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and trivial coaction
K → A ⊗ K
λ ↦→ 1 A ⊗ λ
is trivially a Yetter-Drinfeld module and is a tensor unit for the tensor product.
Proposition 4.5.3.
Let A be a bialgebra. Then the category of Yetter-Drinfeld modules has a natural structure of
a tensor category.
Proof.
Let V, W be Yetter-Drinfeld modules. We only have to show that the vector space V ⊗ W
with action and coaction defined above obeys the Yetter-Drinfeld condition. This is done
graphically.
✷
Proposition 4.5.4.
Let H be a Hopf algebra. Then the category of Yetter-Drinfeld modules has a natural structure
of a braided tensor category with the braiding of two Yetter-Drinfeld modules V, W ∈ H H YD
given by
c V,W : V ⊗ W → W ⊗ V
v ⊗ w ↦→ v (−1) .w ⊗ v (0)
Proof.
The following statements are shown graphically:
• The linear map c V,W is a morphism of modules and comodules and thus a morphism of
Yetter-Drinfeld modules.
• The morphisms c V,W obey the hexagon axioms.
• Based on lemma 2.5.7, we show that the morphisms c V,W have inverses.
We define as in proposition 2.5.13 the right dual action of H on V ∗ = Hom K (V, K) as
the pullback along S of the transpose of the action on V and the left dual action as the
pullback of the transpose along S −1 . The right dual coaction maps β ∈ V ∗ to the linear map
Δ ∨ V (β) ∈ H ⊗ V ∗ ∼ = HomK (V, H)
while the left dual coaction maps to
Δ ∨ V (β) : V ↦→ H
v ↦→ S −1 (v (−1) )β(v (0) )
∨ Δ V (β) : V ↦→ H
v ↦→ S(v (−1) )β(v (0) )
Proposition 4.5.5.
Let H be a Hopf algebra. Then the category of finite-dimensional Yetter-Drinfeld modules H H YD
is rigid.
✷
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