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Proof.
We show the following statements by graphical calculations:
• The above definition indeed defines H-coactions on V ∗ .
• The coaction defined with S −1 has the property that the right evaluation and right coevaluation
are morphisms of H-comodules. The statement that the coaction defined with
S on the left dual is compatible with left evaluation and coevaluation follows in complete
analogy.
• The left and right dual actions and coactions obey the Yetter-Drinfeld axiom.
This raises the question whether for any given Hopf algebra H the category H HYD can be
seen as the category of left modules over a quasi-triangular Hopf algebra D(H).
Observation 4.5.6.
1. To investigate this in more detail, assume that the Hopf algebra H is finite-dimensional
and recall from example 2.2.8.1 that a coaction of H then amounts to an action of H ∗ .
Thus the quasi-triangular Hopf algebra D(H) should account for an action of H and H ∗ .
If the two actions would commute, H ∗ ⊗ H with the product structure for algebra and
coalgebra would be an obvious candidate. This is not the case, and we have to encode the
Yetter-Drinfeld condition in the product on D(H).
2. It will be convenient to consider a slightly different category H YD H of Yetter-Drinfeld
modules: these are triples (V, ρ V , Δ V ) such that
• (V, ρ V ) is a unital left A-module.
• (V, Δ V ) is a counital right A-comodule:
Δ V : V → V ⊗ H
v ↦→ v (V ) ⊗ v (H)
• The Yetter-Drinfeld condition holds in the form
h (1) .v (V ) ⊗ h (2) ∙ v (H) = (h (2) .v) (V ) ⊗ (h (2) .v) (H) ∙ h (1) .
Morphisms of Yetter-Drinfeld modules in H YD H are morphisms of left modules and right
comodules.
This observation leads to the following definition:
Definition 4.5.7
Let H be a finite-dimensional Hopf algebra. Endow the vector space D(H) := H ∗ ⊗ H
• with the structure of a counital coalgebra using the tensor product structure, i.e. for
f ∈ H ∗ and a ∈ H, we have
ɛ(f ⊗ a) := ɛ(a)f(1)
Δ(f ⊗ a) := (f (1) ⊗ a (1) ) ⊗ (f (2) ⊗ a (2) )
This encodes the fact that the tensor product of Yetter-Drinfeld modules is the ordinary
tensor product of modules and comodules.
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