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Lemma 2.3.5.
Let (A, μ, η, Δ, ɛ) be a finite-dimensional (weak) bialgebra and A ∗ = Hom K (A, K) its linear
dual. Then the dual maps
Δ ∗ : (A ⊗ A) ∗ ∼ = A ∗ ⊗ A ∗ → A ∗
ɛ ∗ : K → A ∗
μ ∗ : A ∗ → (A ⊗ A) ∗ = A ∗ ⊗ A ∗
η ∗ : A ∗ → K
define the structure of a (weak) bialgebra (A ∗ , Δ ∗ , ɛ ∗ , μ ∗ , η ∗ ).
Remark 2.3.6.
For any (weak) bialgebra (A, μ, η, Δ, ɛ), we have three more (weak) bialgebras:
A opp = (A, μ opp , η, Δ, ɛ) A opp,copp = (A, μ opp , η, Δ copp , ɛ)
A copp = (A, μ, η, Δ copp , ɛ)
2.4 Tensor categories
We wish to understand more structure that is present on the categories of modules over bialgebras.
In particular, from our experience with Lie algebras and group algebras, we expect
that the tensor product V ⊗ W of two modules V and W over a bialgebra A carries again the
structure of an A-module. This turns a pair of objects (V, W ) of the category A−mod into an
object V ⊗ W , and a pair of morphisms (f, g) into a morphism f ⊗ g.
Definition 2.4.1
The Cartesian product of two categories C, D is defined as the category C × D whose objects
are pairs (V, W ) ∈ Obj(C) × Obj(D) and whose morphisms are given by the Cartesian product
of sets:
Hom C×D ((V, W ), (V ′ , W ′ )) = Hom C (V, V ′ ) × Hom D (W, W ′ ) .
We are now ready to discuss the structure induced by the tensor product of modules:
Definition 2.4.2
1. Let C be a category and ⊗ : C × C → C a functor, called a tensor product.
Note that this associates to any pair (V, W ) of objects an object V ⊗W and to any pair of
morphisms (f, g) a morphism f ⊗ g with source and target given by the tensor products
of the source and target objects. In particular, id V ⊗W = id V ⊗ id W and for composable
morphisms
(f ′ ⊗ g ′ ) ◦ (f ⊗ g) = (f ′ ◦ f) ⊗ (g ′ ◦ g) .
2. A monoidal category or tensor category consists of a category (C, ⊗) with tensor product,
an object I ∈ C, called the tensor unit, and natural isomorphisms, called the associator,
of functors C × C × C → C and
such that the following axioms hold:
a : ⊗(⊗ × id) → ⊗(id × ⊗) .
r : id ⊗ I → id and l : I ⊗ id → id
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