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Proposition 4.5.22.
This defines a strict monoidal category Z(C), called the Drinfeld center of the category C. It is
braided with braiding given by the half-braiding
The forgetful functor
c V,W : (V, c −,V ) ⊗ (W, c −,W ) → (W, c −,W ) ⊗ (V, c −,V ) .
F : Z(C) → C
(V, c −,V ) ↦→ V
is strict monoidal. (It is not, in general essentially surjective nor full, but faithful by definition.)
For the proof of this statement, we refer to the book [Kassel]. We now make contact to the
double construction:
Theorem 4.5.23.
For any finite-dimensional Hopf algebra H, the braided tensor categories Z(H−mod) and
D(H)−mod are equivalent.
Proof.
• We indicate how to construct a functor
Z(H−mod) → H YD H .
To this end, we define on any object (V, c −V ) of Z(H−mod) a right H-coaction. Consider
Δ V : V → V ⊗ H
v ↦→ c H,V (1 ⊗ v) .
One checks that this defines a coassociative coaction.
Again the naturality of the braiding allows us to express the braiding in terms of the
coaction Δ V : consider for x ∈ X the morphism x : H → X with x(1) = x. Then
c X,V (x ⊗ v) = c X,V ◦ (x ⊗ id V )(1 ⊗ v) = (id V ⊗ x) ◦ c H,V (1 ⊗ v)
= v (V ) ⊗ v (H) ∙ x = Δ V (v)(1 H ⊗ x)
which is exactly the braiding on the category H YD H .
• Next, we use the fact that the braiding is H-linear:
a.c X,V (x ⊗ v) = c X,V (a.(x ⊗ v))
for all a ∈ H and v ∈ V, x ∈ X. Replacing the braiding c X,V by the expression just derived
yields the equation
Setting X = H and x = 1 H yields
Δ(a)Δ V (v)(1 ⊗ x) = Δ V (a (2) v)(1 ⊗ a (1) )(1 ⊗ x) .
a (1) .v (V ) ⊗ a (2) ∙ v (H) = (a (2) v) V ⊗ (a (2) v) (H) ∙ a (1)
which is just the Yetter-Drinfeld condition in H YD H .
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