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4.5 Yetter-Drinfeld modules
We now present an important class of examples of braided categories. It will lead us to a class
of factorizable Hopf algebras of particular importance. Using this class, we will also be able
to show that any factorizable Hopf algebra is unimodular. This class of Hopf algebras will
also be naturally realized in the Turaev-Viro construction of three-dimensional topological field
theories.
Some of the theory can be formulated for bialgebras. Whenever we deal with Hopf algebras,
we will continue to assume that the antipode S is invertible (or that a skew antipode exists).
Definition 4.5.1
1. Let A be a bialgebra. A Yetter-Drinfeld module is a triple (V, ρ V , Δ V ) such that
• (V, ρ V ) is a unital left A-module.
• (V, Δ V ) is a counital left A-comodule:
• The Yetter-Drinfeld condition
holds for all h ∈ H.
Δ V : V → H ⊗ V
v ↦→ v (−1) ⊗ v (0)
h (1) ∙ v (−1) ⊗ h (2) .v (0) = (h (1) .v) (−1) ∙ h (2) ⊗ (h (1) .v) (0)
2. Morphisms of Yetter-Drinfeld modules are morphisms of left modules and left comodules.
We write A AYD for the category of Yetter-Drinfeld modules over the bialgebra A.
Example 4.5.2.
Let us consider quite explicitly Yetter Drinfeld modules over the group algebra K[G] of a finite
group G.
Since we have the structure of a comodule, any Yetter-Drinfeld module has a natural structure
of a G-graded vector space,
V = ⊕ g∈G V g ,
which is moreover endowed with a G-action. We evaluate the Yetter-Drinfeld condition for
g ∈ H and v h ∈ V h . We find for the left hand side using Δ(g) = g ⊗ g and Δ V (v h ) = h ⊗ v h
that gh ⊗ g.v h . For the right hand side, we find
∑
xg ⊗ (g.v) x .
x∈G
The equality of the two terms implies that only the term with x such that xg = gh contributes.
Thus the Yetter-Drinfeld condition amounts to g.v h ∈ V ghg −1. Thus the G-action has to cover
for the grading the action of G on itself by conjugation.
We know that modules over a bialgebra form a tensor category, and so do comodules over
a bialgebra. We can thus define as in proposition 2.4.7 and remark 2.4.8 on the tensor product
of the vector spaces underlying two Yetter-Drinfeld modules V, W the structure of a module
and of a comodule. We also note that the ground field with trivial action
A ⊗ K → K
a ⊗ λ ↦→ ɛ(a)λ
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