CHAPTER 3 Hopf Algebras, Algebraic, Formal, and Quantum Groups.

1. Hopf Algebras

Definition 3.1.1. A left Hopf algebra H is a bialgebra H together with a left antipode S : H ⟶ H, i.e. a linear map S such that the following diagram commutes:

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

A Hopf algebra is a left and right Hopf algebra.

Problems 3.1.2. 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 1.6.5). 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 3.1.3. Because of 3.1.2(3) 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-Hopfalg.

Proposition 3.1.4. 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 3.1.5. 1. Let V be a vector space and T (V) the tensor algebra over V. We have seen in 1.5.10 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 ⊗1)δ(v) = −v+v = 0 = ηε(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 1.5.17 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 ⊗ 1)δ(v) = −v + v = 0 = ηε(v), hence S(V) is a Hopf algebra by the preceding proposition.

Example 3.1.6. (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 the preceding proposition shows that 𝕂G is a Hopf algebra.

Example 3.1.7. (Universal Enveloping Algebras) A Lie algebra consists of a vector space 𝔤 together with a 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 (1)

is a Lie algebra denoted by A^{L}. This 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 map ι : 𝔤 ⟶ U(𝔤)^{L} is injective.

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 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 the preceding proposition shows 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.)

Problems 3.1.8. 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 set of primitive elements P (H) = {x ∈ H|Δ(x) = x ⊗ 1 + 1 ⊗ x} of a Hopf algebra H are a Lie subalgebra of H^{L}.

Proposition 3.1.9. 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 3.1.2.

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

Remark 3.1.11. 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 Dringeld 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 dimen-

sion.

(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 Sweed1er 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

Definition 3.2.1. 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.

Problems 3.2.2. 1) If a set Z together with a multiplication m : Z × Z ⟶ Z is a monoid, then the unit element e ∈ Z is uniquely determined. If it is a group then also the inverse i : Z ⟶ Z is uniquely determined. 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) If (G, m) is a group in C and i_{X} : G(X) ⟶ G(X) is the inverse, then i is a natural transformation. The Yoneda Lemma provides a morphism S : G ⟶ G such that i_{X} = Mor_{C}(X, S) = S(X) for all X ∈ C.

Proposition 3.2.3. 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]

commute where Δ is the morphism defined in 1.3.6. The multiplications are related by m_{X} = Mor_{C}(X, m) = m(X).

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

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).

Problems 3.2.4. 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 3.2.5. 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 3.2.6. 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 1.3.6. 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 3.2.7. 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.

Problems 3.2.8. 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 3.2.9. 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 1.8.2 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 3.2.10. 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 (1.5.4), but in 𝕂-cAlg we have

Proposition 3.3.1. 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 2.2.12.

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 3.3.2. An affine algebraic group is a cogroup in the category of 𝕂-cAlg of commutative algebras.

For an arbitrary affine algebraic group H we get by Corollary 3.2.10

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 3.2.10 lead to

Proposition 3.3.3. 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 3.3.4. An algebra H ∈ 𝕂-cAlg is 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 3.3.5. 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 3.3.6. 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 3.3.7. 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 3.3.8. Since U(A) has the structure of a (multiplicative) group the sets 𝕂-Alg(𝕂[x, x^{−1}], A) are also groups.

Lemma 3.3.9. 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 3.3.10. 1. The affine algebraic group called additive group

𝔾_{a} : 𝕂-cAlg ⟶ Ab

with 𝔾_{a}(A) := A^{+} from Lemma 3.3.5 is represented by the Hopf algebra 𝕂[x]. We determine comoltiplication, counit, and antipode.

By Corollary 3.2.10 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 3.3.7 is represented by the Hopf algebra 𝕂[x, x^{−1}] = 𝕂[x, y]/(xy − 1). We determine comoltiplication, counit, and antipode.

By Corollary 3.2.10 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 3.3.9). 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.

Problems 3.3.11. 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. 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 Problem 1.6.3.4. In fact this holds also for non-commutative 𝕂-algebras, but in 𝕂-cCoalg we have

Proposition 3.4.1. 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 3.4.2. 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 3.4.3. 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 3.2.9 lead to the proof of the proposition.

Remark 3.4.4. 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 3.4.5. 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 3.4.6. 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 3.4.7. 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 embed 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 3.4.8. Let H be a finite dimensional Hopf algebra. Then by 1.6.7 and 1.6.9 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-hopfalg ⟶ 𝕂-hopfalg 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 dual 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 to the category of finite commutative formal groups.

5. Integrals and Fourier Transforms

Assume for the rest of this chapter that 𝕂 is a field.

Lemma 3.5.1. 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 3.5.2. 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 group-like 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 3.5.3. Let C be a finite dimensional coalgebra with a group like 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 group-like 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 3.5.4. 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 3.5.5. In 3.4.8 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 3.5.6. 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 3.5.7. 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 3.5.8. Maschke’s Theorem has an extension to finite dimensional Hopf algebras: ε(∫) ≠ 0 iff H^{∗} is semisimple.

Corollary 3.5.9. 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 3.5.10. Let G be a finite group. Let H := Map(G, 𝕂) be the Hopf algebra defined by the 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

∫ := ∑_{x∈G} x ∈ H^{∗} = 𝕂G (2)

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^{∗}.

Problems 3.5.11. Describe the group valued functor 𝕂-cAlg(𝕂^{G}, −) in terms of sets and their group structure.

Definition and Remark 3.5.12. 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 3.5.10.

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 3.5.13. Let H be a finite dimensional Hopf algebra. Then the set Gr(H^{∗}) of group like 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 3.1.6.

Proposition 3.5.14. The Hopf algebra homomorphism φ : 𝕂Ĝ ⟶ 𝕂^{G} is bijective.

Proof. We give the proof by several lemmas.

Lemma 3.5.15. Any set of group like elements in a Hopf algebra H is linearly independent.

Proof. Assume there is a linearly dependent set {x_{0}, x_{1}, …, x_{n}} of group like 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 3.5.16. (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 3.5.17. (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 3.5.14.

Definition 3.5.18. 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⊗π⊗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 3.5.19. 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. Futhermore 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:

\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} (3)

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

(f ⇀ a) = ∑ a_{(1)}⟨a_{(2)}, f⟩. (4)

Analogously we have

(a ↼ f) = ∑⟨a_{(1)}, f⟩a_{(2)}. (5)

Proposition 3.5.20. 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 3.5.1 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 3.5.21. 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 3.5.20 H^{∗} is a right Hopf module over H. By Proposition 3.5.19 there is an isomorphism α : (H^{∗})^{coH} ⊗ H ∋ a ⊗ f ↦ (a ↽ f) = (S(f) ⇀ a) ∈ H^{∗}. Since (H^{∗})^{coH} ≅ Int_{l}(H^{∗}) by 3.5.9 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

⟨a, f⟩ = ∫ f(x)S(g)(x)dx = ∑_{x∈G} f(x)g(x^{−1}) (6)

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

⟨∫ , fg⟩ = ⟨∫ , gν(f)⟩ (7)

or

∫ f(x)g(x)dx = ∫ g(x)ν(f)(x)dx. (8)

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 3.5.22. The map ν : H ⟶ H is an algebra automorphism, called the Nakayama automorphism.

86 3. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS.

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 3.5.23. 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 3.5.24. 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 ∫ ^{∗} a(x)dx = ∫^{∗}a := ⟨a, δ⟩. (9)

Proposition 3.5.25..

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 3.5.26. Let H be a finite dimensional Hopf algebra. Then ∫ ∈ H^{∗} is a left integral iff a(∑ ∫_{(1)} ⊗ S(∫_{(2)})) = (∑ ∫_{(1)} ⊗ S(∫_{(2)}))a (10) iff ∑ S(a)∫_{(1)} ⊗ ∫_{(2)} = ∑ ∫_{(1)} ⊗ a∫_{(2)} (11) iff ∑ f_{(1)}⟨∫ , f_{(2)}⟩ = ⟨∫ , f⟩1_{H}. (12)

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 (12) 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

δ(x) = \begin{cases}0 if x ≠ e; \\ 1 if x = e. \end{cases} (13)

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

δ = |G|^{−1} ∑_{χ∈Ĝ} χ. (14)

Let H be finite dimensional for the rest of this section. In Corollary 5.23 we have seen that the map H ∋ f ↦ (∫ ↼ f) ∈ H^{∗} is an isomorphism. This map will be called the Fourier transform.

88 3. HOPF ALGEBRAS, ALGEBRAIC, FORMAL, AND QUANTUM GROUPS.

Theorem 3.5.27. The Fourier transform H ∋ f ↦ f̃ ∈ H^{∗} is bijective with f̃ = (∫ ↼ f) = ∑⟨∫_{(1)}, f⟩∫_{(2)} (15) The inverse Fourier transform is defined by ã = ∑ S^{−1}(δ_{(1)})⟨a, δ_{(2)}⟩. (16) Since these maps are inverses of each other the following formulas hold

\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} (17)

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 ⟨a, b̂⟩ = ⟨a, (b ⇀ δ)⟩ = ⟨ab, δ⟩ (18) and ⟨f̃, g⟩ = ⟨(∫ ↼ f), g⟩ = ⟨∫ , fg⟩ (19) 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 5.26 ) \\ & = ∑⟨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 (2) and (14) we get

\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} (20)

This implies

f̃(χ) = ∑_{x∈G} f(x)χ(x) = ∫ f(x)χ(x)dx (21)

with inverse transform

ã(x) = |G|^{−1} ∑_{χ∈Ĝ} χ(a)χ^{−1}(x). (22)

Corollary 3.5.28. The Fourier transforms of the left invariant integrals in H and H^{∗} are

δ̃ = εν^{−1} ∈ H^{∗}  and  ˜∫ = 1 ∈ H. (23)

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 3.5.29. Define a convolution multiplication on H^{∗} by

⟨a ∗ b, f⟩ := ∑⟨a, S^{−1}(δ_{(1)})f⟩⟨b, δ_{(2)}⟩.

Then the following transformation rule holds for f, g ∈ H:

f̃g = f̃∗ g̃. (24)

In particular H^{∗} with the convolution multiplication is an associative algebra with unit ˜1_{H} = ∫, i.e.

∫ ∗ a = a ∗ ∫ = a. (25)

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 (23) 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 3.5.30. (The Plancherel formula)

⟨a, f⟩ = ⟨f̃, ν(ã)⟩. (26)

Proof. First we have from (17)

⟨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 3.5.31. 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 3.5.32.

⟨a, f⟩ = ∫ ã(x)f(x)dx = ∫ ^{∗} S^{−1}(a)(x)f̃(x)dx. (27)

Proof. We have the rules for the Fourier transform. From (19) we get ⟨a, f⟩ = ⟨∫ , ãf⟩ = ∫ ã(x)f(x)dx and from (18) ⟨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 3.5.33. 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⟩.

6. Quantum Groups

Definition 3.6.1. (Drinfel’d) A quantum group is a noncommutative noncocommutative Hopf algebra.

Remark 3.6.2. 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 3.6.3. 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 3.6.4. 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 3.6.5. Let M_{q}(2) = 𝕂 ⟨(\begin{matrix}a & b \\ c & d \end{matrix})⟩ /I as in 2.3.6 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 arbitray 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

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}) .

(28)

In particular we have Δ(det_{q}) = det_{q} ⊗ det_{q} and ε(det_{q}) = 1. The quantum determinant is a group like element (see 3.5.13).

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 3.1.4. Symmetrically S is a right antipode. Thus the bialgebra SL_{q}(2) is a Hopf algebra or a quantum group.

Example 3.6.6. 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 3.6.7. Let M_{q}(2) be as in the example 3.6.5. 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 3.6.5 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 3.1.4. Symmetrically S is a right antipode. Thus the bialgebra GL_{q}(2) is a Hopf algebra or a quantum group.

6. QUANTUM GROUPS 95

To be continued