ON SYMBOLIC COMPUTATIONS

IN BRAIDED MONOIDAL CATEGORIES

BODO PAREIGIS

Abstract. There are some powerful notations and tools to perform computations in with tensors, the Sweedler notation for coalgebras, the Einstein convention to reduce the number of summation signs in computations with tensors, the Penrose notation that has been further developed by Joyal and Street to a graphic calculus in braided monoidal categories. In 1977 I introduced a method of computation that looks very much like computation with ordinary elements or tensors, but can be performed in arbitrary monoidal categories, by using a Yoneda Lemma like technique. In the dual of the category of vector spaces this allows to work with ordinary coalgebras as if they were algebras. I will show how to expand this technique to braided monoidal categories, and develop some of the general rules of computation. As an application I will derive the well known result that the antipode of a Hopf algebra in a braided monoidal category is an algebra antihomomorphism which is expressed by the formulas S(1) = 1 and S(ab) = ⟨S(b)S(a), τ⟩.

1. The Beginnings: The Sweedler-Heyneman notation

To describe the comultiplication of a 𝕂-coalgebra in terms of elements we introduce a notation first introduced by Sweedler and Heyneman [H-Sw62] similar to the notation ∇(a ⊗ b) = ab used for algebras. Instead of Δ(c) = ∑ c_{i} ⊗ c′_{i} we write

(1)  Δ(c) = ∑ c_{(1)} ⊗ c_{(2)}.

Observe that only the complete expression on the right hand side makes sense, not the components c_{(1)} or c_{(2)} which are not considered as families of elements of C. This notation alone does not help much in the calculations we have to perform later on. So we introduce a more general notation. Definition 1.1. (Sweedler Notation) Let M be an arbitrary 𝕂-module and C be a 𝕂-coalgebra. Then there is a bijection between all multilinear maps

f : C × … × C ⟶ M

and all linear maps

f^{♯} : C ⊗ … ⊗ C ⟶ M.

These maps are associated to each other by the formula

(2)  f(c_{1}, …, c_{n}) = f^{♯}(c_{1} ⊗ … ⊗ c_{n})

or

f = f^{♯} ∘ ⊗.

Date: February 3, 2004. 1991 Mathematics Subject Classification. Primary 16A10. \begin{matrix}c \\ ◯ \end{matrix}2004 MARCEL DEKKER, INC., 270 MADISON AVE., NEW YORK, NY 10016

This follows from the universal property of the tensor product. For c ∈ C we define

(3)  ∑ f(c_{(1)}, …, c_{(n)}) := f^{♯}(Δ^{n−1}(c)),

where Δ^{n−1} denotes the (n − 1)-fold application of Δ, for example Δ^{n−1} = (Δ ⊗ 1 ⊗ … ⊗ 1) ∘ … ∘ (Δ ⊗ 1) ∘ Δ.

In particular we obtain for the bilinear map ⊗ : C × C ∋ (c, d) ↦ c ⊗ d ∈ C ⊗ C (with associated identity map)

(4)  ∑ c_{(1)} ⊗ c_{(2)} = Δ(c),

and for the multilinear map ⊗^{2} : C × C × C ⟶ C ⊗ C ⊗ C

∑ c_{(1)} ⊗ c_{(2)} ⊗ c_{(3)} = (Δ ⊗ 1)Δ(c) = (1 ⊗ Δ)Δ(c).

With this notation one verifies easily

∑ c_{(1)} ⊗ … ⊗ Δ(c_{(i)}) ⊗ … ⊗ c_{(n)} = ∑ c_{(1)} ⊗ … ⊗ c_{(n+1)}

and

\begin{aligned}∑ c_{(1)} ⊗ … ⊗ ϵ(c_{(i)}) ⊗ … ⊗ c_{(n)} & = ∑ c_{(1)} ⊗ … ⊗ 1 ⊗ … ⊗ c_{(n−1)} \\ & = ∑ c_{(1)} ⊗ … ⊗ c_{(n−1)} \end{aligned}

This notation and its application to multilinear maps will also be used in more general contexts like comodules.

2. Symbolic Computations with tensors

Let C be a monoidal category. For objects A, X ∈ C define

A(X) := Mor_{C}(X, A).

We consider A as a “graded” or “variable” set with component A(X) of “degree” X. Actually A is a (representable) functor from C into Set .

Let f : A ⟶ B be a morphism in C. Then we get “maps of variable sets” written by abuse of notation as f : A(X) ⟶ B(X) with

(5)  f(a) := f ∘ a.

This defines a natural transformation and by the Yoneda Lemma there is a bijection between the morphisms from A to B and the natural transformations from the functor A to the functor B.

In particular two morphisms f, g : A ⟶ B are equal iff

∀X ∈ C, ∀a ∈ A(X) : f(a) = g(a).

Let A, B, C ∈ C. Then C(X ⊗ Y) is a functor in two variables X and Y. Furthermore A(X) × B(Y) is also a functor in two variables denoted by A × B. A natural transformation of functors in two variables f : A × B ⟶ C is called a bimorphism.

A special example of a bimorphism is

⊗ : A(X) × B(Y) ⟶ A ⊗ B(X ⊗ Y) with ⊗ (a, b) := a ⊗ b

where a ⊗ b : X ⊗ Y ⟶ A ⊗ B. An element a ⊗ b ∈ A ⊗ B(X ⊗ Y) coming from two morphisms a, b is called a decomposable tensor.

If f : A × B ⟶ C is a bimorphism and g : C ⟶ D is a morphism then gf : A × B ⟶ D is a bimorphism.

If f : A × B ⟶ C is a bimorphism and g : U ⟶ A and h : V ⟶ B are morphisms then f(g × h) : U × V ⟶ C is a bimorphism. Lemma 2.1. For each bimorphism f : A × B ⟶ C there is exactly one morphism f^{♯} : A ⊗ B ⟶ C such that

[figure: A × B; A ⊗ B; ⊗; C; f; f^{♯}]

commutes.

Proof. This uses a Yoneda Lemma type argument. For details see [Pa77, Lemma 1.1]. □

Occasionally if h = f^{♯} is given then we write the associated bimorphism as h^{♭} := h ∘ ⊗, so that (f^{♯})^{♭} = f and (h^{♭})^{♯} = h.

Given a bimorphism f = f^{♯} ∘ ⊗ and a ∈ A(X), b ∈ B(Y). Let t = a ⊗ b ∈ A ⊗ B(X ⊗ Y) be a decomposable tensor. Then f(a, b) = f^{♯}(a ⊗ b) = f^{♯}(t).

Similar remarks as above hold for multimorphisms f : A_{1} × … × A_{n} ⟶ C and associated morphisms f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ C. In particular we have for a_{i} ∈ A_{i}(X_{i}), i = 1, …, n and t = a_{1} ⊗ … ⊗ a_{n}

f(a_{1}, …, a_{n}) = f^{♯}(t).

We introduce a first symbolic expression for all t ∈ A_{1} ⊗ … ⊗ A_{n}(X) by

(6)  f(t_{1}, …, t_{n}) := f^{♯}(t).

Observe that t is not a decomposable tensor in general. We have, however:

For the multimorphism ⊗^{n−1} : A_{1} × … × A_{n} ⟶ A_{1} ⊗ … ⊗ A_{n} and the associated morphism ⊗^{♯} = id : A_{1} ⊗ … ⊗ A_{n} ⟶ A_{1} ⊗ … ⊗ A_{n} we get

(7)  t_{1} ⊗ … ⊗ t_{n} = t

for all “tensors” t ∈ A_{1} ⊗ … ⊗ A_{n}(X). In particular we have then

(8)  f(t_{1}, …, t_{n}) = f^{♯}(t_{1} ⊗ … ⊗ t_{n}).

Given f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B_{1} ⊗ … ⊗ B_{m} and t ∈ A_{1} ⊗ … ⊗ A_{n}(X). Then we may consider f^{♯}(t) as an element of B_{1} ⊗ … ⊗ B_{m}(X) hence

(9)  \begin{matrix}f^{♯}(t) = f^{♯}(t)_{1} ⊗ … ⊗ f^{♯}(t)_{m} = \\ = f(t_{1}, …, t_{n}) = f(t_{1}, …, t_{n})_{1} ⊗ … ⊗ f(t_{1}, …, t_{n})_{m}. \end{matrix}

Since f^{♯} is also an element in B_{1}⊗…⊗B_{m}(A_{1}⊗…⊗A_{n}) we can write f^{♯} = f_{1}^{♯}⊗…⊗f_{m}^{♯} and get

(10)  \begin{matrix}(f_{1}^{♯} ⊗ … ⊗ f_{m}^{♯})(t) = f^{♯}(t) = f^{♯}(t)_{1} ⊗ … ⊗ f^{♯}(t)_{m}  or \\ (f_{1}^{♯} ⊗ … ⊗ f_{m}^{♯})(t_{1} ⊗ … ⊗ t_{n}) = f(t_{1}, …, t_{n})_{1} ⊗ … ⊗ f(t_{1}, …, t_{n})_{m}. \end{matrix}

If in addition g^{♯} : B_{1} ⊗ … ⊗ B_{m} ⟶ C is given then we get

g(f^{♯}(t)_{1}, …, f^{♯}(t)_{m}) = g^{♯}f(t_{1}, …, t_{n}).

If f_{i} : A_{i} ⟶ B_{i}, i = 1, …, n, f^{♯} := f_{1} ⊗ … ⊗ f_{n}, and t ∈ A_{1} ⊗ … ⊗ A_{n}(X) are given then we have

f_{1}(t_{1}) ⊗ … ⊗ f_{n}(t_{n}) = f^{♯}(t) = f^{♯}(t)_{1} ⊗ … ⊗ f^{♯}(t)_{n}.

Observe, we do not admit the same notation for an arbitrary morphism f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B_{1} ⊗ … ⊗ B_{m}. The problem is that certain natural transformations will commute with morphisms of the form f_{1} ⊗ … ⊗ f_{n} : A_{1} ⊗ … ⊗ A_{n} ⟶ B_{1} ⊗ … ⊗ B_{n} but not with morphisms of the general form f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B_{1} ⊗ … ⊗ B_{m} even if m = n. Lemma 2.2. Given multimorphisms f, g : A_{1} × … × A_{n} ⟶ C with associated morphisms f^{♯}, g^{♯}. Then the following are equivalent:

(1) f^{♯} = g^{♯},

(2) ∀X_{i}, ∀a_{1}, …, a_{n} ∈ A_{i}(X_{i}) : f(a_{1}, …, a_{n}) = g(a_{1}, …, a_{n}),

(3) ∀X, ∀t ∈ A_{1} ⊗ … ⊗ A_{n}(X) : f(t_{1}, …, t_{n}) = g(t_{1}, …, t_{n}).

Proof. (1) ⟹ f = g ⟹ (3) ⟹ (2) ⟹ f = g ⟹ (1). □

This notation will be used to express and compute certain identities of morphisms. We explain this by the following example. Let (A, μ : A ⊗ A ⟶ A) be given. We want to express associativity by elements. Write ab := μ(a ⊗ b) ∈ A(X ⊗ Y). Then (ab)c = μ(μ(a ⊗ b) ⊗ c) ∈ A((X ⊗ Y) ⊗ Z). Similarly a(bc) = μ(a ⊗ μ(b ⊗ c)) ∈ A(X ⊗ (Y ⊗ Z)). In order to compare these two products we apply A(α) : A(X ⊗ (Y ⊗ Z)) ⟶ A((X ⊗ Y) ⊗ Z) to get (ab)c = a(bc) ∘ α = A(α)(a(bc)) iff (A, μ) is associative.

Since most such computations can be transferred to a strict monoidal category, we are going to assume from now on that C is a strict monoidal category.

Then (A, μ : A ⊗ A ⟶ A) is associative iff (ab)c = a(bc).

3. Braidings and tensors

Let C be a strict monoidal category that is braided. Let ρ ∈ B_{n} be a braid in the braid group with canonical image \overline{ρ} ∈ S_{n}. Let σ := \overline{ρ}^{−1}. Let ρ : A_{1} ⊗ … ⊗ A_{n} ⟶ A_{σ(1)} ⊗ … ⊗ A_{σ(n)} also denote the associated braid action on the n-fold tensor product. So ρ is a natural transformation of functors in n variables.

Let f^{♯} : A_{σ(1)}⊗…⊗A_{σ(n)} ⟶ B be a morphism in C and f := f^{♯}∘⊗^{n} : A_{σ(1)}(X_{σ(1)})× … × A_{σ(n)}(X_{σ(n)}) ⟶ B(X_{σ(1)} ⊗ … ⊗ X_{σ(n)}) be the associated multimorphism.

We want to study the application of f^{♯} on a tensor t ∈ A_{1} ⊗ … ⊗ A_{n}(X) if ρ is first applied to t. First assume that t is a decomposable tensor of the form t = a_{1} ⊗ … ⊗ a_{n} ∈ A_{1} ⊗ … ⊗ A_{n}(X_{1} ⊗ … ⊗ X_{n}) with a_{i} ∈ A_{i}(X_{i}). We get

(11)  f(ρ(t)_{1}, …, ρ(t)_{n}) = f(a_{σ(1)}, …, a_{σ(n)})ρ

since f(ρ(t)_{1}, …, ρ(t)_{n}) = f^{♯}ρ(t) = f^{♯}ρ(a_{1} ⊗ … ⊗ a_{n}) = f^{♯}(a_{σ(1)} ⊗ … ⊗ a_{σ(n)})ρ = f(a_{σ(1)}, …, a_{σ(n)})ρ where we used that ρ is a natural transformation.

Observe that in the symbolic notation ρ is not really applied to a_{1} ⊗ … ⊗ a_{n} as it is in ordinary computations in braided categories, it changes only the order of the components with σ ∈ S_{n}. We would only be interested in the expression f(a_{σ(1)}, …, a_{σ(n)}) in ordinary computations, but some information about ρ is lost, if we study this term in symbolic calculations. View ρ as an index for this expression and write

(12)  ⟨f(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ := f(a_{σ(1)}, …, a_{σ(n)})ρ.

In particular we have ⟨f(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ = f^{♯}(ρ(a_{1} ⊗ … ⊗ a_{n})).

We extend this notation to arbitrary tensors t = t_{1} ⊗ … ⊗ t_{n} ∈ A_{1} ⊗ … ⊗ A_{n}(X) (see equation (7)). Definition 3.1. We define the map

⟨., ., .⟩ : Nat(A_{σ(1)} × … × A_{σ(n)}, B) × (A_{1} ⊗ … ⊗ A_{n})(X) × B_{n} ⟶ B(X)

by ⟨f, t, ρ⟩ := f^{♯}(ρ(t)) = f^{♯}∘ρ∘t, where f : A_{σ(1)}×…×A_{σ(n)} ⟶ B is a multimorphism, t = t_{1} ⊗ … ⊗ t_{n} ∈ A_{1} ⊗ … ⊗ A_{n}(X) is a variable or argument, and ρ ∈ B_{n} is a braid. We write for ⟨f, t, ρ⟩ also ⟨f(t_{σ(1)}, …, t_{σ(n)}), ρ⟩ and define

(13)  ⟨f(t_{σ(1)}, …, t_{σ(n)}), ρ⟩ := f^{♯}(ρ(t)).

The expression f(t_{σ(1)}, …, t_{σ(n)}) taken separately is clearly not defined, except in the case where t is a decomposable tensor. Observe that t : X ⟶ A_{1} ⊗ … ⊗ A_{n} and f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B are morphisms so that ρ can operate on the range of t and on the domain of f^{♯}. We tacitly assume in writing down an expression ⟨f(t_{σ(1)}, …, t_{σ(n)}), ρ⟩ that the range and domain of f^{♯} and t are given and fixed. (Otherwise an operation of ρ would not be well defined.) If this separation is not quite clear we also use the notation

⟨f[t_{σ(1)}, …, t_{σ(n)}], ρ⟩ := f^{♯}(ρ(t)).

In some cases one has to name the arguments explicitly, which are used in a concrete computation. Theorem 3.2. (Comparison theorem): Given f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B and g^{♯} : A_{σ(1)} ⊗ … ⊗ A_{σ(n)} ⟶ B. Then the following are equivalent:

(1) g^{♯} ∘ ρ = f^{♯},

(2) ∀a_{1}, …, a_{n} ∈ A_{i}(X_{i}) : f(a_{1}, …, a_{n}) = ⟨g(a_{σ(1)}, …, a_{σ(n)}), ρ⟩,

(3) ∀t ∈ A_{1} ⊗ … ⊗ A_{n}(X) : f(t_{1}, …, t_{n}) = ⟨g(t_{σ(1)}, …, t_{σ(n)}), ρ⟩.

Proof. (1) ⟹ (3): f(t_{1}, …, t_{n}) = f^{♯}(t) = g^{♯}(ρ(t)) = ⟨g(t_{σ(1)}, …, t_{σ(n)}), ρ⟩.

(3) ⟹ (2): Take t := a_{1} ⊗ … ⊗ a_{n}. Then f(a_{1}, …, a_{n}) = f(t_{1}, …, t_{n}) = ⟨g(t_{σ(1)}, …, t_{σ(n)}), ρ⟩ = g^{♯}(ρ(t)) = g^{♯}(ρ(a_{1} ⊗ … ⊗ a_{n})) = ⟨g(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ as in equation (12).

(2) ⟹ (1): Take X_{i} = A_{i}, a_{i} = id_{i}. Then f(a_{1}, …, a_{n}) = ⟨g(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ implies f^{♯} = g^{♯} ∘ ρ. □

4. Rules of Computation

4.1. Special cases: With this symbolic notation we get the following rules of computation.

If ρ = id then

(14)  ⟨f(t_{1}, …, t_{n}), id⟩ = f^{♯}(t) = f(t_{1}, …, t_{n}).

So the identity braid in the pairing of our notation can be omitted.

If f^{♯} = id_{A_{σ(1)}⊗...⊗A_{σ(n)}} then we get

(15)  ⟨t_{σ(1)} ⊗ … ⊗ t_{σ(n)}, ρ⟩ = ρ(t) = ρ ∘ t.

4.2. Equality and Substitution: We begin with a warning. Usually certain terms in more complex expressions may be substituted by equal terms. However, separate components of the form f(t_{σ(1)}, …, t_{σ(n)}) in our expression ⟨f(t_{σ(1)}, …, t_{σ(n)}), ρ⟩ may not be replaced, even if it looks as if they could be equal.

For an example let indecomposable tensors a_{1} ⊗ a_{2}, b_{1} ⊗ b_{2} ∈ A ⊗ A(X) be given, and let m^{♯} : A ⊗ A ⟶ A be a multiplication. Assume m^{♯}(a_{1} ⊗ a_{2}) = a_{1}a_{2} = b_{1}b_{2} = m^{♯}(b_{1} ⊗ b_{2}). Then in general

⟨a_{1}a_{2}, τ^{2}⟩ ≠ ⟨b_{1}b_{2}, τ^{2}⟩.

We will find a certain replacement or substitution rule in (17). This expression, however, differs from (17) in that here we have “elements” (or a “function applied to specific elements”) whereas we have “functions” in (17). In terms of morphisms we may have

(X ⟶^{a} A ⊗ A ⟶^{m^{♯}} A) = (X ⟶^{b} A ⊗ A ⟶^{m^{♯}} A)

and at the same time

(X ⟶^{a} A ⊗ A ⟶^{τ^{2}} A ⊗ A ⟶^{m^{♯}} A) ≠ (X ⟶^{b} A ⊗ A ⟶^{τ^{2}} A ⊗ A ⟶^{m^{♯}} A).

If a = a_{1} ⊗ a_{2}, b = b_{1} ⊗ b_{2} are decomposable tensors then we have indeed

(X ⊗ X ⟶^{a} A ⊗ A ⟶^{τ^{2}} A ⊗ A ⟶^{m^{♯}} A) = (X ⊗ X ⟶^{b} A ⊗ A ⟶^{τ^{2}} A ⊗ A ⟶^{m^{♯}} A)

since τ^{2} is a natural transformation.

By the definition of ⟨., ., .⟩ we may certainly substitute equal expressions for the separate components f, ρ, and t. The following gives a somewhat more general rule for substitutions in case we have decomposable tensors as arguments. Proposition 4.1. Given f, g : A_{σ(1)} × … × A_{σ(1)} ⟶ B, ρ ∈ B and a_{i}, b_{i} ∈ A_{i}(X_{i}) defining decomposable tensors a = a_{1} ⊗ … ⊗ a_{n} and b = b_{1} ⊗ … ⊗ b_{n}. If

f(a_{σ(1)}, …, a_{σ(n)}) = g(b_{σ(1)}, …, b_{σ(n)})

then

⟨f(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ = ⟨g(b_{σ(1)}, …, b_{σ(n)}), ρ⟩.

Proof. This is a simple computation:

⟨f(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ = f^{♯} ∘ ρ ∘ a
= f^{♯} ∘ (a_{σ(1)} ⊗ … ⊗ a_{σ(n)}) ∘ ρ
= f(a_{σ(1)}, …, a_{σ(n)}) ∘ ρ
= g(b_{σ(1)}, …, b_{σ(n)}) ∘ ρ
= g^{♯} ∘ (b_{σ(1)} ⊗ … ⊗ b_{σ(n)}) ∘ ρ
= g^{♯} ∘ ρ ∘ b
= ⟨g(b_{σ(1)}, …, b_{σ(n)}), ρ⟩.
□

The proposition shows that ⟨f(a_{σ(1)}, …, a_{σ(n)}), ρ⟩ does indeed only depend on the value of f(a_{σ(1)}, …, a_{σ(n)}) whereas in general it depends separately on f and a. So we may replace the term f(a_{σ(1)}, …, a_{σ(n)}) by its value because the result does not depend on the particular representation.

If t = id : A_{1} ⊗ … ⊗ A_{n} ⟶ A_{1} ⊗ … ⊗ A_{n} then t = id_{A_{1}} ⊗ … ⊗ id_{A_{n}} is a decomposable tensor. In this case we may apply Proposition 4.1 and have f(id_{A_{σ(1)}}, …, id_{A_{σ(n)}}) alone in ⟨f(id_{A_{σ(1)}}, …, id_{A_{σ(n)}}), ρ⟩ is defined and we have f(id_{A_{σ(1)}}, …, id_{A_{σ(n)}}) = f^{♯}(id_{A_{σ(1)}} ⊗ … ⊗ id_{A_{σ(n)}}) = f^{♯} so that we we can write ⟨f^{♯}, ρ⟩ and get

(16)  ⟨f^{♯}, ρ⟩ = f^{♯} ∘ ρ.

Hence the expression ⟨f^{♯}, ρ⟩ makes sense without the argument t. The argument can safely be assumed to be t = id.

If f_{1}^{♯}, f_{2}^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B then we have

(17)  ⟨f_{1}^{♯}, ρ⟩ = ⟨f_{2}^{♯}, ρ⟩ ⟺ f_{1}^{♯} = f_{2}^{♯},

since ρ is an isomorphism, and

(18)  ⟨f_{1}^{♯}, ρ_{1}⟩ = ⟨f_{2}^{♯}, ρ_{2}⟩ ⟺ f_{1}^{♯} = ⟨f_{1}^{♯}, id⟩ = ⟨f_{2}^{♯}, ρ_{2}ρ_{1}^{−1}⟩.

4.3. Compatibility with elements of the braid group: If t = a_{1} ⊗ … ⊗ a_{n} with (a_{1}, …, a_{n}) ∈ A_{1}(X_{1}) × … × A_{n}(X_{n}) we get

(19)  a_{σ(1)} ⊗ … ⊗ a_{σ(n)} = ρ(a_{1} ⊗ … ⊗ a_{n})ρ^{−1}

since ρ is a natural transformation where the expression a_{σ(1)} ⊗ … ⊗ a_{σ(n)} is the morphism a_{σ(1)} ⊗ … ⊗ a_{σ(n)} : X_{σ(1)} ⊗ … ⊗ X_{σ(n)} ⟶ A_{σ(1)} ⊗ … ⊗ A_{σ(n)}.

If t ∈ A_{1}⊗…⊗A_{n}(X_{1}⊗…⊗X_{n}) then we can use equation (16) to define ⟨ρtρ^{−1}, ρ⟩ and get

⟨ρtρ^{−1}, ρ⟩ = ρtρ^{−1} ∘ ρ = ρ ∘ t = ⟨t_{σ(1)} ⊗ … ⊗ t_{σ(n)}, ρ⟩

In view of equations (17) and (19) we define for t ∈ A_{1} ⊗ … ⊗ A_{n}(X_{1} ⊗ … ⊗ X_{n})

(20)  t_{σ(1)} ⊗ … ⊗ t_{σ(n)} := ρtρ^{−1}.

We will write [ρ](t) := ρtρ^{−1} = t_{σ(1)} ⊗ … ⊗ t_{σ(n)} if t ∈ A_{1} ⊗ … ⊗ A_{n}(X_{1} ⊗ … ⊗ X_{n}). Then [ρ](a_{1} ⊗ … ⊗ a_{n}) = a_{σ(1)} ⊗ … ⊗ a_{σ(n)}.

4.4. Composition: Remark: The terms encountered in this section are complicated and usually have no simplification. They are given for completeness and will not be used in the sequal.

Certain of these expressions can be composed or applied to each other. In particular we get the following. Given ρ_{1}, ρ_{2} ∈ B_{n}. If f^{♯} : A_{σ_{2}σ_{1}(1)} ⊗ … ⊗ A_{σ_{2}σ_{1}(n)} ⟶ B and t ∈ A_{1} ⊗ … ⊗ A_{n}(X) then

(21)  ⟨f^{♯}, ρ_{1}⟩ (⟨t_{σ_{2}(1)} ⊗ … ⊗ t_{σ_{2}(n)}, ρ_{2}⟩) = ⟨f(t_{σ_{2}σ_{1}(1)}, …, t_{σ_{2}σ_{1}(n)}), ρ_{1}ρ_{2}⟩.

The following are immediately clear

(22)  ⟨f^{♯}, ρ_{1}ρ_{2}⟩ = ⟨⟨f^{♯}, ρ_{1}⟩, ρ_{2}⟩,
(23)  ⟨f_{1}^{♯}, ρ_{1}⟩ ⊗ ⟨f_{2}^{♯}, ρ_{2}⟩ = ⟨f_{1}^{♯} ⊗ f_{2}^{♯}, ρ_{1} ⊗ ρ_{2}⟩,

If t ∈ A_{1} ⊗ … ⊗ A_{n}(X_{1} ⊗ … ⊗ X_{n}) then

(24)  ⟨f^{♯}, ρ_{1}⟩ ∘ ⟨t, ρ_{2}⟩ = ⟨f^{♯} ∘ [ρ_{1}](t), ρ_{1} ∘ ρ_{2}⟩.

If f^{♯} : A_{σ(1)} ⊗ … ⊗ A_{σ(n)} ⟶ B and g : B ⟶ C are given then

(25)  g(⟨f(t_{σ(1)}, …, t_{σ(n)}), ρ⟩) = ⟨gf(t_{σ(1)}, …, t_{σ(n)}), ρ⟩.

If t ∈ A_{1} ⊗ … ⊗ A_{n}(X) then we get from (21)

(26)  f^{♯}(⟨t_{σ(1)} ⊗ … ⊗ t_{σ(n)}, ρ⟩) = ⟨f(t_{σ(1)}, …, t_{σ(n)}), ρ⟩.

The naturality of braids leads to a very usefull rule for a change of variables or of arguments. Given f_{i} : A_{i} ⟶ B_{i}, i = 1, …, n, g^{♯} : B_{1} ⊗ … ⊗ B_{n} ⟶ C, and t ∈ A_{1} ⊗ … ⊗ A_{n}(X). Then we get (27)

\begin{matrix}⟨g^{♯}(f_{σ(1)}(t_{σ(1)}) ⊗ … ⊗ f_{σ(n)}(t_{σ(n)})), ρ⟩ = ⟨g(f_{σ(1)}(t_{σ(1)}), …, f_{σ(n)}(t_{σ(n)})), ρ⟩ \\ = ⟨(g^{♯}(f_{σ(1)} ⊗ … ⊗ f_{σ(n)}))^{♭}(t_{σ(1)}, …, t_{σ(n)}), ρ⟩ = ⟨(g(f_{σ(1)}, …, f_{σ(n)}))(t_{σ(n)}, …, t_{σ(n)}), ρ⟩. \end{matrix}

or

⟨g[f_{σ(1)}(t_{σ(1)}), …, f_{σ(n)}(t_{σ(n)})], ρ⟩ = ⟨(g^{♯}(f_{σ(1)} ⊗ … ⊗ f_{σ(n)})⊗^{n})[t_{σ(1)}, …, t_{σ(n)}], ρ⟩.

where we change from the “arguments” f_{1}(t_{1}), …, f_{n}(t_{n}) to the “arguments” t_{1}, …, t_{n}.

More general terms for composition are obtained from f^{♯} : A_{1} ⊗ … ⊗ A_{n} ⟶ B_{1} ⊗ … ⊗ B_{m} as

⟨f(t_{σ(1)}, …, t_{σ(n)})_{1} ⊗ … ⊗ f(t_{σ(1)}, …, t_{σ(n)})_{m}, ρ⟩.

We get compositions of such terms which in general cannot be simplified. For g^{♯} : B_{1} ⊗ … ⊗ B_{m} ⟶ C we get (28)

\begin{matrix}⟨g(⟨f(t_{σ_{2}(1)}, …, t_{σ_{2}(n)})_{σ_{1}(1)}, …, f(t_{σ_{2}(1)}, …, t_{σ_{2}(n)})_{σ_{1}(m)}, ρ_{2}⟩), ρ_{1}⟩ = \\ ⟨g(f_{σ_{1}(1)}, …, f_{σ_{1}(m)}), ρ_{1}⟩ ∘ ⟨f^{♯}(t_{σ_{2}(1)}, …, t_{σ_{2}(n)})_{1} ⊗ … ⊗ f^{♯}(t_{σ_{2}(1)}, …, t_{σ_{2}(n)})_{m}, ρ_{2}⟩. \end{matrix}

Furthermore we may take tensor products of terms as follows:

(29)  \begin{matrix}⟨f(t_{σ_{1}(1)}, …, t_{σ_{1}(n)})_{1} ⊗ … ⊗ f(t_{σ_{1}(1)}, …, t_{σ_{1}(n)})_{m}, ρ_{1}⟩⊗ \\ ⟨g(s_{σ_{2}(1)}, …, s_{σ_{2}(r)})_{1} ⊗ … ⊗ g(u_{σ_{2}(1)}, …, u_{σ_{2}(r)})_{s}, ρ_{2}⟩ = \\ ⟨f(t_{σ(1)}, …, t_{σ(n)}, s_{σ(n+1)}, …, s_{σ2(n+r)})_{1} ⊗ … \\ … ⊗ g(t_{σ(1)}, …, t_{σ(n)}, s_{σ(n+1)}, …, s_{σ2(n+r)})_{m+s}, ρ_{1} ⊗ ρ_{2}⟩. \end{matrix}

5. Coalgebras, Hopf Algebras, and beyond

5.1. Linear algebra: First we observe some rules from Linear Algebra. Let κ ∈ I(X), a_{i} ∈ A_{i}(Y_{i}), and f : A_{1} × … × A_{n} ⟶ B be a multimorphism. Let κa_{i} resp a_{i}κ denote the multiplication given by λ : I ⊗ A_{i} = A_{i} and ρ : A_{i} ⊗ I = A_{i}. Then we have

(30)  f(a_{1}, …, a_{i}κ, a_{i+1}, …, a_{n}) = f(a_{1}, …, a_{i}, κa_{i+1}, …, a_{n})

and for any morphism g : A ⟶ B

(31)  κf(a) = f(κa) and f(a)κ = f(aκ).

A more interesting formula for a braiding is obtained as

(32)  κa = ⟨aκ, τ⟩ and κa = ⟨aκ, τ^{−1}⟩.

Let (A, ∇, η) be an algebra then

(33)  η(κ) · a = κa and a · η(κ) = aκ.

5.2. The Sweedler-Heyneman Notation: Let H be a Hopf algebra in C. For a ∈ H(X) we want to have Δ(a) = a_{(1)} ⊗ a_{(2)}.

Let f : H × … × H ⟶ M with associated morphism f^{♯} : H ⊗ … ⊗ H ⟶ M be given. Let a ∈ H(X), then Δ^{n−1}(a) ∈ H ⊗ … ⊗ H(X). Using the definition in equation (6) we define

(34)  f(a_{(1)}, …, a_{(n)}) := f^{♯}(Δ^{n−1}(a)).

As in equation (7) (and also as in (4)) this gives the formula Δ(a) = a_{(1)} ⊗ a_{(2)}. Then by equation (13)

⟨f(a_{(σ(1))}, …, a_{(σ(n))}), ρ⟩ = f^{♯}(ρ(a_{(1)} ⊗ … ⊗ a_{(n)})).

Using the coassociativity of Δ we get the following rule for a change of the number of arguments

(35) ⟨f(a_{(σ(1))}, …, Δ(a_{(σ(i))}), …, a_{(σ(n))}), ρ⟩ = ⟨f(a_{(σ(1))}, …, a_{(σ(n+1))}), ρ_{i}⟩

where ρ_{i} acts like ρ but switches the braids i and i + 1 in parallel.

Observe, however, that the braid does not change in

(36)  \begin{matrix}⟨f(a_{(σ(1))}, …, a_{(σ(i))}, …, a_{(σ(n+1))}), ρ⟩ \\ = ⟨f(a_{(σ(1))}, …, a_{(σ(i))(1)}, …, a_{(σ(i))(2)}, …, a_{(σ(n))}), ρ⟩. \end{matrix}

5.3. Hopf Algebras: As usual one gets

(37)  ηε(a_{(1)})a_{(2)} = a = a_{(1)}ηε(a_{(2)})

and

(38)  a_{(1)}S(a_{(2)}) = ηε(a) = S(a_{(1)})a_{(2)}.

These last equations are to be considered as functions in one argument a, so they allow substitution at any position where a occurs.

The compatibility of multiplication and comultiplication is expressed by

(39)  (ab)_{(1)} ⊗ (ab)_{(2)} = ⟨a_{(1)}b_{(1)} ⊗ a_{(2)}b_{(2)}, τ_{23}⟩

where a ⊗ b ∈ H ⊗ H(X) and τ is the basic braid map interchanging two factors. Furthermore we have from (32)

(40)  ηε(a_{(1)})a_{(2)} = a = ⟨a_{(2)}ηε(a_{(1)}), τ⟩.

Theorem 5.1. If H is a braided Hopf algebra then the antipode S of the Hopf algebra H is an algebra τ-antihomomorphism, i.e.

(41)  S(ab) = ⟨S(b)S(a), τ⟩.

Proof. We compute

S(ab) = S((ab)_{(1)}ηε((ab)_{(2)}))

(by (37), the arguments are a, b)

= S((ab)_{(1)}) ηε((ab)_{(2)})

(by (31) and (33))

= ⟨S(a_{(1)}b_{(1)}) ηε(a_{(2)}b_{(2)}), τ_{23}⟩ (by (39), change to 4 arguments a_{(1)}, a_{(2)}, b_{(1)}, b_{(2)})

= ⟨S(a_{(1)}b_{(1)}) ε(a_{(2)}b_{(2)}), τ_{23}⟩ (by (33))

= ⟨S(a_{(1)}b_{(1)}) ε(a_{(2)}) ε(b_{(2)}), τ_{23}⟩ (ε is multiplicative for all elements)

= ⟨S(a_{(1)}b_{(1)}) ηε(a_{(2)}) ε(b_{(2)}), τ_{23}⟩ (by (33))

= ⟨S(a_{(1)}b_{(1)}) a_{(2)(1)}S(a_{(2)(2)}) ε(b_{(2)}), τ_{23}⟩ (by (38), the arguments are still a_{(1)}, a_{(2)}, b_{(1)}, b_{(2)})

= ⟨S(a_{(1)}b_{(1)}) a_{(2)}S(a_{(3)}) ε(b_{(2)}), τ_{23}τ_{34}⟩ (by (35), change of arguments to a_{(1)}, a_{(2)}, a_{(3)}, b_{(1)}, b_{(2)}) (change of arguments by (27) to a_{(1)}, a_{(2)}, S(a_{(3)}), b_{(1)}, ε(b_{(2)}))

= ⟨S(a_{(1)}b_{(1)}) a_{(2)} ε(b_{(2)}) S(a_{(3)}), τ_{45}τ_{23}τ_{34}⟩ (by (32), change arguments back to a_{(1)}, a_{(2)}, a_{(3)}, b_{(1)}, b_{(2)} by (27))

= ⟨S(a_{(1)}b_{(1)}) a_{(2)} ηε(b_{(2)}) S(a_{(3)}), τ_{45}τ_{23}τ_{34}⟩ (by (33))

= ⟨S(a_{(1)}b_{(1)}) a_{(2)} b_{(2)} S(b_{(3)}) S(a_{(3)}), τ_{56}τ_{45}τ_{23}τ_{34}⟩ (as above by (38), (35))

= ⟨S(a_{(1)}b_{(1)}) a_{(2)} b_{(2)} S(b_{(3)}) S(a_{(3)}), τ_{23}τ_{56}τ_{45}τ_{34}⟩ (change of braid map)

= ⟨S(a_{(1)(1)}b_{(1)(1)}) a_{(1)(2)} b_{(1)(2)} S(b_{(2)}) S(a_{(2)}), τ_{23}τ_{56}τ_{45}τ_{34}⟩ (by (36), the arguments are a_{(1)(1)}, a_{(1)(2)}, a_{(2)}, b_{(1)(1)}, b_{(1)(2)}, b_{(2)})

= ⟨S((a_{(1)}b_{(1)})_{(1)}) (a_{(1)}b_{(1)})_{(2)} S(b_{(2)}) S(a_{(2)}), τ_{34}τ_{23}⟩ (39) (change to 4 arguments a_{(1)}, a_{(2)}, b_{(1)}, b_{(2)}, apply (35) twice) (read from lower line to upper line)

= ⟨ηε(a_{(1)}b_{(1)}) S(b_{(2)}) S(a_{(2)}), τ_{34}τ_{23}⟩ (by (38))

= ⟨ε(a_{(1)}) ε(b_{(1)}) S(b_{(2)}) S(a_{(2)}), τ_{34}τ_{23}⟩ (by (33) and multiplicativity of ε)

= ⟨ε(a_{(1)}) S(b) S(a_{(2)}), τ_{23}⟩ (by (38) together with change of arguments to a_{(1)}, a_{(2)}, b)

= ⟨S(b) ε(a_{(1)}) S(a_{(2)}), τ_{12}τ_{23}⟩ (by (32))

= ⟨S(b) S(a), τ⟩

(by (38) together with change of arguments to a, b).

□

References

[JS91] Joyal, A. and Street, R.: The Geometry of Tensor Calculus, I. Adv. Math. 88, (55-112) 1991.

[Pa77] Pareigis, B.: Non-additive ring and module theory I. General theory of monoids. Publicationes Mathematicae 24, Debrecen, (190-204) 1977.

[H-Sw62] Heyneman, R.G. and Sweedler, M.E.: Affine Hopf algebras I, J. Algebra 13, (192-241)

1969

[Pe71] Penrose, R.: Applications of Negative Dimensional Tensors. In: Combinatorial Mathematics and its Applications. Academic Press. (221-244) 1971.

Mathematisches Institut der Universität München, Germany

  E-mail address: pareigis@rz.mathematik.uni-muenchen.de