This is a direct consequence of the universal property. □

Thus the Tambara bialgebra that represents the universal quantum monoid acting on a finite quantum space may be reconstructed by the Tannaka-Krein reconstruction from representation theory. Similar reconstructions can be given for more complicated quantum spaces such as so called quadratic quantum spaces.

8. RECONSTRUCTION AND C-CATEGORIES 103

8. Reconstruction and C-categories

Now we show that an arbitrary coalgebra C can be reconstructed by the methods introduced above from its (co-)representations or more precisely from the underlying functor ω : Comod -C ⟶ Vec . In this case one can not use the usual construction of coend(ω) that is restricted to finite dimensional comodules.

The following Theorem is an example that shows that the restriction to finite dimensional comodules in general is too strong for Tannaka reconstruction. There may be universal coendomorphism bialgebras for more general diagrams. On the other hand the following Theorem also holds if one only considers finite dimensional corepresentations of C. However the proof then becomes somewhat more complicated.

Definition 3.8.1. Let C be a monoidal category. A category D together with a bifunctor ⊗ : C ×D ⟶ D and natural isomorphisms β : (A⊗B)⊗M ⟶ A⊗(B ⊗M), η : I ⊗ M ⟶ M is called a C-category if the following diagrams commute

[figure: ((A ⊗ B) ⊗ C) ⊗ M; (A ⊗ (B ⊗ C)) ⊗ M; α(A,B,C)⊗1; A ⊗ ((B ⊗ C) ⊗ M); β(A,B⊗C,M); β(A⊗B,C,M); 1⊗β(B,C,M); (A ⊗ B) ⊗ (C ⊗ M); A ⊗ (B ⊗ (C ⊗ M)); β(A,B,C⊗M)]

[figure: (A ⊗ I) ⊗ M; A ⊗ (I ⊗ M); β(A,I,M); A ⊗ M; ρ(A)⊗1; 1⊗η(M)]

A C-category is called strict if the morphisms β, η are the identities.

Let (D, ⊗) and (D′, ⊗) be C-categories. A functor F : D ⟶ D′ together with a natural transformation ζ(A, M) : A ⊗ F(M) ⟶ F(A ⊗ M) is called a weak C-functor if the following diagrams commute

[figure: (A ⊗ B) ⊗ F(M); F((A ⊗ B) ⊗ M); ζ; β; F(β); A ⊗ (B ⊗ F(M)); A ⊗ F(B ⊗ M); 1⊗ζ; F(A ⊗ (B ⊗ M)); ζ]

[figure: I ⊗ F(M); F(I ⊗ M); ζ; F(M); η; F(η)]

If, in addition, ζ is an isomorphism then we call F a C-functor. The functor is called a strict C-functor if ζ is the identity morphism.

A natural transformation φ : F ⟶ F′ between (weak) C-functors is called a C-transformation if

[figure: A ⊗ F′(M); F′(A ⊗ M); ζ′; A ⊗ F(M); F(A ⊗ M); ζ; 1_{A}⊗φ(M); φ(A⊗M)]

commutes.

Example 3.8.2. Let C be a coalgebra and C := Vec . Then the category Comod -C of right C-comodules is a C-category since N ∈ Comod -C and V ∈ C = Vec implies that V ⊗ N is a comodules with the comodule structure of N.

The underlying functor ω : Comod -C ⟶ Vec is a strict C-functor since we have V ⊗ ω(N) = ω(V ⊗ N). Similarly ω ⊗ M : Comod -C ⟶ Vec is a C-functor since V ⊗ (ω(N) ⊗ M) ≅ ω(V ⊗ N) ⊗ M.

Lemma 3.8.3. Let C be a coalgebra. Let ω : Comod -C ⟶ Vec be the underlying functor. Let φ : ω ⟶ ω ⊗ M be a natural transformation. Then φ is a C-transformation with C = Vec .

Proof. It suffices to show 1_{V} ⊗ φ(N) = φ(V ⊗ N) for an arbitrary comodule N. We show that the diagram

[figure: V ⊗ N; V ⊗ N ⊗ M; 1_{V} ⊗φ(N); V ⊗ N; V ⊗ N ⊗ M; φ(V ⊗N); 1; 1]

commutes. Let (v_{i}) be a basis of V. For an arbitrary vector space W let p_{i} : V ⊗ W ⟶ W be the projections defined by p_{i}(t) = p_{i}(∑_{j} v_{j} ⊗ w_{j}) = w_{i} where ∑_{j} v_{j} ⊗ w_{j} is the unique representation of an arbitrary tensor in V ⊗ W. So we get

t = ∑_{i} v_{i} ⊗ p_{i}(t)

for all t ∈ V ⊗W. Now we consider V ⊗N as a comodule by the comodule structure of N. Then the p_{i} : V ⊗ N ⟶ N are homomorphisms of comodules. Hence all diagrams of the form commute. Expressed in formulas this means φ(N)p_{i}(t) = p_{i}φ(V ⊗ N)(t) for all t ∈ V ⊗ N. Hence we have

[figure: N; N ⊗ M.; φ(N); V ⊗ N; V ⊗ N ⊗ M; φ(V ⊗N); p_{i}; p_{i}⊗M]

\begin{matrix}(1_{V} ⊗ φ(N))(t) = (1_{V} ⊗ φ(N))(∑ v_{i} ⊗ p_{i}(t)) = ∑ v_{i} ⊗ φ(N)p_{i}(t) \\ = ∑_{i} v_{i} ⊗ p_{i}φ(V ⊗ N)(t) = φ(V ⊗ N)(t) \end{matrix}

So we have 1_{V} ⊗ φ(N) = φ(V ⊗ N) as claimed. □

We prove the following Theorem only for the category C = Vec of vector spaces. The Theorem holds in general and says that in an arbitrary symmetric monoidal category C the coalgebra C represents the functor C- Nat(ω, ω ⊗ M) ≅ Mor_{C}(C, M) of natural C-transformations.

Theorem 3.8.4. (Reconstruction of coalgebras) Let C be a coalgebra. Let ω : Comod -C ⟶ Vec be the underlying functor. Then C ≅ coend(ω).

Proof. Let M in Vec and let φ : ω ⟶ ω ⊗ M be a natural transformation. We define the homomorphism φ̃: C ⟶ M by φ̃ = (ϵ ⊗ 1)φ(C) using the fact that C is a comodule.

Let N be a C-comodule. Then N is a subcomodule of N ⊗ C by δ : N ⟶ N ⊗ C since the diagram

[figure: N ⊗ C; N ⊗ C ⊗ C; δ⊗1; N; N ⊗ C; δ; δ; 1⊗Δ]

commutes. Thus the following diagram commutes

In particular we have shown that the diagram

commutes.

[figure: N; N ⊗ C; δ; N ⊗ M; N ⊗ C ⊗ M; δ⊗1; φ(N); φ(N⊗C)=1_{N} ⊗φ(C); N ⊗ M; 1; 1⊗ϵ⊗1; 1⊗φ̃]

[figure: ω; ω ⊗ C; δ; φ; ω ⊗ M; 1⊗φ̃]

To show the uniqueness of φ̃ let g : C ⟶ M be another homomorphism with (1 ⊗ g)δ = φ. For c ∈ C we have g(c) = g(ϵ ⊗ 1)Δ(c) = (ϵ ⊗ 1)(1 ⊗ g)Δ(c) = (ϵ ⊗ 1)φ(C)(c) = φ̃(c).

The coalgebra structure from Corollary 3.5.1 is the original coalgebra structure of C. This can be seen as follows. The comultiplication δ : ω ⟶ ω ⊗ C is a natural transformation hence (δ ⊗ 1_{C})δ : ω ⟶ ω ⊗ C ⊗ C is also a natural transformation. As in Corollary 3.5.1 this induced a unique homomorphism Δ : C ⟶ C ⊗ C so that the diagram

[figure: ω ⊗ coend(ω); ω ⊗ coend(ω) ⊗ coend(ω); δ⊗1; ω; ω ⊗ coend(ω); δ; δ; 1⊗Δ]

commutes. In a similar way the natural isomorphism ω ≅ ω ⊗ 𝕂 induces a unique homomorphism ϵ : C ⟶ 𝕂 so that the diagram

[figure: ω; ω ⊗ coend(ω); δ; id_{ω}; ω ⊗ I; 1⊗ϵ]

commutes. Because of the uniqueness these must be the structure homomorphisms of C. □

We need a more general version of this Theorem in the next chapter. So let C be a coalgebra. Let ω : Comod -C ⟶ Vec be the underlying functor and δ : ω ⟶ ω ⊗ C the universal natural transformation for C ≅ coend(ω).

We use the permutation map τ on the tensor product that gives the natural isomorphism

τ : N_{1} ⊗ T_{1} ⊗ N_{2} ⊗ T_{2} ⊗ … ⊗ N_{n} ⊗ T_{n} ≅ N_{1} ⊗ N_{2} ⊗ … ⊗ N_{n} ⊗ T_{1} ⊗ T_{2} … ⊗ ⊗T_{n}

which is uniquely determined by the coherence theorems and is constructed by suitable applications of the flip τ : N ⊗ T ≅ T ⊗ N.

Let ω^{n} : Comod -C×Comod -C×…×Comod -C ⟶ Vec be the functor ω^{n}(N_{1}, N_{2}, …, N_{n}) = ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}). For notational convenience we abbreviate {N}^{n} := N_{1} ⊗ N_{2} ⊗ … ⊗ N_{n}, similarly {C}^{n} = C ⊗ C ⊗ … ⊗ C and {f}^{n} := f_{1} ⊗ f_{2} ⊗ … ⊗ f_{n}. So we get τ : {N ⊗ T }^{n} ≅ {N}^{n} ⊗ {T }^{n}.

Lemma 3.8.5. Let φ : ω^{n} ⟶ ω^{n} ⊗ M be a natural transformation. Then φ is a C-transformation in the sense that the diagrams

[figure: {V }^{n} ⊗ {N}^{n}; {V }^{n} ⊗ {N}^{n} ⊗ M; {V }^{n}⊗φ(N_{1},...,N_{n}); {V ⊗ N}^{n}; {V ⊗ N}^{n} ⊗ M; φ(V_{1}⊗N_{1},...,V_{n}⊗N_{n}); τ; τ⊗M]

commute for all vector spaces V_{i} and C-comodules N_{i}.

Proof. Choose bases {v_{ij}} of the vector spaces V_{i} with corresponding projections p_{ij} : V_{i} ⊗ N_{i} ⟶ N_{i}. Then we have τ(t_{1} ⊗ … ⊗ t_{n}) = ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ p_{1i_{1}}(t_{1}) ⊗ … ⊗ p_{ni_{n}}(t_{n}) so τ = ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ {p}^{n}.

The p_{ij_{i}} : V_{i} ⊗N_{i} ⟶ N_{i} are homomorphisms of C-comodules. Hence the diagrams

[figure: {N}^{n}; {N}^{n} ⊗ M; φ(N_{1},...,N_{n}); {V ⊗ N}^{n}; {V ⊗ N}^{n} ⊗ M; φ(V_{1}⊗N_{1},...,V_{n}⊗N_{n}); {p}^{n}; {p}^{n}⊗M]

commute for all choices of {p}^{n} = p_{1i_{1}} ⊗ … ⊗ p_{ni_{n}}.

So we get for all t_{i} ∈ V_{i} ⊗ N_{i}

({V }^{n} ⊗ φ(N_{1}, …, N_{n}))τ(t_{1} ⊗ … ⊗ t_{n}) =

= ({V }^{n} ⊗ φ(N_{1}, …, N_{n}))(∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ p_{1i_{1}}(t_{1}) ⊗ … ⊗ p_{ni_{n}}(t_{n}))

= ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ φ(N_{1}, …, N_{n}){p}^{n}(t_{1} ⊗ … ⊗ t_{n})

= ∑ v_{1i_{1}} ⊗ … ⊗ v_{ni_{n}} ⊗ ({p}^{n} ⊗ M)φ(V_{1} ⊗ N_{1}, …, V_{n} ⊗ N_{n})(t_{1} ⊗ … ⊗ t_{n})

= (τ ⊗ M)φ(V_{1} ⊗ N_{1}, …, V_{n} ⊗ N_{n})(t_{1} ⊗ … ⊗ t_{n}).

□

Theorem 3.8.6. With the notation given above we have

coend(ω^{n}) ≅ C ⊗ C ⊗ … ⊗ C

with the universal natural transformation

δ^{(n)}(N_{1}, N_{2}, …, N_{n}) := τ(δ(N_{1}) ⊗ δ(N_{2}) ⊗ … ⊗ δ(N_{n})) :

ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}) ⟶ ω(N_{1}) ⊗ C ⊗ ω(N_{2}) ⊗ C ⊗ … ⊗ ω(N_{n}) ⊗ C ≅ ω(N_{1}) ⊗ ω(N_{2}) ⊗ … ⊗ ω(N_{n}) ⊗ C ⊗ C ⊗ … ⊗ C.

Proof. We proceed as in the proof of the previous Theorem.

Let M in Vec and let φ : ω^{n} ⟶ ω^{n} ⊗ M be a natural transformation. We define the homomorphism φ̃: C^{n} = ω(C) ⊗ ω(C) ⊗ … ⊗ ω(C) = C ⊗ C ⊗ … ⊗ C ⟶ M by φ̃ = (ε^{n} ⊗ 1_{M})φ(C, …, C) using the fact that C is a comodule.

As in the preceding proof we get that δ : N_{i} ⟶ N_{i} ⊗ C are homomorphisms of C-comodules. Thus the following diagram commutes

[figure: N_{1} ⊗ … ⊗ N_{n}; N_{1} ⊗ C ⊗ … ⊗ N_{n} ⊗ C; δ⊗...⊗δ; N_{1} ⊗ … ⊗ N_{n}⊗; τ; C ⊗ … ⊗ C; N_{1} ⊗ … ⊗ N_{n} ⊗ M; N_{1} ⊗ C ⊗ … ⊗ N_{n} ⊗ C ⊗ M; δ⊗...⊗δ⊗M; N_{1} ⊗ … ⊗ N_{n}⊗; τ⊗M; C ⊗ … ⊗ C ⊗ M; φ(N_{1}⊗...⊗N_{n}); φ(N_{1}⊗C,...,N_{n}⊗C); N_{1}⊗...⊗N_{n}⊗φ(C,...,C); 1; {1⊗ε}^{n}⊗1; N_{1} ⊗ … ⊗ N_{n} ⊗ M; 1⊗{ϵ}^{n}⊗1]

Hence we get the commutative diagram

[figure: ω^{n}; ω^{n} ⊗ {C}^{n}; δ^{(n)}; φ; ω^{n} ⊗ M; 1⊗φ̃]

To show the uniqueness of φ̃ let g : C^{n} ⟶ M be another homomorphism with (1_{ω^{n}} ⊗ g)δ^{(n)} = φ. We have g = g(ε^{n} ⊗ 1_{C^{n}})τΔ^{n} = g(ε^{n} ⊗ 1_{C^{n}})δ^{(n)}(C, …, C) = (ε^{n} ⊗ 1_{M})(1_{C^{n}} ⊗ g)δ^{(n)}(C, …, C) = (ε^{n} ⊗ 1_{M})φ(C, …, C) = φ̃. □

Now we prove the finite dimensional case of reconstruction of coalgebras.

Proposition 3.8.7. (Reconstruction) Let C be a coalgebra. Let Comod_{0}-C be the category of finite dimensional C-comodules and ω : Comod_{0}-C ⟶ Vec be the underlying functor. Then we have C ≅ coend(ω).

Proof. Let M be in Vec and let φ : ω ⟶ ω ⊗ M be a natural transformation. We define the homomorphism φ̃ : C ⟶ M as follows. Let c ∈ C. Let N be a finite dimensional C-subcomodule of C containing c. Then we define g(c) := (ϵ|_{N} ⊗ 1)φ(N)(c). If N′ is another finite dimensional subcomodule of C with c ∈ N′ and with N ⊆ N′ then the following commutes

[figure: N; N ⊗ M; φ(N); N′; N′ ⊗ M; φ(N′); C ⊗ M; M; ϵ⊗1]

Thus the definition of φ̃(c) is independent of the choice of N. Furthermore φ̃ : N ⟶ M is obviously a linear map. For any two elements c, c′ ∈ C there is a finite dimensional subcomodule N ⊆ C with c, c′ ∈ N e. g. the sum of the finite dimensional subcomodules containing c and c′ separately. Thus φ̃ can be extended to all of C.

The rest of the proof is essentially the same as the proof of the first reconstruction theorem. □

The representations allow to reconstruct further structure of the coalgebra. We prove a reconstruction theorem about bialgebras. Recall that the category of Bcomodules over a bialgebra B is a monoidal category, furthermore that the underlying functor ω : Comod -B ⟶ Vec is a monoidal functor. From this information we can reconstruct the full bialgebra structure of B. We have

Theorem 3.8.8. Let B be a coalgebra. Let Comod -B be a monoidal category such that the underlying functor ω : Comod -B ⟶ Vec is a monoidal functor. Then there is a unique bialgebra structure on B that induces the given monoidal structure on the corepresentations.

Proof. First we prove the uniqueness of the multiplication ∇ : B ⊗ B ⟶ B and of the unit η : 𝕂 ⟶ B. The natural transformation δ : ω ⟶ ω ⊗ B becomes a monoidal natural transformation with ∇ : B ⊗ B ⟶ B and η : 𝕂 ⟶ B We show that ∇ and η are uniquely determined by ω and δ.

Let ∇′ : B ⊗ B ⟶ B and η′ : B ⟶ 𝕂 be morphisms that make δ a monoidal natural transformation. The diagrams

and

commute. In particular the following diagrams commute and

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ B ⊗ B; δ(X)⊗δ(Y); ρ; ρ⊗∇′; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ B; δ(X⊗Y)]

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; 1 ⊗ η′; ω(𝕂); ω(𝕂) ⊗ B; δ(𝕂)] [figure: ω(B) ⊗ ω(B); ω(B) ⊗ ω(B) ⊗ B ⊗ B; δ(B)⊗δ(B); ρ; ρ⊗∇′; ω(B ⊗ B); ω(B ⊗ B) ⊗ B; δ(B⊗B)]

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; 1 ⊗ η′; ω(𝕂); ω(𝕂) ⊗ B; δ(𝕂)]

Hence we get ∑ b_{(1)}⊗c_{(1)}⊗b_{(2)}c_{(2)} = ∑ b_{(1)}⊗c_{(1)}⊗∇′(b_{(2)}⊗c_{(2)}) and 1⊗1 = 1⊗η′(1). This implies bc = ∑ ϵ(b_{(1)})ϵ(c_{(1)})b_{(2)}c_{(2)} = ∑ ϵ(b_{(1)})ϵ(c_{(1)})∇′(b_{(2)} ⊗ c_{(2)}) = ∇′(b ⊗ c) and 1 = η′(1).

Now we show the existence of a bialgebra structure. Let B be a coalgebra only and let ω : Comod -B ⟶ Vec be a monoidal functor with ξ : ω(M) ⊗ ω(N) ⟶ ω(M ⊗ N) and ξ_{0} : 𝕂 ⟶ ω(𝕂). First we observe that the new tensor product between the comodules M and N coincides with the tensor product of the underlying vector spaces (up to an isomorphism ξ). Because of the coherence theorems for monoidal categories (that also hold in our situation) we may identify along the maps ξ and ξ_{0}.

We define η := (𝕂 ⟶^{δ(𝕂)} 𝕂 ⊗ B ≅ B) and ∇ := (B ⊗ B ⟶^{δ(B⊗B)} B ⊗ B ⊗ B ⟶^{ϵ⊗ϵ⊗1_{B}}

𝕂 ⊗ 𝕂 ⊗ B ≅ B).

Since the structural morphism for the comodule δ : M ⟶ M ⊗ B is a homomorphism of of B comodules where the comodule structure on M ⊗ B is only given by the diagonal of B that is the C-structure on ω : Comod -B ⟶ Vec we get that also δ(M) ⊗ δ(N) : M ⊗ N ⟶ M ⊗ N ⊗ B is a comodule homomorphism. Hence the first square in the following diagram commutes

[figure: M ⊗ N; M ⊗ B ⊗ N ⊗ B; δ(M)⊗δ(N); M ⊗ N ⊗ B; M ⊗ B ⊗ N ⊗ B ⊗ B; δ(M)⊗δ(N)⊗1_{B}; M ⊗ N ⊗ B ⊗ B; 1⊗τ⊗1; M ⊗ N ⊗ B ⊗ B ⊗ B; 1⊗τ⊗1⊗1; δ(M⊗N); δ(M⊗B⊗N⊗B); 1⊗1⊗δ(B⊗B)]

The second square commutes by a similar reasoning since the comodule structure on M ⊗ B resp. N ⊗ B is given by the diagonal on B hence M ⊗ N can be factored out of the natural (C-)transformation. Now we attach

1_{M} ⊗ 1_{N} ⊗ ϵ ⊗ ϵ ⊗ 1_{B} : M ⊗ N ⊗ B ⊗ B ⊗ B ⟶ M ⊗ N ⊗ B

to the commutative rectangle and obtain δ(M ⊗N) = (1_{M} ⊗1_{N} ⊗∇)(1⊗τ ⊗1)(δ(M)⊗ δ(N). Thus the comodule structure on M ⊗ N is induced by the multiplication ∇ : B ⊗ B ⟶ B defined above.

8. RECONSTRUCTION AND C-CATEGORIES 111

So the following diagrams commute

[figure: B ⊗ B; B ⊗ B ⊗ B ⊗ B; Δ⊗Δ; B ⊗ B ⊗ B ⊗ B; 1⊗τ⊗1; B ⊗ B; B ⊗ B ⊗ B; δ(B⊗B); B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B; 1⊗1⊗Δ; B; B ⊗ B; Δ; 1_{B}⊗1_{B}; 1⊗∇; δ(B⊗B); δ(B⊗B)⊗1; ∇; ∇⊗1; ϵ⊗ϵ⊗1; ϵ⊗ϵ⊗1⊗1]

[figure: B ⊗ B; B; ∇; B ⊗ B; 𝕂; ϵ⊗ϵ; B ⊗ B ⊗ B; δ(B⊗B); ϵ⊗ϵ⊗1; 1; 1⊗1⊗ϵ; ϵ]

[figure: 𝕂; B; η; 𝕂 ⊗ B; 𝕂 ⊗ B ⊗ B; δ(𝕂)⊗1; B ⊗ B; ≅; B; B ⊗ B; η⊗1; 𝕂 ⊗ B; δ(𝕂); δ(𝕂); ≅; ≅; η⊗1; 1_{B}; η; 1⊗Δ; ≅; Δ]

and

[figure: 𝕂; 𝕂; 1; 𝕂 ⊗ B; δ(𝕂); 1⊗ϵ; η; ϵ; B; ≅]

Hence η and ∇ are coalgebra homomorphisms.

To show the associativity of ∇ we identify along the maps α : (M ⊗ N) ⊗ P ≅ M ⊗(N ⊗P) and furthermore simplify the relevant diagram by fixing that σ represents a suitable permutation of the tensor factors. Then the following commute

[figure: B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B ⊗ B ⊗ B; σ(δ(B)⊗δ(B)⊗δ(B); B ⊗ B ⊗ B; ϵ⊗ϵ⊗ϵ⊗1; B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B ⊗ B ⊗ B; σ(δ(B⊗B)⊗δ(B)); σ(δ(B)⊗δ(B⊗B)); B ⊗ B; ϵ⊗ϵ⊗ϵ⊗1; B ⊗ B ⊗ B; B ⊗ B ⊗ B ⊗ B; δ(B⊗B⊗B); B; ϵ⊗ϵ⊗ϵ⊗1; 1; 1; 1⊗(∇⊗1); 1⊗(1⊗∇); ∇⊗1; 1⊗∇; 1⊗∇; ∇]

The upper row is the identity hence we get the associative law.

For the proof that η has the properties of a unit we must explicitly consider the coherence morphisms λ and ρ By reasons of symmetry we will only show one half of of the unit axiom. This axiom follows from the commutativity of the following diagram

□

[figure: B; B ⊗ B; δ(B); B ⊗ B ⊗ 𝕂; ρ^{−1}; B ⊗ B; ρ; B; ϵ⊗1; B ⊗ 𝕂; B ⊗ B ⊗ 𝕂 ⊗ B; δ(B)⊗δ(𝕂); B ⊗ 𝕂 ⊗ B ⊗ B; 1⊗τ⊗1; B ⊗ B ⊗ B; ρ⊗1⊗1; B ⊗ B; ϵ⊗1⊗1; B ⊗ 𝕂; B ⊗ 𝕂 ⊗ B; δ(B⊗𝕂); B; B ⊗ B; δ(B); B; ϵ⊗1; ρ^{−1}; 1⊗1⊗δ(𝕂); 1⊗1⊗η; 1⊗η; =; 1⊗1⊗∇; 1⊗∇; ∇; ρ; δ(B)⊗1; ρ⊗1]

CHAPTER 4 The Infinitesimal Theory

1. Integrals and Fourier Transforms

Assume for this chapter that 𝕂 is a field.

Lemma 4.1.1. Let C be a finite dimensional coalgebra. Every right C-comodule M is a left C^{∗}-module by c^{∗}m = ∑ m_{(M)}⟨c^{∗}, m_{(1)}⟩ and conversely by δ(m) = ∑_{i} c_{i}^{∗}m⊗ c_{i} where ∑ c_{i}^{∗} ⊗ c_{i} is the dual basis.

Proof. We check that M becomes a left C^{∗}-module

\begin{aligned}(c^{∗}c′^{∗})m & = ∑ m_{(M)}⟨c^{∗}c′^{∗}, m_{(1)}⟩ = ∑ m_{(M)}⟨c^{∗}, m_{(1)}⟩⟨c′^{∗}, m_{(2)}⟩ \\ & = c^{∗} ∑ m_{(M)}⟨c′^{∗}, m_{(1)}⟩ = c^{∗}(c′^{∗}m). \end{aligned}

It is easy to check that the two constructions are inverses of each other. In particular assume that M is a right C-comodule. Choose m_{i} such that δ(m) = ∑ m_{i} ⊗ c_{i}. Then c_{j}^{∗}m = ∑ m_{i}⟨c_{j}^{∗}, c_{i}⟩ = m_{j} and ∑ c_{i}^{∗}m ⊗ c_{i} = ∑ m_{i} ⊗ c_{i} = δ(m). □

Definition 4.1.2. 1. Let A be an algebra with augmentation ε : A ⟶ 𝕂, an algebra homomorphism. Let M be a left A-module. Then ^{A}M = {m ∈ M|am = ε(a)m} is called the space of left invariants of M.

This defines a functor ^{A}- : A-Mod ⟶ Vec .

2. Let C be a coalgebra with a grouplike element 1 ∈ C. Let M be a right C-comodule. Then M^{coC} := {m ∈ M|δ(m) = m ⊗ 1} is called the space of right coinvariants of M.

This defines a functor -^{coC} : Comod -C ⟶ Vec .

Lemma 4.1.3. Let C be a finite dimensional coalgebra with a grouplike element 1 ∈ C. Then A := C^{∗} is an augmented algebra with augmentation ε : C^{∗} ∋ a ↦ ⟨a, 1⟩ ∈ 𝕂. Let M be a right C-comodule. Then M is a left C^{∗}-module and we have

C^{∗}M = M^{coC}.

Proof. Since 1 ∈ C is grouplike we have ε_{A}(ab) = ⟨ab, 1⟩ = ⟨a, 1⟩⟨b, 1⟩ = ε_{A}(a)ε_{A}(b) and ε_{A}(1_{A}) = ⟨1_{A}, 1_{C}⟩ = ε_{C}(1_{C}) = 1.

We have m ∈ M^{coH} iff δ(m) = ∑ m_{(M)} ⊗ m_{(1)} = m ⊗ 1 iff ∑ m_{(M)}⟨a, m_{(1)}⟩ = m⟨a, 1⟩ for all a ∈ A = C^{∗} and by identifying C^{∗}⊗C = Hom(C^{∗}, C^{∗}) iff am = ε_{A}(a)m iff m ∈ ^{A}M. □

Remark 4.1.4. The theory of Fourier transforms contains the following statements. Let H be the (Schwartz) space of infinitely differentiable functions f : ℝ ⟶ ℂ, such that f and all derivatives rapidly decrease at infinity. (f decreases rapidly at infinity if |x|^{m}f(x) is bounded for all m.) This space is an algebra (without unit) under the multiplication of values. There is a second multiplication on H, the convolution

(f ∗ g)(x) = (2π)^{−1/2} ∫_{−∞}^{+∞} f(t)g(x − t)dt.

The Fourier transform is a homomorphism ˆ- : H ⟶ H defined by

f̂(x) = (2π)^{−1/2} ∫_{−∞}^{+∞} f(t)e^{−itx}dt.

It satisfies the identity (f ∗ g)ˆ = f̂ĝ hence it is an algebra homomorphism. We want to find an analogue of this theory for finite quantum groups.

A similar example is the following. Let G be a locally compact topological group. Let μ be the (left) Haar measure on G and ∫ f := ∫_{G} f(x)dμ(x) be the Haar integral.

The Haar measure is left invariant in the sense that μ(E) = μ(gE) for all g ∈ G and all compact subsets E of G. The Haar measure exists and is unique up to a positive factor. The Haar integral is translation invariant i.e. for all y ∈ G we have ∫ f(yx)dμ(x) = ∫ f(x)dμ(x).

If μ is a left-invariant Haar measure then there is a continuous homomorphism mod : G ⟶ (ℝ^{+}, ·) such that ∫ f(xy^{−1})dμ(x) = mod (y) ∫ (f(x)dμ(x). The homomorphism μ does not depend on f and is called the modulus of G. The group G is called unimodular if the homomorphism mod is the identity.

If G is a compact, or discrete, or Abelian group, or a connected semisimple or nilpotent Lie group, then G is unimodular.

Let G be a quantum group (or a quantum monoid) with function algebra H an arbitrary Hopf algebra. We also use the algebra of linear functionals H^{∗} = Hom(H, 𝕂) (called the bialgebra of G in the French literature). The operation H^{∗} ⊗ H ∋ a ⊗ f ↦ ⟨a, f⟩ ∈ 𝕂 is nondegenerate on both sides. We denote the elements of H by f, g, h ∈ H, the elements of H^{∗} by a, b, c ∈ H^{∗}, the (non existing) elements of the quantum group G by x, y, z ∈ G.

Remark 4.1.5. In 2.4.8 we have seen that the dual vector space H^{∗} of a finite dimensional Hopf algebra H is again a Hopf algebra. The Hopf algebra structures are connected by the evaluation bilinear form

ev : H^{∗} ⊗ H ∋ a ⊗ f ↦ ⟨a, f⟩ ∈ 𝕂

as follows:

\begin{matrix}⟨a ⊗ b, ∑ f_{(1)} ⊗ f_{(2)}⟩ = ⟨ab, f⟩, & ⟨∑ a_{(1)} ⊗ a_{(2)}, f ⊗ g⟩ = ⟨a, fg⟩, \\ ⟨a, 1⟩ = ε(a), & ⟨1, f⟩ = ε(f), \end{matrix} ; ⟨a, S(f)⟩ = ⟨S(a), f⟩.

Definition 4.1.6. 1. The linear functionals a ∈ H^{∗} are called generalized integrals on H ([Riesz-Nagy] S.123).

2. An element ∫ ∈ H^{∗} is called a left (invariant) integral on H if

a∫ = ⟨a, 1_{H}⟩∫

or a ∫ = ε_{H^{∗}}(a) ∫ for all a ∈ H^{∗}.

3. An element δ ∈ H is called a left integral in H if

fδ = ε(f)δ

for all f ∈ H.

4. The set of left integrals in H is denoted by Int_{l}(H), the set of right integrals by Int_{r}(H). The set of left (right) integrals on H is Int_{l}(H^{∗}) (Int_{r}(H^{∗})).

5. A Hopf algebra H is called unimodular if Int_{l}(H) = Int_{r}(H).

Lemma 4.1.7. The left integrals Int_{l}(H^{∗}) form a two sided ideal of H^{∗}. If the antipode S is bijective then S induces an isomorphism S : Int_{l}(H^{∗}) ⟶ Int_{r}(H^{∗}).

Proof. For ∫ in Int_{l}(H^{∗}) we have a ∫ = ε(a) ∫ ∈ Int_{l}(H^{∗}) and a ∫ b = ε(a) ∫ b hence ∫ b ∈ Int_{l}(H^{∗}). If S is bijective then the induced map S : H^{∗} ⟶ H^{∗} is also bijective and satisfies S(∫)b = S(∫)S(S^{−1}(b)) = S(S^{−1}(b) ∫) = S(∫)ε(b) hence S(∫) ∈ Int_{r}(H^{∗}). □

Remark 4.1.8. Maschke’s Theorem has an extension to finite dimensional Hopf algebras: ε(∫) ≠ 0 iff H^{∗} is semisimple.

Corollary 4.1.9. Let H be a finite dimensional Hopf algebra. Then H^{∗} is a left H^{∗}-module by the usual multiplication, hence a right H-comodule. We have

(H^{∗})^{coH} = Int_{l}(H^{∗}).

Proof. By definition we have Int_{l}(H^{∗}) = ^{H^{∗}}H^{∗}. □

Example 4.1.10. Let G be a finite group. Let H := Map(G, 𝕂) be the Hopf algebra defined by the following isomorphism

𝕂^{G} = Map(G, 𝕂) ≅ Hom(𝕂G, 𝕂) = (𝕂G)^{∗}.

This isomorphism between the vector space 𝕂^{G} of all set maps from the group G to the base ring 𝕂 and the dual vector space (𝕂G)^{∗} of the group algebra 𝕂G defines the structure of a Hopf algebra on 𝕂^{G}.

We regard H := 𝕂^{G} as the function algebra on the set G. In the sense of algebraic geometry this is not quite true. The algebra 𝕂^{G} represents a functor from 𝕂-cAlg to Set that has G as value for all connected algebras A in particular for all field extensions of 𝕂.

As before we use the map ev : 𝕂G ⊗ 𝕂^{G} ⟶ 𝕂. The multiplication of 𝕂^{G} is given by pointwise multiplication of maps since ⟨x, ff′⟩ = ⟨∑ x_{(1)} ⊗ x_{(2)}, f ⊗ f′⟩ = ⟨x ⊗ x, f ⊗ f′⟩ = ⟨x, f⟩⟨x, f′⟩ for all f, f′ ∈ 𝕂^{G} and all x ∈ G. The unit element 1_{𝕂^{G}} of 𝕂^{G} is the map ε : 𝕂G ⟶ 𝕂 restricted to G, hence ε(x) = 1 = ⟨x, 1_{𝕂^{G}}⟩ for all x ∈ G. The antipode of f ∈ 𝕂^{G} is given by S(f)(x) = ⟨x, S(f)⟩ = f(x^{−1}).

The elements of the dual basis (x^{∗}|x ∈ G) with ⟨x, y^{∗}⟩ = δ_{x,y} considered as maps from G to 𝕂 form a basis of 𝕂^{G}. They satisfy the conditions

x^{∗}y^{∗} = δ_{x,y}x^{∗} and ∑_{x∈G} x^{∗} = 1_{𝕂^{G}}

since ⟨z, x^{∗}y^{∗}⟩ = ⟨z, x^{∗}⟩⟨z, y^{∗}⟩ = δ_{z,x}δ_{z,y} = δ_{x,y}⟨z, x^{∗}⟩ and ⟨z, ∑_{x∈G} x^{∗}⟩ = 1 = ⟨z, 1_{𝕂^{G}}⟩.

Hence the dual basis (x^{∗}|x ∈ G) is a decomposition of the unit into a set of minimal orthogonal idempotents and the algebra of 𝕂^{G} has the structure

𝕂^{G} = ⊕_{x∈G}𝕂x^{∗} ≅ 𝕂 × … × 𝕂.

In particular 𝕂^{G} is commutative and semisimple.

The diagonal of 𝕂^{G} is

Δ(x^{∗}) = ∑_{y∈G} y^{∗} ⊗ (y^{−1}x)^{∗} = ∑_{y,z∈G,yz=x} y^{∗} ⊗ z^{∗}

since

\begin{matrix}⟨z ⊗ u, Δ(x^{∗})⟩ = ⟨zu, x^{∗}⟩ = δ_{x,zu} = δ_{z^{−1}x,u} = ∑_{y∈G} δ_{y,z}δ_{y^{−1}x,u} \\ = ∑_{y∈G}⟨z, y^{∗}⟩⟨u, (y^{−1}x)^{∗}⟩ = ⟨z ⊗ u, ∑_{y∈G} y^{∗} ⊗ (y^{−1}x)^{∗}⟩. \end{matrix}

Let a ∈ 𝕂G. Then a defines a map ã : G ⟶ 𝕂 ∈ 𝕂^{G} by a = ∑_{x∈G} ã(x)x. For arbitrary f ∈ 𝕂^{G} and a ∈ 𝕂G this gives

⟨a, f⟩ = f(∑_{x∈G} ã(x)x) = ∑_{x∈G} ã(x)f(x).

The counit of 𝕂^{G} is given by ε(x^{∗}) = δ_{x,e} where e ∈ G is the unit element.

The antipode is, as above, S(x^{∗}) = (x^{−1})^{∗}.

We consider H = 𝕂^{G} as the function algebra on the finite group G and 𝕂G as the dual space of H = 𝕂^{G} hence as the set of distributions on H.

Then

(7)  ∫ := ∑_{x∈G} x ∈ H^{∗} = 𝕂G

is a (two sided) integral on H since ∑_{x∈G} yx = ∑_{x∈G} x = ε(y) ∑_{x∈G} x = ∑_{x∈G} yx. We write

∫ f(x)dx := ⟨∫ , f⟩ = ∑_{x∈G} f(x).

We have seen that there is a decomposition of the unit 1 ∈ 𝕂^{G} into a set of primitive orthogonal idempotents {x^{∗}|x ∈ G} such that every element f ∈ 𝕂^{G} has a unique representation f = ∑ f(x)x^{∗}. Since ∫ y^{∗} = ∑_{x∈G}⟨x, y^{∗}⟩ we get ∫ fy^{∗} = ∑_{x∈G}⟨x, fy^{∗}⟩ = ∑ f(x)y^{∗}(x) = f(y) hence

f = ∑(∫ f(x)y^{∗}(x)dx)y^{∗}.

Problem 4.1.30. Describe the group valued functor 𝕂-cAlg (𝕂^{G}, −) in terms of sets and their group structure.

Definition and Remark 4.1.11. Let 𝕂 be an algebraicly closed field and let G be a finite abelian group (replacing ℝ above). Assume that the characteristic of 𝕂 does not divide the order of G. Let H = 𝕂^{G}. We identify 𝕂^{G} = Hom(𝕂G, 𝕂) along the linear expansion of maps as in Example 2.1.10.

Let us consider the set Ĝ := {χ : G ⟶ 𝕂^{∗}|χ group homomorphism}. Since 𝕂^{∗} is an abelian group, the set Ĝ is an abelian group by pointwise multiplication.

The group Ĝ is called the character group of G.

Obviously the character group is a multiplicative subset of 𝕂^{G} = Hom(𝕂G, 𝕂). Actually it is a subgroup of 𝕂-cAlg (𝕂G, 𝕂) ⊆ Hom(𝕂G, 𝕂) since the elements χ ∈ Ĝ expand to algebra homomorphisms: χ(ab) = χ(∑ α_{x}x ∑ β_{y}y) = ∑ α_{x}β_{y}χ(xy) = χ(a)χ(b) and χ(1) = χ(e) = 1. Conversely an algebra homomorphism f ∈ 𝕂-cAlg (𝕂G, 𝕂) restricts to a character f : G ⟶ 𝕂^{∗}. Thus Ĝ = 𝕂-cAlg (𝕂G, 𝕂), the set of rational points of the affine algebraic group represented by 𝕂G.

There is a more general observation behind this remark.

Lemma 4.1.12. Let H be a finite dimensional Hopf algebra. Then the set Gr(H^{∗}) of grouplike elements of H^{∗} is equal to 𝕂-Alg (H, 𝕂).

Proof. In fact f : H ⟶ 𝕂 is an algebra homomorphism iff ⟨f ⊗ f, a ⊗ b⟩ = ⟨f, a⟩⟨f, b⟩ = ⟨f, ab⟩ = ⟨Δ(f), a ⊗ b⟩ and 1 = ⟨f, 1⟩ = ε(f). □

Hence there is a Hopf algebra homomorphism φ : 𝕂Ĝ ⟶ 𝕂^{G} by 2.1.5.

Proposition 4.1.13. The Hopf algebra homomorphism φ : 𝕂Ĝ ⟶ 𝕂^{G} is bijective.

Proof. We give the proof by several lemmas.

Lemma 4.1.14. Any set of grouplike elements in a Hopf algebra H is linearly independent.

Proof. Assume there is a linearly dependent set {x_{0}, x_{1}, …, x_{n}} of grouplike elements in H. Choose such a set with n minimal. Obviously n ≥ 1 since all elements are non zero. Thus x_{0} = ∑_{i=1}^{n} α_{i}x_{i} and {x_{1}, …, x_{n}} linearly independent. We get

∑_{i,j} α_{i}α_{j}x_{i} ⊗ x_{j} = x_{0} ⊗ x_{0} = Δ(x_{0}) = ∑_{i} α_{i}x_{i} ⊗ x_{i}.

Since all α_{i} ≠ 0 and the x_{i} ⊗ x_{j} are linearly independent we get n = 1 and α_{1} = 1 so that x_{0} = x_{1}, a contradiction. □

Corollary 4.1.15. (Dedekind’s Lemma) Any set of characters in 𝕂^{G} is linearly independent.

Thus φ : 𝕂Ĝ ⟶ 𝕂^{G} is injective. Now we prove that the map φ : 𝕂Ĝ ⟶ 𝕂^{G} is surjective.

Lemma 4.1.16. (Pontryagin duality) The evaluation Ĝ × G ⟶ 𝕂^{∗} is a nondegenerate bilinear map of abelian groups.

Proof. First we observe that Hom(C_{n}, 𝕂^{∗}) ≅ C_{n} for a cyclic group of order n since 𝕂 has a primitive n-th root of unity (char(𝕂)−̸ |G|).

Since the direct product and the direct sum coincide in Ab we can use the fundamental theorem for finite abelian groups G ≅ C_{n_{1}} × … × C_{n_{t}} to get Hom(G, 𝕂^{∗}) ≅ G for any abelian group G with char(𝕂)−̸ |G|. Thus Ĝ ≅ G and ˆ Ĝ = G. In particular χ(x) = 1 for all x ∈ G iff χ = 1. By the symmetry of the situation we get that the bilinear form ⟨., .⟩ : Ĝ × G ⟶ 𝕂^{∗} is non-degenerate. □

Thus |Ĝ| = |G| hence dim(𝕂Ĝ) = dim(𝕂^{G}). This proves Proposition 2.1.13. □

Definition 4.1.17. Let H be a Hopf algebra. A 𝕂-module M that is a right H-module by ρ : M ⊗ H ⟶ M and a right H-comodule by δ : M ⟶ M ⊗ H is called a Hopf module if the diagram

[figure: M ⊗ H; H; ρ; M ⊗ H; δ; M ⊗ H ⊗ H ⊗ H; M ⊗ H ⊗ H ⊗ H; 1⊗τ⊗1; δ⊗Δ; ρ⊗∇]

commutes, i.e. if δ(mh) = ∑ m_{(M)}h_{(1)} ⊗ m_{(1)}h_{(2)} holds for all h ∈ H and all m ∈ M.

Observe that H is an Hopf module over itself. Furthermore each module of the form V ⊗ H is a Hopf module by the induced structure. More generally there is a functor Vec ∋ V ↦ V ⊗ H ∈ Hopf-Mod-H.

Proposition 4.1.18. The two functors -^{coH} : Hopf-Mod-H ⟶ Vec and - ⊗ H : Vec ∋ V ↦ V ⊗ H ∈ Hopf-Mod-H are inverse equivalences of each other.

Proof. Define natural isomorphisms

α : M^{coH} ⊗ H ∋ m ⊗ h ↦ mh ∈ M

with inverse map

α^{−1} : M ∋ m ↦ ∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)} ∈ M^{coH} ⊗ H

and

β : V ∋ v ↦ v ⊗ 1 ∈ (V ⊗ H)^{coH}

with inverse map

(V ⊗ H)^{coH} ∋ v ⊗ h ↦ vε(h) ∈ V.

Obviously these homomorphisms are natural transformations in M and V. Furthermore α is a homomorphism of H-modules. α^{−1} is well-defined since

δ(∑ m_{(M)}S(m(1))) = ∑ m_{(M)}S(m_{(3)}) ⊗ m_{(1)}S(m_{(2)})

(since M is a Hopf module) = ∑ m_{(M)}S(m_{(2)}) ⊗ ηε(m_{(1)}) = ∑ m_{(M)}S(m_{(1)}) ⊗ 1

hence ∑ m_{(M)}S(m_{(1)}) ∈ M^{coH}. Furthermore α^{−1} is a homomorphism of comodules since

\begin{aligned}δα^{−1}(m) & = δ(∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)}) = ∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)} ⊗ m_{(3)} \\ & = ∑ α^{−1}(m_{(M)}) ⊗ m_{(1)} = (α^{−1} ⊗ 1)δ(m). \end{aligned}

Finally α and α^{−1} are inverse to each other by

αα^{−1}(m) = α(∑ m_{(M)}S(m_{(1)}) ⊗ m_{(2)}) = ∑ m_{(M)}S(m_{(1)})m_{(2)} = m

and

\begin{aligned}α^{−1}α(m ⊗ h) & = α^{−1}(mh) = ∑ m_{(M)}h_{(1)}S(m_{(1)}h_{(2)}) ⊗ m_{(2)}h_{(3)} \\ & = ∑ mh_{(1)}S(h_{(2)}) ⊗ h_{(3)} ( by δ(m) = m ⊗ 1 ) = m ⊗ h. \end{aligned}

Thus α and α^{−1} are mutually inverse homomorphisms of Hopf modules.

The image of β is in (V ⊗ H)^{coH} by δ(v ⊗ 1) = v ⊗ Δ(1) = (v ⊗ 1) ⊗ 1. Both β and β^{−1} are 𝕂-linear maps. Furthermore we have

β^{−1}β(v) = β^{−1}(v ⊗ 1) = vε(1) = v

and

\begin{aligned}ββ^{−1} & (∑ v_{i} ⊗ h_{i}) = β(∑ v_{i}ε(h_{i})) = ∑ v_{i}ε(h_{i}) ⊗ 1 = ∑ v_{i} ⊗ ε(h_{i})1 \\ & = ∑ v_{i} ⊗ ε(h_{i(1)})h_{i(2)} ( since ∑ v_{i} ⊗ h_{i} ∈ (V ⊗ H)^{coH} ) = ∑ v_{i} ⊗ h_{i}. \end{aligned}

Thus β and β^{−1} are mutually inverse homomorphisms. □

Since H^{∗} = Hom(H, 𝕂) and S : H ⟶ H is an algebra antihomomorphism, the dual H^{∗} is an H-module in four different ways:

(8)  \begin{matrix}⟨(f ⇀ a), g⟩ := ⟨a, gf⟩, & ⟨(a ↼ f), g⟩ := ⟨a, fg⟩, \\ ⟨(f ⇁ a), g⟩ := ⟨a, S(f)g⟩, & ⟨(a ↽ f), g⟩ := ⟨a, gS(f)⟩. \end{matrix}

If H is finite dimensional then H^{∗} is a Hopf algebra. The equality ⟨(f ⇀ a), g⟩ = ⟨a, gf⟩ = ∑⟨a_{(1)}, g⟩⟨a_{(2)}, f⟩ implies

(9)  (f ⇀ a) = ∑ a_{(1)}⟨a_{(2)}, f⟩.

Analogously we have

(10)  (a ↼ f) = ∑⟨a_{(1)}, f⟩a_{(2)}.

Proposition 4.1.19. Let H be a finite dimensional Hopf algebra. Then H^{∗} is a right Hopf module over H.

Proof. H^{∗} is a left H^{∗}-module by left multiplication hence by 2.1.1 a right Hcomodule by δ(a) = ∑_{i} b_{i}^{∗}a⊗b_{i}. Let f, g ∈ H and a, b ∈ H^{∗}. The (left) multiplication of H^{∗} satisfies

ab = ∑ b_{(H^{∗})}⟨a, b_{(1)}⟩.

We use the right H-module structure

(a ↽ f) = ∑ a_{(1)}⟨S(f), a_{(2)}⟩.

on H^{∗} = Hom(H, 𝕂).

Now we check the Hopf module property. Let a, b ∈ H^{∗} and f, g ∈ H. We apply H^{∗} ⊗ H to its dual H ⊗ H^{∗} and get

δ(a ↽f)(g ⊗ b) = ∑⟨(a ↽ f)_{(H^{∗})}, g⟩⟨b, (a ↽ f)_{(1)}⟩ = ⟨b(a ↽ f), g⟩
= ∑⟨b, g_{(1)}⟩⟨(a ↽ f), g_{(2)}⟩ = ∑⟨b, g_{(1)}⟩⟨a, g_{(2)}S(f)⟩

= ∑⟨b, g_{(1)}ε(f_{(2)})⟩⟨a, g_{(2)}S(f_{(1)})⟩ = ∑⟨(f_{(3)} ⇀ b), g_{(1)}S(f_{(2)})⟩⟨a, g_{(2)}S(f_{(1)})⟩

= ∑⟨(f_{(2)} ⇀ b)a, gS(f_{(1)})⟩ = ∑⟨((f_{(2)} ⇀ b)a) ↽ f_{(1)}, g⟩
= ∑⟨(a_{(H^{∗})}⟨(f_{(2)} ⇀ b), a_{(1)}⟩) ↽ f_{(1)}, g⟩
= ∑⟨(a_{(H^{∗})} ↽ f_{(1)})⟨(f_{(2)} ⇀ b), a_{(1)}⟩, g⟩ = ∑⟨(a_{(H^{∗})} ↽ f_{(1)})⟨b, a_{(1)}f_{(2)}⟩, g⟩

hence δ(a ↽ f) = ∑(a_{(H^{∗})} ↽ f_{(1)}) ⊗ a_{(1)}f_{(2)}. □

Theorem 4.1.20. Let H be a finite dimensional Hopf algebra. Then the antipode S is bijective, the space of left integrals Int_{l}(H^{∗}) has dimension 1, and the homomorphism

H ∋ f ↦ (f ⇀ ∫) = ∑_{∫(1)}⟨_{∫(2)}, f⟩ ∋ H^{∗}

is bijective for any 0 ≠ ∫ ∈ Int_{l}(H^{∗}).

Proof. By Proposition 2.1.19 H^{∗} is a right Hopf module over H. By Proposition 2.1.18 there is an isomorphism α : (H^{∗})^{coH} ⊗ H ∋ a ⊗ f ↦ (a ↽ f) = (S(f) ⇀ a) ∈ H^{∗}. Since (H^{∗})^{coH} ≅ Int_{l}(H^{∗}) by 2.1.9 we get

Int_{l}(H^{∗}) ⊗ H ≅ H^{∗}

as right H-Hopf modules by the given map. This shows dim(Int_{l}(H^{∗})) = 1. So we get an isomorphism H ∋ f ↦ (∫ ↽ f) ∈ H^{∗} that is a composition of S and f ↦ (f ⇀ ∫). Since H is finite dimensional both of these maps are bijective. □

If G is a finite group then every generalized integral a ∈ 𝕂G can be written with a uniquely determined g ∈ H = 𝕂^{G} as

(11)  ⟨a, f⟩ = ∫ f(x)S(g)(x)dx = ∑_{x∈G} f(x)g(x^{−1})

for all f ∈ H.

If G is a finite Abelian group then each group element (rational integral) y ∈ G ⊆ 𝕂G can be written as

y = ∑_{x∈G} ∑_{χ∈Ĝ} β_{χ}⟨x^{−1}, χ⟩x

since

\begin{aligned}⟨y, f⟩ & = ⟨(∫ ↽ ∑_{χ∈Ĝ} β_{χ}χ), f⟩ = ⟨∫ , fS(∑_{χ∈Ĝ} β_{χ}χ)⟩ \\ & = ∑_{x∈G}⟨x, f⟩ ∑_{χ∈Ĝ} β_{χ}⟨x, S(χ)⟩ = ⟨∑_{x∈G} ∑_{χ∈Ĝ} β_{χ}⟨x^{−1}, χ⟩x, f⟩. \end{aligned}

In particular the matrix (⟨x^{−1}, χ⟩) is invertible.

Let H be finite dimensional. Since ⟨∫ , fg⟩ = ⟨(∫ ↼ f), g⟩ as a functional on g is a generalized integral, there is a unique ν(f) ∈ H such that

(12)  ⟨∫ , fg⟩ = ⟨∫ , gν(f)⟩

or

(13)  ∫ f(x)g(x)dx = ∫ g(x)ν(f)(x)dx.

Although the functions f, g ∈ H of the quantum group do not commute under multiplication, there is a simple commutation rule if the product is integrated.

Proposition and Definition 4.1.21. The map ν : H ⟶ H is an algebra automorphism, called the Nakayama automorphism.

Proof. It is clear that ν is a linear map. We have ∫ fν(gh) = ∫ ghf = ∫ hfν(g) = ∫ fν(g)ν(h) hence ν(gh) = ν(g)ν(h) and ∫ fν(1) = ∫ f hence ν(1) = 1. Furthermore if ν(g) = 0 then 0 = ⟨∫ , fν(g)⟩ = ⟨∫ , gf⟩ = ⟨(f ⇀ ∫), g⟩ for all f ∈ H hence ⟨a, g⟩ = 0 for all a ∈ H^{∗} hence g = 0. So ν is injective hence bijective. □

Corollary 4.1.22. The map H ∋ f ↦ (∫ ↼ f) ∈ H^{∗} is an isomorphism.

Proof. We have

(∫ ↼ f) = (ν(f) ⇀ ∫)

since ⟨(∫ ↼ f), g⟩ = ⟨∫ , fg⟩ = ⟨∫ , gν(f)⟩ = ⟨(ν(f) ⇀ ∫), g⟩. This implies the corollary. □

If G is a finite group and H = 𝕂^{G} then H is commutative hence ν = id.

Definition 4.1.23. An element δ ∈ H is called a Dirac δ-function if δ is a left invariant integral in H with ⟨∫ , δ⟩ = 1, i.e. if δ satisfies

fδ = ε(f)δ  and  ∫ δ(x)dx = 1

for all f ∈ H. If H has a Dirac δ-function then we write

(14)  ∫ ^{∗} a(x)dx = ∫^{∗}a := ⟨a, δ⟩.

Proposition 4.1.24..

(1) If H is finite dimensional then there exists a unique Dirac δ-function δ.

(2) If H is infinite dimensional then there exists no Dirac δ-function.

Proof. 1. Since H ∋ f ↦ (f ⇀ ∫) ∈ H^{∗} is an isomorphism there is a δ ∈ H such that (δ ⇀ ∫) = ε. Then (fδ ⇀ ∫) = (f ⇀ (δ ⇀ ∫)) = (f ⇀ ε) = ε(f)ε = ε(f)(δ ⇀ ∫) which implies fδ = ε(f)δ. Furthermore we have ⟨∫ , δ⟩ = ⟨∫ , 1_{H}δ⟩ = ⟨(δ ⇀ ∫), 1_{H}⟩ = ε(1_{H}) = 1_{𝕂}.

2. is [Sweedler] exercise V.4. □

Lemma 4.1.25. Let H be a finite dimensional Hopf algebra. Then ∫ ∈ H^{∗} is a left integral iff

(15)  a(∑ ∫_{(1)} ⊗ S(∫_{(2)})) = (∑ ∫_{(1)} ⊗ S(∫_{(2)}))a

iff

(16)  ∑ S(a)∫_{(1)} ⊗ ∫_{(2)} = ∑ ∫_{(1)} ⊗ a∫_{(2)}

iff

(17)  ∑ f_{(1)}⟨∫ , f_{(2)}⟩ = ⟨∫ , f⟩1_{H}.

Proof. Let ∫ be a left integral. Then

∑ a_{(1)}∫_{(1)} ⊗ S(∫_{(2)})S(a_{(2)}) = ∑(a∫)_{(1)} ⊗ S((a∫)_{(2)}) = ε(a)(∑ ∫_{(1)} ⊗ S(∫_{(2)}))

for all a ∈ H. Hence

(∑ ∫_{(1)} ⊗ S(∫_{(2)}))a = ∑ ε(a_{(1)})(∫_{(1)} ⊗ S(∫_{(2)}))a_{(2)}
= ∑ a_{(1)}∫_{(1)} ⊗ S(∫_{(2)})S(a_{(2)})a_{(3)}
= ∑ a_{(1)}∫_{(1)} ⊗ S(∫_{(2)})ε(a_{(2)}) = a(∑ ∫_{(1)} ⊗ S(∫_{(2)})).

Conversely a(∑ ∫_{(1)} ε(S(∫_{(2)}))) = (∑ ∫_{(1)} ε(S(∫_{(2)})a)) = ε(a)(∑ ∫_{(1)} ε(S(∫_{(2)}))), hence ∫ = ∑ ∫_{(1)} ε(S(∫_{(2)})) is a left integral.

Since S is bijective the following holds

\begin{matrix}∑ S(a) ∫_{(1)} ⊗ ∫_{(2)} = ∑ S(a) ∫_{(1)} ⊗S^{−1}(S(∫_{(2)})) \\ = ∑ ∫_{(1)} ⊗S^{−1}(S(∫_{(2)})S(a)) = ∑ ∫_{(1)} ⊗a ∫_{(2)} . \end{matrix}

The converse follows easily.

If ∫ ∈ Int_{l}(H) is a left integral then ∑⟨a, f_{(1)}⟩⟨∫ , f_{(2)}⟩ = ⟨a ∫ , f⟩ = ⟨a, 1_{H}⟩⟨∫ , f⟩. Conversely if λ ∈ H^{∗} with (17) is given then ⟨aλ, f⟩ = ∑⟨a, f_{(1)}⟩⟨λ, f_{(2)}⟩ =

⟨a, 1_{H}⟩⟨λ, f⟩ hence aλ = ε(a)λ. □

If G is a finite group then

(18)  δ(x) = \begin{cases}0 if x ≠ e; \\ 1 if x = e. \end{cases}

In fact since δ is left invariant we get f(x)δ(x) = f(e)δ(x) for all x ∈ G and f ∈ 𝕂^{G}. Since G ⊂ H^{∗} = 𝕂G is a basis, we get δ(x) = 0 if x ≠ e. Furthermore ∫ δ(x)dx = ∑_{x∈G} δ(x) = 1 implies δ(e) = 1. So we have δ = e^{∗}.

If G is a finite Abelian group we get δ = α ∑_{χ∈Ĝ} χ for some α ∈ 𝕂. The evaluation gives 1 = ⟨∫ , δ⟩ = α ∑_{x∈G,χ∈Ĝ}⟨χ, x⟩. Now let λ ∈ Ĝ. Then ∑_{χ∈Ĝ}⟨χ, x⟩ = ∑_{χ∈Ĝ}⟨λχ, x⟩ = ⟨λ, x⟩ ∑_{χ∈Ĝ}⟨χ, x⟩. Since for each x ∈ G \ {e} there is a λ such that ⟨λ, x⟩ ≠ 1 and we get

∑_{χ∈Ĝ}⟨χ, x⟩ = |G|δ_{e,x}.

Hence ∑_{x∈G,χ∈Ĝ}⟨χ, x⟩ = |G| = α^{−1} and

(19)  δ = |G|^{−1} ∑_{χ∈Ĝ} χ.

Let H be finite dimensional for the rest of this section. In Corollary 1.22 we have seen that the map H ∋ f ↦ (∫ ↼ f) ∈ H^{∗} is an isomorphism. This map will be called the Fourier transform.

Theorem 4.1.26. The Fourier transform H ∋ f ↦ f̃ ∈ H^{∗} is bijective with

(20)  f̃ = (∫ ↼ f) = ∑⟨∫_{(1)}, f⟩∫_{(2)}

The inverse Fourier transform is defined by

(21)  ã = ∑ S^{−1}(δ_{(1)})⟨a, δ_{(2)}⟩.

Since these maps are inverses of each other the following formulas hold

(22)  \begin{matrix}⟨f̃, g⟩ = ∫ f(x)g(x)dx & ⟨a, b̃⟩ = ^{∫ ∗}S^{−1}(a)(x)b(x)dx \\ f = ∑ S^{−1}(δ_{(1)})⟨f̃, δ_{(2)}⟩ & a = ∑⟨∫_{(1)}, ã⟩ ∫_{(2)} . \end{matrix}

Proof. We use the isomorphisms H ⟶ H^{∗} defined by f̂ := f̃ = (∫ ↼ f) = ∑⟨∫_{(1)}, f⟩ ∫_{(2)} and H^{∗} ⟶ H defined by â := (a ⇀ δ) = ∑ δ_{(1)}⟨a, δ_{(2)}⟩. Because of

(23)  ⟨a, b̂⟩ = ⟨a, (b ⇀ δ)⟩ = ⟨ab, δ⟩

and

(24)  ⟨f̃, g⟩ = ⟨(∫ ↼ f), g⟩ = ⟨∫ , fg⟩

we get for all a ∈ H^{∗} and f ∈ H

\begin{aligned}⟨a, ˆ f̂⟩ & = ⟨af̂, δ⟩ = ∑⟨a, δ_{(1)}⟩⟨f̂, δ_{(2)}⟩ = ∑⟨a, δ_{(1)}⟩⟨∫ , fδ_{(2)}⟩ ( by Lemma 1.25 ) \\ & = ∑⟨a, S(f)δ_{(1)}⟩⟨∫ , δ_{(2)}⟩ = ⟨a, S(f)⟩⟨∫ , δ⟩ = ⟨a, S(f)⟩. \end{aligned}

This gives ˆf̂ = S(f). So the inverse map of H ⟶ H^{∗} with f̂ = (∫ ↼ f) = f̃ is H^{∗} ⟶ H with S^{−1}(â) = ∑ S^{−1}(δ_{(1)})⟨a, δ_{(2)}⟩ = ã. Then the given inversion formulas are clear.

We note for later use ⟨a, b̃⟩ = ⟨a, S^{−1}(b̂)⟩ = ⟨S^{−1}(a), b̂⟩ = ⟨S^{−1}(a)b, δ⟩. □

If G is a finite group and H = 𝕂^{G} then

f̃ = ∑_{x∈G} f(x)x.

Since Δ(δ) = ∑_{x∈G} x^{−1}^{∗} ⊗ x^{∗} where the x^{∗} ∈ 𝕂^{G} are the dual basis to the x ∈ G, we get

ã = ∑_{x∈G}⟨a, x^{∗}⟩x^{∗}.

If G is a finite Abelian group then the groups G and Ĝ are isomorphic so the Fourier transform induces a linear automorphism ˜- : 𝕂^{G} ⟶ 𝕂^{G} and we have

ã = |G|^{−1} ∑_{χ∈Ĝ}⟨a, χ⟩χ^{−1}

By substituting the formulas for the integral and the Dirac δ-function (7) and (19) we get

(25)  \begin{matrix}f̃ = ∑_{x∈G} f(x)x, & ã = |G|^{−1} ∑_{χ∈Ĝ} a(χ)χ^{−1}, \\ f = |G|^{−1} ∑_{χ∈Ĝ} f̃(χ)χ^{−1}, & a = ∑_{x∈G} ã(x)x. \end{matrix}

This implies

(26)  f̃(χ) = ∑_{x∈G} f(x)χ(x) = ∫ f(x)χ(x)dx

with inverse transform

(27)  ã(x) = |G|^{−1} ∑_{χ∈Ĝ} χ(a)χ^{−1}(x).

Corollary 4.1.27. The Fourier transforms of the left invariant integrals in H and H^{∗} are

(28)  δ̃ = εν^{−1} ∈ H^{∗}  and  ˜∫ = 1 ∈ H.

Proof. We have ⟨δ̃, f⟩ = ⟨∫ , δf⟩ = ⟨∫ , ν^{−1}(f)δ⟩ = εν^{−1}(f)⟨∫ , δ⟩ = εν^{−1}(f) hence δ̃ = εν^{−1}. From ˜1 = (∫ ↼ 1) = ∫ we get ˜∫ = 1. □

Proposition 4.1.28. Define a convolution multiplication on H^{∗} by

⟨a ∗ b, f⟩ := ∑⟨a, S^{−1}(δ_{(1)})f⟩⟨b, δ_{(2)}⟩.

1. INTEGRALS AND FOURIER TRANSFORMS 125

Then the following transformation rule holds for f, g ∈ H:

(29) f̃g = f̃∗ g̃.

In particular H^{∗} with the convolution multiplication is an associative algebra with unit ˜1_{H} = ∫, i.e.

(30)  ∫ ∗ a = a ∗ ∫ = a.

Proof. Given f, g, h ∈ H^{∗}. Then

⟨f̃g, h⟩ = ⟨∫ , fgh⟩ = ⟨∫ , fS^{−1}(1_{H})gh⟩⟨∫ , δ⟩
= ∑⟨∫ , fS^{−1}(δ_{(1)})gh⟩⟨∫ , δ_{(2)}⟩ = ∑⟨∫ , fS^{−1}(δ_{(1)})h⟩⟨∫ , gδ_{(2)}⟩
= ∑⟨f̃, S^{−1}(δ_{(1)})h⟩⟨g̃, δ_{(2)}⟩ = ⟨f̃∗ g̃, h⟩.

From (28) we get ˜1_{H} = ∫. So we have f̃ = ˜1f = ˜1 ∗ f̃ = ∫ ∗f̃. □

If G is a finite Abelian group and a, b ∈ H^{∗} = 𝕂Ĝ. Then

(a ∗ b)(μ) = |G|^{−1} ∑_{χ,λ∈Ĝ,χλ=μ} a(λ)b(χ).

In fact we have

\begin{aligned}(a ∗ b)(μ) & = ⟨a ∗ b, μ⟩ = ∑⟨a, S^{−1}(δ_{(1)})μ⟩⟨b, δ_{(2)}⟩ \\ & = |G|^{−1} ∑_{χ∈Ĝ}⟨a, χ^{−1}μ⟩⟨b, χ⟩ = |G|^{−1} ∑_{χ,λ∈Ĝ,χλ=μ} a(λ)b(χ). \end{aligned}

One of the most important formulas for Fourier transforms is the Plancherel formula on the invariance of the inner product under Fourier transforms. We have

Theorem 4.1.29. (The Plancherel formula)

(31)  ⟨a, f⟩ = ⟨f̃, ν(ã)⟩.

Proof. First we have from (22)

⟨a, f⟩ = ∑⟨∫_{(1)}, ã⟩⟨∫_{(2)}, S^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩ = ∑⟨∫ , ãS^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , S^{−1}(δ_{(1)})ν(ã)⟩⟨f̃, δ_{(2)}⟩ = ∑⟨∫ , S^{−1}(S(ν(ã))δ_{(1)})⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , S^{−1}(δ_{(1)})⟩⟨f̃, ν(ã)δ_{(2)}⟩ = ∑⟨∫ , S^{−1}(δ)_{(2)}⟩⟨f̃, ν(ã)S(S^{−1}(δ)_{(1)})⟩
= ⟨∫ , S^{−1}(δ)⟩⟨f̃, ν(ã)⟩.

Apply this to ⟨∫ , δ⟩. Then we get

1 = ⟨∫ , δ⟩ = ⟨∫ , S^{−1}(δ)⟩⟨δ̃, ν(˜∫)⟩ = ⟨∫ , S^{−1}(δ)⟩εν^{−1}ν(1) = ⟨∫ , S^{−1}(δ)⟩. Hence we get ⟨a, f⟩ = ⟨f̃, ν(ã)⟩. □

Corollary 4.1.30. If H is unimodular then ν = S^{2}.

Proof. H unimodular means that δ is left and right invariant. Thus we get

⟨a, f⟩ = ∑⟨∫_{(1)}, ã⟩⟨∫_{(2)}, S^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , ãS^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}⟩ = ∑⟨∫ , S^{−1}(δ_{(1)}S(ã))⟩⟨f̃, δ_{(2)}⟩
= ∑⟨∫ , S^{−1}(δ_{(1)})⟩⟨f̃, δ_{(2)}S^{2}(ã)⟩ ( since δ is right invariant)
= ⟨∫ , S^{−1}(δ)⟩⟨f̃, S^{2}(ã)⟩ = ⟨f̃, S^{2}(ã)⟩.

Hence S^{2} = ν. □

We also get a special representation of the inner product H^{∗} ⊗ H ⟶ 𝕂 by both integrals:

Corollary 4.1.31.

(32)  ⟨a, f⟩ = ∫ ã(x)f(x)dx = ∫ ^{∗} S^{−1}(a)(x)f̃(x)dx.

Proof. We have the rules for the Fourier transform. From (24) we get ⟨a, f⟩ = ⟨∫ , ãf⟩ = ∫ ã(x)f(x)dx and from (23) ⟨a, f⟩ = ⟨S^{−1}(a)f̃, δ⟩ = ∫^{∗} S^{−1}(a)(x)f̃(x)dx.

□

The Fourier transform leads to an interesting integral transform on H by double application.

Proposition 4.1.32. The double transform f̆ := (δ ↼ (∫ ↼ f)) defines an automorphism H ⟶ H with

f̆(y) = ∫ f(x)δ(xy)dx.

Proof. We have

⟨y, f̆⟩ = ⟨y, (δ ↼ (∫ ↼ f))⟩ = ⟨(∫ ↼ f)y, δ⟩
= ∑⟨(∫ ↼ f), δ_{(1)}⟩⟨y, δ_{(2)}⟩ = ∑⟨∫ , fδ_{(1)}⟩⟨y, δ_{(2)}⟩
= ∑⟨∫_{(1)}, f⟩⟨∫_{(2)}, δ_{(1)}⟩⟨y, δ_{(2)}⟩ = ∑⟨∫_{(1)}, f⟩⟨∫_{(2)} y, δ⟩
= ∑⟨∫_{(1)}, f⟩⟨∫_{(2)}, (y ⇀ δ)⟩ = ⟨∫ , f(y ⇀ δ)⟩
= ∫ f(x)δ(xy)dx

since ⟨x, (y ⇀ δ)⟩ = ⟨xy, δ⟩. □

2. Derivations

Definition 4.2.1. Let A be a 𝕂-algebra and _{A}M_{A} be an A-A-bimodule (with identical 𝕂-action on both sides). A linear map D : A ⟶ M is called a derivation if

D(ab) = aD(b) + D(a)b.

The set of derivations Der_{𝕂}(A,_{A}M_{A}) is a 𝕂-module and a functor in _{A}M_{A}.

By induction one sees that D satisfies

D(a_{1} … a_{n}) = ∑_{i=1}^{n} a_{1} … a_{i−1}D(a_{i})a_{i+1} … a_{n}.

Let A be a commutative 𝕂-algebra and _{A}M be an A-module. Consider M as an A-A-bimodule by ma := am. We denote the set of derivations from A to M by Der_{𝕂}(A, M)_{c}.

Proposition 4.2.2. 1. Let A be a 𝕂-algebra. Then the functor Der_{𝕂}(A, -) : A-Mod -A ⟶ Vec is representable by the module of differentials Ω_{A}.

2. Let A be a commutative 𝕂-algebra. Then the functor Der_{𝕂}(A, -)_{c} : A-Mod ⟶ Vec is representable by the module of commutative differentials Ω_{A}^{c}.

Proof. 1. Represent A as a quotient of a free 𝕂-algebra A := 𝕂⟨X_{i}|i ∈ J⟩/I where B = 𝕂⟨X_{i}|i ∈ J⟩ is the free algebra with generators X_{i}. We first prove the theorem for free algebras.

a) A representing module for Der_{𝕂}(B, -) is (Ω_{B}, d : B ⟶ Ω_{B}) with

Ω_{B} := B ⊗ F (dX_{i}|i ∈ J) ⊗ B

where F (dX_{i}|i ∈ J) is the free 𝕂-module on the set of formal symbols {dX_{i}|i ∈ J} as a basis.

We have to show that for every derivation D : B ⟶ M there exists a unique homomorphisms φ : Ω_{B} ⟶ M of B-B-bimodules such that the diagram

[figure: B; Ω_{B}; d; D; M; φ]

commutes. The module Ω_{B} is a B-B-bimodule in the canonical way. The products X_{1} … X_{n} of the generators X_{i} of B form a basis for B. For any product X_{1} … X_{n} we define d(X_{1} … X_{n}) := ∑_{i=1}^{n} X_{1} … X_{i−1} ⊗ dX_{i} ⊗ X_{i+1} … X_{n} in particular d(X_{i}) = 1⊗dX_{i} ⊗1. To see that d is a derivation it suffices to show this on the basis elements:

d(X_{1} … X_{k}X_{k+1} … X_{n})
= ∑_{j=1}^{k} X_{1} … X_{j−1} ⊗ dX_{j} ⊗ X_{j+1} … X_{k}X_{k+1} … X_{n}
+ ∑_{j=k+1}^{n} X_{1} … X_{k}X_{k+1} … X_{j−1} ⊗ dX_{j} ⊗ X_{j+1} … X_{n}
= d(X_{1} … X_{k})X_{k+1} … X_{n} + X_{1} … X_{k}d(X_{k+1} … X_{n})

Now let D : B ⟶ M be a derivation. Define φ by φ(1 ⊗ dX_{i} ⊗ 1) := D(X_{i}). This map obviously extends to a homomorphism of B-B-bimodules. Furthermore we have

\begin{aligned}φd(X_{1} … X_{n}) & = φ(∑_{j} X_{1} … X_{j−1} ⊗ dX_{j} ⊗ X_{j+1} … X_{n}) \\ & = ∑_{j} X_{1} … X_{j−1}φ(1 ⊗ dX_{j} ⊗ 1)X_{j+1} … X_{n} = D(X_{1} … X_{n}) \end{aligned}

hence φd = D.

To show the uniqueness of φ let ψ : Ω_{B} ⟶ M be a bimodule homomorphism such that ψd = D. Then ψ(1 ⊗ dX_{i} ⊗ 1) = ψd(X_{i}) = D(X_{i}) = φ(1 ⊗ dX_{i} ⊗ 1). Since ψ and φ are B-B-bimodules homomorphisms this extends to ψ = φ.

b) Now let A := 𝕂⟨X_{i}|i ∈ J⟩/I be an arbitrary algebra with B = 𝕂⟨X_{i}|i ∈ J⟩ free. Define

Ω_{A} := Ω_{B}/(IΩ_{B} + Ω_{B}I + Bd_{B}(I) + d_{B}(I)B).

We first show that IΩ_{B}+Ω_{B}I+Bd_{B}(I)+d_{B}(I)B is a B-B-subbimodule. Since Ω_{B} and I are B-B-bimodules the terms IΩ_{B} and Ω_{B}I are bimodules. Furthermore we have bd_{B}(i)b′ = bd_{B}(ib′) − bid_{B}(b′) ∈ Bd_{B}(I) + IΩ_{B} hence IΩ_{B} + Ω_{B}I + Bd_{B}(I) + d_{B}(I)B is a bimodule.

Now IΩ_{B} and Ω_{B}I are subbimodules of IΩ_{B} + Ω_{B}I + Bd_{B}(I) + d_{B}(I)B. Hence A = B/I acts on both sides on Ω_{A} so that Ω_{A} becomes an A-A-bimodule.

Let ν : Ω_{B} ⟶ Ω_{A} and also ν : B ⟶ A be the residue homomorphisms. Since νd_{B}(i) ∈ νd_{B}(I) = 0 ⊆ Ω_{A} we get a unique factorization map d_{A} : A ⟶ Ω_{A} such that

[figure: A; Ω_{A}; d_{A}; B; Ω_{B}; d_{B}; ν; ν]

commutes. Since d_{A}(\overline{b}) = \overline{d_{B}(b)} it is clear that d_{A} is a derivation.

Let D : A ⟶ M be a derivation. The A-A-bimodule M is also a B-B-bimodule by bm = \overline{b}m. Furthermore Dν : B ⟶ A ⟶ M is again a derivation. Let φ_{B} : Ω_{B} ⟶ M be the unique factorization map for the B-derivation Dν. Consider the following diagram

[figure: B; Ω_{B}; d_{B}; A; Ω_{A}; d_{A}; ψ; φ; M; D]

We want to construct ψ such that the diagram commutes. Let iω ∈ IΩ_{B}. Then φ(iω) = \overline{i}φ(ω) = 0 and similarly φ(ωi) = 0. Let bd_{B}(i) ∈ Bd_{B}(I) then φ(bd_{B}(i)) = \overline{b}φd_{B}(i) = \overline{b}D(\overline{i}) = 0 and similarly φ(d_{B}(i)b) = 0. Hence φ vanishes on IΩ_{B} + Ω_{B}I + Bd_{B}(I)+d_{B}(I)B and thus factorizes through a unique map ψ : Ω_{A} ⟶ M. Obviously ψ is a homomorphism of A-A-bimodules. Furthermore we have Dν = φd_{B} = ψνd_{B} = ψd_{A}ν and, since ν is surjective, D = ψd_{A}. It is clear that ψ is uniquely determined by this condition.

2. If A is commutative then we can write A = 𝕂[X_{i}|i ∈ J]/I and Ω_{B}^{c} = B ⊗ F (dX_{i}). With Ω_{A}^{c} = Ω_{B}^{c}/(IΩ_{B}^{c} + Bd_{B}(I)) the proof is analogous to the proof in the noncommutative situation. □

Remark 4.2.3. 1. Ω_{A} is generated by d(A) as a bimodule, hence all elements are of the form ∑_{i} a_{i}d(a′_{i})a′′_{i}. These elements are called differentials.

2. If A = 𝕂⟨X_{i}⟩/I, then Ω_{A} is generated as a bimodule by the elements \{\overline{d(X_{i})}\}.

3. Let f ∈ B = 𝕂⟨X_{i}⟩. Let B^{op} be the algebra opposite to B (with opposite multiplication). Then Ω_{B} = B ⊗ F (dX_{i}) ⊗ B is the free B ⊗ B^{op} left module over the free generating set {d(X_{i})}. Hence d(f) has a unique representation

d(f) = ∑_{i} \frac{∂f}{∂X_{i}} d(X_{i})

with uniquely defined coefficients

\frac{∂f}{∂X_{i}} ∈ B ⊗ B^{op}.

In the commutative situation we have unique coefficients

\frac{∂f}{∂X_{i}} ∈ 𝕂[X_{i}].

4. We give the following examples for part 3:

\frac{∂X_{i}}{∂X_{j}} = δ_{ij},

\frac{∂X_{1}X_{2}}{∂X_{1}} = 1 ⊗ X_{2},

\frac{∂X_{1}X_{2}}{∂X_{2}} = X_{1} ⊗ 1,

\frac{∂X_{1}X_{2}X_{3}}{∂X_{2}} = X_{1} ⊗ X_{3},

\frac{∂X_{1}X_{3}X_{2}}{∂X_{2}} = X_{1}X_{3} ⊗ 1.

This is obtained by direct calculation or by the product rule

\frac{∂fg}{∂X_{i}} = (1 ⊗ g)\frac{∂f}{∂X_{i}} + (f ⊗ 1)\frac{∂g}{∂X_{i}} .

The product rule follows from

d(fg) = d(f)g + fd(g) = ∑((1 ⊗ g)\frac{∂f}{∂X_{i}} + (f ⊗ 1)\frac{∂g}{∂X_{i}})d(X_{i}).

Let A = 𝕂⟨X_{i}⟩/I. If f ∈ I then \overline{d(f)} = d_{A}(\overline{f}) = 0 hence

∑ \frac{∂f}{∂X_{i}} d_{A}(\overline{X_{i}}) = 0.

These are the defining relations for the A-A-bimodule Ω_{A} with the generators d_{A}(\overline{X_{i}}).

For motivation of the quantum group case we consider an affine algebraic group G with representing commutative Hopf algebra A. Recall that Hom(A, R) is an algebra with the convolution multiplication for every R ∈ 𝕂-cAlg and that G(R) = 𝕂-cAlg (A, R) ⊆ Hom(A, R) is a subgroup of the group of units of the algebra Hom(A, R).

Definition and Remark 4.2.4. A linear map T : A ⟶ A is called left translation invariant, if the following diagram functorial in R ∈ 𝕂-cAlg commutes:

[figure: G(R) × Hom(A, R); Hom(A, R); ∗; G(R) × Hom(A, R); Hom(A, R); ∗; 1⊗Hom(T,R); Hom(T,R)]

i. e. if we have

∀g ∈ G(R), ∀x ∈ Hom(A, R) : g ∗ (x ∘ T) = (g ∗ x) ∘ T.

This condition is equivalent to

(33)  Δ_{A} ∘ T = (1_{A} ⊗ T) ∘ Δ_{A}.

In fact if (33) holds then g ∗ (x ∘ T) = ∇_{R}(g ⊗ x)(1_{A} ⊗ T)Δ_{A} = ∇_{R}(g ⊗ x)Δ_{A}T = (g ∗ x) ∘ T.

Conversely if the diagram commutes, then take R = A, g = 1_{A} and we get ∇_{A}(1_{A} ⊗ x)(1_{A} ⊗ T)Δ_{A} = 1_{A} ∗ (x ∘ T) = (1_{A} ∗ x) ∘ T = ∇_{A}(1_{A} ⊗ x)Δ_{A}T for all x ∈ Hom(A, A). To get (33) it suffices to show that the terms ∇_{A}(1_{A}⊗x) can be cancelled in this equation. Let ∑_{i=1}^{n} a_{i}⊗b_{i} ∈ A⊗A be given such that ∇_{A}(1_{A}⊗x)(∑ a_{i}⊗b_{i}) = 0 for all x ∈ Hom(A, A) and choose such an element with a shortest representation (n minimal). Then ∑ a_{i}x(b_{i}) = 0 for all x. Since the b_{i} are linearly independent in such a shortest representation, there are x_{i} with x_{j}(b_{i}) = δ_{ij}. Hence a_{j} = ∑ a_{i}x_{j}(b_{i}) = 0 and thus ∑ a_{i} ⊗ b_{i} = 0. From this follows (33).

Definition 4.2.5. Let H be an arbitrary Hopf algebra. An element T ∈ Hom(H, H) is called left translation invariant if it satisfies

Δ_{H}T = (1_{H} ⊗ T)Δ_{H}.

Proposition 4.2.6. Let H be an arbitrary Hopf algebra. Then Φ : H^{∗} ⟶ End(H) with Φ(f) := id ∗u_{H}f is an algebra monomorphism satisfying

Φ(f ∗ g) = Φ(f) ∘ Φ(g).

The image of Φ is precisely the set of left translation invariant elements T ∈ End(H).

Proof. For f ∈ Hom(H, 𝕂) we have u_{H}f ∈ End(H) hence id ∗u_{H}f ∈ End(H). Thus Φ is a well defined homomorphism. Observe that

Φ(f)(a) = (id_{H} ∗ u_{H}f)(a) = ∑ a_{(1)}f(a_{(2)}).

Φ is injective since it has a retraction End(H) ∋ g ↦ ε_{H} ∘g ∈ Hom(H, 𝕂). In fact we have (εΦ(f))(a) = ε(∑ a_{(1)}f(a_{(2)})) = ∑ ε(a_{(1)})f(a_{(2)}) = f(∑ ε(a_{(1)})a_{(2)}) = f(a) hence εΦ(f) = f.

The map Φ preserves the algebra unit since Φ(1_{H^{∗}}) = Φ(ε_{H}) = id_{H} ∗u_{H}ε_{H} = id_{H}. The map Φ is compatible with the multiplication: Φ(f ∗ g)(a) = ∑ a_{(1)}(f ∗

g)(a_{(2)}) = ∑ a_{(1)}f(a_{(2)})g(a_{(3)}) = ∑(id ∗u_{H}f)(a_{(1)})g(a_{(2)}) = Φ(f)(∑ a_{(1)}g(a_{(2)})) = Φ(f)Φ(g)(a) so that Φ(f ∗ g) = Φ(f) ∘ Φ(g).

For each f ∈ H^{∗} the element Φ(f) is left translation invariant since ΔΦ(f)(a) = Δ(∑ a_{(1)}f(a_{(2)})) = ∑ a_{(1)} ⊗ a_{(2)}f(a_{(3)}) = (1 ⊗ Φ(f))Δ(a).

Let T ∈ End(H) be left translation invariant then S ∗T = ∇_{H}(S ⊗1)(1⊗T)Δ_{H} = ∇_{H}(S ⊗ 1)Δ_{H}T = u_{H}ε_{H}T. Thus Φ(εT) = id ∗u_{H}ε_{H}T = id ∗S ∗ T = T, so that T is in the image of Φ. □

Proposition 4.2.7. Let d ∈ Hom(H, 𝕂) and Φ(d) = D ∈ Hom(H, H) be given. The following are equivalent:

(1) d : H ⟶ _{ε}𝕂_{ε} is a derivation.

(2) D : H ⟶ _{H}H_{H} is a (left translation invariant) derivation. In particular Φ induces an isomorphism between the set of derivations d : H ⟶ _{ε}𝕂_{ε} and the set of left translation invariant derivations D : H ⟶ _{H}H_{H}.

Proof. Assume that 1. holds so that d satisfies d(ab) = ε(a)d(b) + d(a)ε(b). Then we get D(ab) = Φ(d)(ab) = ∑ a_{(1)}b_{(1)}d(a_{(2)}b_{(2)}) = ∑ a_{(1)}b_{(1)}ε(a_{(2)})d(b_{(2)}) + ∑ a_{(1)}b_{(1)}d(a_{(2)})ε(b_{(2)}) = aD(b) + D(a)b. Conversely assume that D(ab) = aD(b) + D(a)b. Then d(ab) = εD(ab) = ε(a)εD(b) + εD(a)ε(b) = ε(a)d(b) + d(a)ε(b). □

3. The Lie Algebra of Primitive Elements

Lemma 4.3.1. Let H be a Hopf algebra and H^{o} be its Sweedler dual. If d ∈ Der_{𝕂}(H,_{ε}𝕂_{ε}) ⊆ Hom(H, 𝕂) is a derivation then d is a primitive element of H^{o}. Furthermore every primitive element d ∈ H^{o} is a derivation in Der_{𝕂}(H,_{ε}𝕂_{ε}).

Proof. Let d : H ⟶ 𝕂 be a derivation and let a, b ∈ H. Then (b ⇀ d)(a) = d(ab) = ε(a)d(b) + d(a)ε(b) = (d(b)ε + ε(b)d)(a) hence (b ⇀ d) = d(b)ε + ε(b)d. Consequently we have Hd = (H ⇀ d) ⊆ 𝕂ε + 𝕂d so that dim Hd ≤ 2 < ∞. This shows d ∈ H^{o}. Furthermore we have ⟨Δd, a ⊗ b⟩ = ⟨d, ab⟩ = d(ab) = d(a)ε(b) + ε(a)d(b) = ⟨d ⊗ ε, a ⊗ b⟩ + ⟨ε ⊗ d, a ⊗ b⟩ = ⟨1_{H^{o}} ⊗ d + d ⊗ 1_{H^{o}}, a ⊗ b⟩ hence Δ(d) = d ⊗ 1_{H^{o}} + 1_{H^{o}} ⊗ d so that d is a primitive element in H^{o}.

Conversely let d ∈ H^{o} be primitive. then d(ab) = ⟨Δ(d), a ⊗ b⟩ = d(a)ε(b) + ε(a)d(b). □

Proposition and Definition 4.3.2. Let H be a Hopf algebra. The set of primitive elements of H will be denoted by Lie (H) and is a Lie algebra. If char(𝕂) = p > 0 then Lie (H) is a restricted Lie algebra or a p-Lie algebra.

Proof. Let a, b ∈ H be primitive elements. Then Δ([a, b]) = Δ(ab − ba) = (a⊗1+1⊗a)(b⊗1+1⊗b)−(b⊗1+1⊗b)(a⊗1+1⊗a) = (ab−ba)⊗1+1⊗(ab−ba) hence Lie (H) ⊆ H^{L} is a Lie algebra. If the characteristic of 𝕂 is p > 0 then we have (a ⊗ 1 + 1 ⊗ a)^{p} = a^{p} ⊗ 1 + 1 ⊗ a^{p}. Thus Lie (H) is a restricted Lie subalgebra of H^{L} with the structure maps [a, b] = ab − ba and a^{[p]} = a^{p}. □

Corollary 4.3.3. Let H be a Hopf algebra. Then the set of left translation invariant derivations D : H ⟶ H is a Lie algebra under [D, D′] = DD′ − D′D. If char = p then these derivations are a restricted Lie algebra with D^{[p]} = D^{p}.

Proof. The map Ψ : H^{o} ⟶ H^{∗} ⟶^{Φ} End(H) is a homomorphism of algebras by 4.2.6. Hence Ψ(d ∗ d′ − d′ ∗ d) = Φ(d ∗ d′ − d′ ∗ d) = Φ(d)Φ(d′) − Φ(d′)Φ(d). If d is a primitive element in H^{o} then by 4.2.7 and 4.3.1 the image D := Ψ(d) in End(H) is a left translation invariant derivation and all left translation invariant derivations are of this form. Since [d, d′] = d ∗ d′ − d′ ∗ d is again primitive we get that [D, D′] = DD′ − D′D is a left translation invariant derivation so that the set of left translation invariant derivations Der_{𝕂}^{H}(H, H) is a Lie algebra resp. a restricted Lie algebra. □

Definition 4.3.4. Let H be a Hopf algebra. An element c ∈ H is called cocommutative if τΔ(c) = Δ(c), i. e. if ∑ c_{(1)} ⊗ c_{(2)} = ∑ c_{(2)} ⊗ c_{(1)}. Let C(H) := {c ∈ H|c is cocommutative }.

Let G(H) denote the set of grouplike elements of H.

Lemma 4.3.5. Let H be a Hopf algebra. Then the set of cocommutative elements C(H) is a subalgebra of H and the grouplike elements G(H) form a linearly independent subset of C(H). Furthermore G(H) is a multiplicative subgroup of the group of units U(C(H)).

Proof. It is clear that C(H) is a linear subspace of H. If a, b ∈ C(H) then Δ(ab) = Δ(a)Δ(b) = (τΔ)(a)(τΔ)(b) = τ(Δ(a)Δ(b)) = τΔ(ab) and Δ(1) = 1 ⊗ 1 = τΔ(1). Thus C(H) is a subalgebra of H.

The grouplike elements obviously are cocommutative and form a multiplicative group, hence a subgroup of U(C(H)). They are linearly independent by Lemma 2.1.14. □

Proposition 4.3.6. Let H be a Hopf algebra with S^{2} = id_{H}. Then there is a left module structure

C(H) ⊗ Lie (H) ∋ c ⊗ a ↦ c · a ∈ Lie (H)

with c · a := ∇_{H}(∇_{H} ⊗ 1)(1 ⊗ τ)(1 ⊗ S ⊗ 1)(Δ ⊗ 1)(c ⊗ a) = ∑ c_{(1)}aS(c_{(2)}) such that

c · [a, b] = ∑[c_{(1)} · a, c_{(2)} · b].

In particular G(H) acts by Lie automorphisms on Lie (H).

Proof. The given action is actually the action H ⊗ H ⟶ H with h · a = ∑ h_{(1)}aS(h_{(2)}), the so-called adjoint action.

We first show that the given map has image in Lie (H). For c ∈ C(H) and a ∈ Lie (H) we have Δ(c · a) = Δ(∑ c_{(1)}aS(c_{(2)})) = ∑ Δ(c_{(1)})(a ⊗ 1 + 1 ⊗ a)Δ(S(c_{(2)})) = ∑ Δ(c_{(1)})(a⊗1)Δ(S(c_{(2)}))+∑ Δ(c_{(2)})(1⊗a)Δ(S(c_{(1)})) = ∑ c_{(1)}aS(c_{(4)})⊗c_{(2)}S(c_{(3)})+ ∑ c_{(3)}S(c_{(2)}) ⊗ c_{(4)}aS(c_{(1)}) = c · a ⊗ 1 + 1 ⊗ c · a since c is cocommutative, S^{2} = id_{H} and a is primitive.

We show now that Lie (H) is a C(H)-module. (cd) · a = ∑ c_{(1)}d_{(1)}aS(c_{(2)}d_{(2)}) = ∑ c_{(1)}d_{(1)}aS(d_{(2)})S(c_{(2)}) = c · (d · a). Furthermore we have 1 · a = 1aS(1) = a.

To show the given formula let a, b ∈ Lie (H) and c ∈ C(H). Then c · [a, b] = ∑ c_{(1)}(ab−ba)S(c_{(2)}) = ∑ c_{(1)}aS(c_{(2)})c_{(3)}bS(c_{(4)})−∑ c_{(1)}bS(c_{(2)})c_{(3)}aS(c_{(4)}) = ∑(c_{(1)}· a)(c_{(2)}·b)−∑(c_{(1)}·b)(c_{(2)}·a) = ∑[c_{(1)}·a, c_{(2)}·b] again since c ∈ C(H) is cocommutative.

Now let g ∈ G(H). Then g · a = gaS(g) = gag^{−1} since S(g) = g^{−1} for any grouplike element. Furthermore g · [a, b] = [g · a, g · b] hence g defines a Lie algebra automorphism of Lie (H). □

Problem 4.3.31. Show that the adjoint action H⊗H ∋ h⊗a ↦ ∑ h_{(1)}aS(h_{(2)}) ∈ H makes H an H-module algebra.

Definition and Remark 4.3.7. The algebra 𝕂(δ) = 𝕂[δ]/(δ^{2}) is called the algebra of dual numbers. Observe that 𝕂(δ) = 𝕂 ⊕ 𝕂δ as a 𝕂-module.

We consider δ as a ”small quantity“ whose square vanishes.

The maps p : 𝕂(δ) ⟶ K with p(δ) = 0 and j : 𝕂 ⟶ 𝕂(δ) are algebra homomorphism satisfying pj = id.

Let 𝕂(δ, δ′) := 𝕂[δ, δ′]/(δ^{2}, δ′^{2}). Then 𝕂(δ, δ′) = 𝕂 ⊕ 𝕂δ ⊕ 𝕂δ′ ⊕ 𝕂δδ′. The map 𝕂(δ) ∋ δ ↦ δδ′ ∈ 𝕂(δ, δ′) is an injective algebra homomorphism. Furthermore for every α ∈ 𝕂 we have an algebra homomorphism φ_{α} : 𝕂(δ) ∋ δ ↦ αδ ∈ 𝕂(δ).

These algebra homomorphisms induce algebra homomorphisms H ⊗ 𝕂(δ) ⟶ H ⊗ 𝕂(δ) resp. H ⊗ 𝕂(δ) ⟶ H ⊗ 𝕂(δ, δ′) for every Hopf algebra H.

Proposition 4.3.8. The map

e^{δ-} : Lie (H) ⟶ H ⊗ 𝕂(δ) ⊆ H ⊗ 𝕂(δ, δ′)

with e^{δa} := 1 + a ⊗ δ = 1 + δa is called the exponential map and satisfies e^{δ(a+b)} = e^{δa}e^{δb},

e^{δαa} = φ_{α}(e^{δa}),
e^{δδ′[a,b]} = e^{δa}e^{δ′b}(e^{δa})^{−1}(e^{δ′b})^{−1}.

Furthermore all elements e^{δa} ∈ H ⊗ 𝕂(δ) are grouplike in the 𝕂(δ)-Hopf algebra H ⊗ 𝕂(δ).

Proof. 1. e^{δ(a+b)} = (1 + δ(a + b)) = (1 + δa)(1 + δb) = e^{δa}e^{δb}.

2. e^{δαa} = 1 + δαa = φ_{α}(1 + δa) = φ_{α}(e^{δa}).

3. Since (1+δa)(1−δa) = 1 we have (e^{δa}) = 1−δa. So we get e^{δδ′[a,b]} = 1+δ[a, b] = 1 + δ(a − a) + δ′(b − b) + δδ′(ab − ab − ba + ab) = (1 + δa)(1 + δ′b)(1 − δa)(1 − δ′b) = e^{δa}e^{δ′b}(e^{δa})^{−1}(e^{δ′b})^{−1}.

4. Δ_{𝕂(δ)}(e^{δa}) = Δ(1 + a ⊗ δ) = 1 ⊗_{𝕂(δ)} 1 + (a ⊗ 1 + 1 ⊗ a) ⊗ δ = 1 ⊗_{𝕂(δ)} 1 + δa ⊗_{𝕂(δ)} 1 + 1 ⊗_{𝕂(δ)} δa + δa ⊗_{𝕂(δ)} δa = (1 + δa) ⊗_{𝕂(δ)} (1 + δa) = e^{δa} ⊗_{𝕂(δ)} e^{δa} and ε_{𝕂(δ)}(e^{δa}) = ε_{𝕂(δ)}(1 + δa) = 1 + δε(a) = 1. □

Corollary 4.3.9. (Lie (H), e^{δ-}) is the kernel of the group homomorphism p : G_{𝕂(δ)}(H ⊗ 𝕂(δ)) ⟶ G(H).

Proof. p = 1 ⊗ p : H ⊗ 𝕂(δ) ⟶ H ⊗ 𝕂 = H is a homomorphism of 𝕂-algebras. We show that it preserves grouplike elements. Observe that grouplike elements in H ⊗ 𝕂(δ) are defined by the Hopf algebra structure over 𝕂(δ). Let g ∈ G_{𝕂(δ)}(H ⊗ 𝕂(δ)). Then (Δ_{H} ⊗ 1)(g) = g ⊗_{𝕂(δ)} g and (ε_{H} ⊗ 1)(g) = 1 ∈ 𝕂(δ).

Since p : 𝕂(δ) ⟶ 𝕂 is an algebra homomorphism the following diagram commutes

[figure: (H ⊗ 𝕂) ⊗ (H ⊗ 𝕂); H ⊗ H ⊗ 𝕂.; ≅; (H ⊗ 𝕂(δ)) ⊗_{𝕂(δ)} (H ⊗ 𝕂(δ)); H ⊗ H ⊗ 𝕂(δ); ≅; (1⊗p)⊗(1⊗p); 1⊗p]

We identify elements along the isomorphisms. Thus we get (Δ_{H} ⊗ 1_{𝕂})(1_{H} ⊗ p)(g) = (1_{H⊗H} ⊗ p)(Δ_{H} ⊗ 1_{𝕂(δ)})(g) = ((1_{H} ⊗ p) ⊗_{𝕂(δ)} (1_{H} ⊗ p))(g ⊗_{𝕂(δ)} g) = (1_{H} ⊗ p)(g) ⊗ (1_{H} ⊗ p)(g), so that 1_{H} ⊗ p : G_{𝕂(δ)}(H ⊗ 𝕂(δ)) ⟶ G(H). Now we have (1_{H} ⊗ p)(gg′) = (1_{H} ⊗ p)(g)(1_{H} ⊗ p)(g′) so that 1_{H} ⊗ p is a group homomorphism.

Now let g = g_{0} ⊗ 1 + g_{1} ⊗ δ ∈ G_{𝕂(δ)}(H ⊗ 𝕂(δ)) ⊆ H ⊗ 𝕂 ⊕ H ⊗ 𝕂δ. Then we have (1_{H} ⊗ p)(g) = 1 iff g_{0} = 1 iff g = 1_{H} ⊗ 1_{𝕂(δ)} + g_{1} ⊗ δ. Furthermore we have

Δ_{H⊗𝕂(δ)}(g) = g ⊗_{𝕂(δ)} g ⟺

1_{H} ⊗ 1_{H} ⊗ 1_{𝕂(δ)} + Δ_{H}(g_{1}) ⊗ δ = (1_{H} ⊗ 1_{𝕂(δ)} + g_{1} ⊗ δ) ⊗_{𝕂(δ)} (1_{H} ⊗ 1_{𝕂(δ)} + g_{1} ⊗ δ)

= 1_{H} ⊗ 1_{H} ⊗ 1_{𝕂(δ)} + (g_{1} ⊗ 1_{H} + 1_{H} ⊗ g_{1}) ⊗ δ ⟺

Δ_{H}(g_{1}) = g_{1} ⊗ 1_{H} + 1_{H} ⊗ g_{1}.

Similarly we have ε_{𝕂(δ)})(g) = 1 iff 1 ⊗ 1 + ε(g_{1}) ⊗ δ = 1 iff ε(g_{1}) = 0. □

4. Derivations and Lie Algebras of Affine Algebraic Groups

Lemma and Definition 7.4.1. Let G : 𝕂-cAlg ⟶ Set be a group valued functor. The kernel Lie(G)(R) of the sequence

[figure: 0; Lie(G)(R); G(R(δ)); G(p); G(j); G(R); 0]

is called the Lie algebra of G and is a group valued functor in R.

Proof. For every algebra homomorphism f : R ⟶ S the following diagram of groups commutes

[figure: 0; Lie(G)(R); G(R(δ)); G(p); G(j); G(R); 0; G(f(δ)); G(f); 0; Lie(G)(S); G(S(δ)); G(p); G(j); G(S); 0]
□

Proposition 4.4.2. Let G : 𝕂-cAlg ⟶ Set be a group valued functor with multiplication ∗. Then there are functorial operations

G(R) × Lie(G)(R) ∋ (g, x) ↦ g · x ∈ Lie(G)(R)
R × Lie(G)(R) ∋ (a, x) ↦ ax ∈ Lie(G)(R)

such that

g · (x + y) = g · x + g · y,
h · (g · x) = (h ∗ g) · x,
a(x + y) = ax + ay,
(ab)x = a(bx),
g · (ax) = a(g · x).

Proof. First observe that the composition + on Lie(G)(R) is induced by the multiplication ∗ of G(R(δ)) so it is not necessarily commutative.

We define g·x := G(j)(g)∗x∗G(j)(g)^{−1}. Then G(p)(g·x) = G(p)G(j)(g)∗G(p)(x)∗ G(p)G(j)(g)^{−1} = g ∗ 1 ∗ g^{−1} = 1 hence g · x ∈ Lie(G)(R).

Now let a ∈ R. To define a : Lie(G)(R) ⟶ Lie(G)(R) we use u_{a} : R(δ) ⟶ R(δ) defined by u_{a}(δ) := aδ and thus u_{a}(b+cδ) := b+acδ. Obviously u_{a} is a homomorphism of R-algebras. Furthermore we have pu_{a} = p and u_{a}j = j. Thus we get a commutative diagram

[figure: 0; Lie(G)(R); G(R(δ)); G(p); G(j); G(R); 0; a; G(u_{a}); id; 0; Lie(G)(R); G(R(δ)); G(p); G(j); G(R); 0]

that defines a group homomorphism a : Lie(G)(R) ⟶ Lie(G)(R) on the kernel of the exact sequences. In particular we have then a(x + y) = ax + ay.

Furthermore we have u_{ab} = u_{a}u_{b} hence (ab)x = a(bx).

The next formula follows from g · (x + y) = G(j)(g) ∗ x ∗ y ∗ G(j)(g)^{−1} = G(j)(g) ∗ x ∗ G(j)(g)^{−1} ∗ G(j)(g) ∗ y ∗ G(j)(g)^{−1} = g · x + g · y.

We also see (h∗g)·x = G(j)(h∗g)∗x∗G(j)(h∗g)^{−1} = G(j)(h)∗G(j)(g)∗x∗G(j)(g)^{−1}∗ G(j)(h)^{−1} = h · (g · x). Finally we have g · (ax) = G(j)(g) ∗ G(u_{a})(x) ∗ G(j)(g^{−1}) = G(u_{a})(G(j)(g) ∗ x ∗ G(j)(g^{−1})) = a(g · x). □

Proposition 4.4.3. Let G = 𝕂-cAlg (H, -) be an affine algebraic group. Then Lie(G)(𝕂) ≅ Lie (H^{o}) as additive groups. The isomorphism is compatible with the operations given in 4.4.2 and 4.3.6.

Proof. We consider the following diagram

[figure: 0; Lie(G)(𝕂); 𝕂-cAlg (H, 𝕂(δ)); p; 𝕂-cAlg (H, 𝕂); 0; e; ω; ≅; 0; Lie (H^{o}); G_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)); e^{δ-}; p; G(H^{o}); 0]

We know by definition that the top sequence is exact. The bottom sequence is exact by Corollary 4.3.9.

Let f ∈ 𝕂-cAlg (H, 𝕂). Since Ker(f) is an ideal of codimension 1 we get f ∈ H^{o}. The map f is an algebra homomorphism iff ⟨f, ab⟩ = ⟨f ⊗ f, a ⊗ b⟩ and ⟨f, 1⟩ = 1 iff Δ_{H^{o}}(f) = f ⊗ f and ε_{H^{o}}(f) = 1 iff f ∈ G(H^{o}). Hence we get the right hand vertical isomorphism 𝕂-cAlg (H, 𝕂) ≅ G(H^{o}).

Consider an element f ∈ 𝕂-cAlg (H, 𝕂(δ)) ⊆ Hom(H, 𝕂(δ)). It can be written as f = f_{0}+f_{1}δ with f_{0}, f_{1} ∈ Hom(H, 𝕂). The linear map f is an algebra homomorphism iff f_{0} : H ⟶ 𝕂 is an algebra homomorphism and f_{1} satisfies f_{1}(1) = 0 and f_{1}(ab) = f_{0}(a)f_{1}(b) + f_{1}(a)f_{0}(b). In fact we have f(1) = f_{0}(1) + f_{1}(1)δ = 1 iff f_{0}(1) = 1 and f_{1}(1) = 0 (by comparing coefficients). Furthermore we have f(ab) = f(a)f(b) iff f_{0}(ab) + f_{1}(ab)δ = (f_{0}(a) + f_{1}(a)δ)(f_{0}(b) + f_{1}(b)δ) = f_{0}(a)f_{0}(b) + f_{0}(a)f_{1}(b)δ + f_{1}(a)f_{0}(b)δ iff f_{0}(ab) = f_{0}(a)f_{0}(b) and f_{1}(ab) = f_{0}(a)f_{1}(b) + f_{1}(a)f_{0}(b).

Since f_{0} is an algebra homomorphism we have as above f_{0} ∈ H^{o}. For f_{1} we have (b ⇀ f_{1})(a) = f_{1}(ab) = f_{0}(a)f_{1}(b) + f_{1}(a)f_{0}(b) = (f_{1}(b)f_{0} + f_{0}(b)f_{1})(a) hence (b ⇀ f_{1}) = f_{1}(b)f_{0} +f_{0}(b)f_{1} ∈ 𝕂f_{0} +𝕂f_{1}, a two dimensional subspace. Thus f_{1} ∈ H^{o}.

In the following computations we will identify (H^{o} ⊗ 𝕂(δ)) ⊗_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)) with H^{o} ⊗ H^{o} ⊗ |K(δ).

Let f = f_{0} + f_{1}δ = f_{0} ⊗ 1 + f_{1} ⊗ δ ∈ H^{o} ⊕ H^{o}δ = H^{o} ⊗ 𝕂(δ). Then f is a homomorphism of algebras iff f(ab) = f(a)f(b) and f(1) = 1 iff f_{0}(ab) = f_{0}(a)f_{0}(b) and f_{1}(ab) = f_{0}(a)f_{1}(b)+f_{1}(a)f_{0}(b) and f_{0}(1) = 1 and f_{1}(1) = 0 iff Δ_{H^{o}}(f_{0}) = f_{0} ⊗f_{0} and Δ_{H^{o}}(f_{1}) = f_{0}⊗f_{1}+f_{1}⊗f_{0} and ε_{H^{o}}(f_{0}) = 1 and ε_{H^{o}}(f_{1}) = 0 iff (Δ_{H^{o}} ⊗id_{𝕂(δ)})(f_{0}⊗ 1+f_{1} ⊗δ) = f_{0} ⊗f_{0} ⊗1+f_{0} ⊗f_{1} ⊗δ+f_{1} ⊗f_{0} ⊗δ = (f_{0} ⊗1+f_{1} ⊗δ)⊗_{𝕂(δ)}(f_{0} ⊗1+f_{1} ⊗δ) and (ε_{H^{o}} ⊗ id_{𝕂(δ)})(f_{0} ⊗ 1 + f_{1} ⊗ δ) = 1 ⊗ 1 iff (Δ_{H^{o}} ⊗ id_{𝕂(δ)})(f) = f ⊗_{𝕂(δ)} f and (ε_{H^{o}} ⊗ id_{𝕂(δ)})(f) = 1 iff f ∈ G_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)).

Hence we have a bijective map ω : 𝕂-cAlg (H, 𝕂(δ)) ∋ f = f_{0} + f_{1}δ ↦ f_{0} ⊗ 1 + f_{1} ⊗ δ ∈ G_{𝕂(δ)}(H^{o} ⊗ 𝕂(δ)). Since the group multiplication in 𝕂-cAlg (H, 𝕂(δ)) ⊆ Hom(H, 𝕂(δ)) is the convolution ∗ and the group multiplication in G_{𝕂(δ)}(H^{o}⊗𝕂(δ)) ⊆ H^{o} ⊗ 𝕂(δ) is the ordinary algebra multiplication, where the multiplication of H^{o} again is the convolution, it is clear that ω is a group homomorphism. Furthermore the right hand square of the above diagram commutes. Thus we get an isomorphism e : Lie (H^{o}) ⟶ Lie(G)(𝕂) on the kernels. This map is defined by e(d) = 1 + dδ ∈ 𝕂-cAlg (H, 𝕂(δ)).

To show that this isomorphism is compatible with the actions of 𝕂 resp. G(H^{o}) let α ∈ 𝕂, a ∈ H, and d ∈ Lie (H^{o}). We have e(αd)(a) = ε(a) + αd(a)δ = u_{α}(ε(a) + d(a)δ) = (u_{α} ∘ (1 + dδ))(a) = (u_{α} ∘ e(d))(a) = (αe(d))(a) hence e(αd) = αe(d).

Furthermore let g ∈ G(H^{o}) = 𝕂-cAlg (H, 𝕂), a ∈ H, and d ∈ Lie (H^{o}). Then we have e(g · d)(a) = e(gdg^{−1})(a) = (1 + gdg^{−1}δ)(a) = ε(a) + gdg^{−1}(a)δ = ∑ g(a_{(1)})ε(a_{(2)})gS(a_{(3)}) + ∑ g(a_{(1)})d(a_{(2)})gS(a_{(3)})δ = ∑ g(a_{(1)})e(d)(a_{(2)})gS(a_{(3)}) = (j ∘ g ∗ e(d) ∗ j ∘ g^{−1})(a) = (g · e(d))(a) hence e(g · d) = g · e(d). □

Proposition 4.4.4. Let H be a Hopf algebra and let I := Ker(ε). Then Der_{ε}(H, -) : Vec ⟶ Vec is representable by I/I^{2} and d : H ⟶^{1−ε} I ⟶^{ν} I/I^{2}, in particular

Der_{ε}(H, -) ≅ Hom(I/I^{2}, -)  and  Lie (H^{o}) ≅ Hom(I/I^{2}, 𝕂).

Proof. Because of ε(id −uε)(a) = ε(a) − εuε(a) = 0 we have Im(id −ε) ⊆ I. Let i ∈ I. Then we have i = i−ε(i) = (id −ε)(i) hence Im(id −ε) = Ker(ε). We have I^{2} ∋ (id −ε)(a)(id −ε)(b) = ab − ε(a)b − aε(b) + ε(a)ε(b) = (id −ε)(ab) − ε(a)(id −ε)(b) − (id −ε)(b). Hence we have in I/I^{2} the equation (id −ε)(ab) = ε(a)(id −ε)(b) + (id −ε)(a)ε(b) so that ν(id −ε) : H ⟶ I ⟶ I/I^{2} is an ε-derivation.

Now let D : H ⟶ M be an ε-derivation. Then D(1) = D(1 1) = 1D(1) + D(1)1 hence D(1) = 0. It follows D(a) = D(id −ε)(a). From ε(I) = 0 we get D(I^{2}) ⊆ ε(I)D(I) + D(I)ε(I) = 0 hence there is a unique factorization

[figure: H; I; id −ε; I/I^{2}; ν; D; D; M.; f]
□

Corollary 4.4.5. Let H be a Hopf algebra that is finitely generated a s an algebra. Then Lie (H^{o}) is finite dimensional.

Proof. Let H = 𝕂⟨a_{1}, …, a_{n}⟩. Since H = 𝕂 ⊕ I we can choose a_{1} = 1 and a_{2}, …, a_{n} ∈ I. Thus any element in i ∈ I can be written as ∑ α_{J}a_{j_{1}} … a_{j_{k}} so that I/I^{2} = 𝕂\overline{a_{2}} + … + \overline{a_{n}}. This gives the result. □

Proposition 4.4.6. Let H be a commutative Hopf algebra and _{H}M be an Hmodule. Then we have Ω_{H} ≅ H ⊗ I/I^{2} and d : H ⟶ H ⊗ I/I^{2} is given by d(a) = ∑ a_{(1)} ⊗ \overline{(id −ε)(a_{(2)})}.

Proof. Consider the algebra B := H ⊕ M with (a, m)(a′, m′) = (aa′, am′ + a′m). Let G = 𝕂-cAlg (H, -). Then we have G(B) ⊆ Hom(H, B) ≅ Hom(H, H) ⊕ Hom(H, M). An element (φ, D) ∈ Hom(H, B) is in G(B) iff (φ, D)(1) = (φ(1), D(1)) = (1, 0), hence φ(1) = 1 and D(1) = 0, and (φ(ab), D(ab)) = (φ, D)(ab) = (φ, D)(a)(φ, D)(b) = (φ(a), D(a))(φ(b), D(b)) = (φ(a)φ(b), φ(a)D(b) + D(a)φ(b), hence φ(ab) = φ(a)φ(b) and D(ab) = φ(a)D(b) + D(a)φ(b). So (φ, D) is in G(B) iff φ ∈ G(H) and D is a φ-derivation. The ∗-multiplication in Hom(H, B) is given by (φ, D) ∗ (φ′, D′) = (φ ∗ φ′, φ ∗ D′ + D ∗ φ′) by applying this to an element a ∈ H. Since (φ, 0) ∈ G(B) and (uε, D) ∈ G(B) for every ε-derivation D, there is a bijection Der_{ε}(H, M) ≅ {(uε, D_{ε}) ∈ G_{ε}(B)} ≅ {(1_{H}, D_{1}) ∈ G_{1}(B)} ≅ Der_{𝕂}(H, M) by (uε, D_{ε}) ↦ (1, 0) ∗ (uε, D_{ε}) = (1, 1 ∗ D_{ε}) ∈ G_{1}(B) with inverse map (1, D_{1}) ↦ (S, 0) ∗ (1, D_{1}) = (uε, S ∗ D_{1}) ∈ G_{ε}(B). Hence we have isomorphisms Der_{𝕂}(H, M) ≅ Der_{ε}(H, M) ≅ Hom(I/I^{2}, M) ≅ Hom_{H}(H ⊗ I/I^{2}, M).

The universal ε-derivation for vector spaces is \overline{id −ε} : A ⟶ I/I^{2}. The universal ε-derivation for H-modules is D_{ε}(a) = 1 ⊗ \overline{(id −ε)(a)} ∈ A ⊗ I/I^{2}. The universal 1-derivation for H-modules is 1 ∗ D_{ε} with (1 ∗ D_{ε})(a) = ∑ a_{(1)} ⊗ \overline{(id −ε)(a_{(2)})} ∈ A ⊗ I/I^{2}. □

Bibliography

[Advanced Algebra] Bodo Pareigis: Advanced Algebra. Lecture Notes WS 2001/02.