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Proposition 5.1.17.
1. Let (H, R, ν) be a ribbon Hopf algebra. For V ∈ H−mod fd , consider the endomorphism
θ V : V → V
v ↦→ ν.v
This defines a twist that is compatible with the dualities.
2. Conversely, suppose that (H, R) is a quasi-triangular Hopf algebra and that there is an
element ν ∈ H such that for any V ∈ H−mod fd the endomorphism θ V (v) := ν.v is a
twist on the braided category H−mod fd . Then ν is a ribbon element.
Proof.
• If ν is central and invertible, then all θ V are H-linear isomorphisms. Conversely, for any
algebra A any natural transformation θ : id A−mod → id A−mod is given by the action of an
element of the center Z(A) of A, End(id A−mod ) = Z(A).
• We compute for x ∈ V ⊗ W :
c W,V c V,W (θ V ⊗ θ W )(x) = R 21 R(νx (1) ⊗ νx (2) ) = Δ(ν) ∙ x = θ V ⊗W (x) .
The compatibility of twist and braiding is thus equivalent to the property R 21 R(ν ⊗ ν) =
Δ(ν).
• It remains to show that
(θ V ∗ ⊗ id V )b V (1) = (id V ∗ ⊗ θ V )b V (1) .
With {e i } a basis of V , this amounts to
∑
νe ∗ i ⊗ e i = ∑
i
i
e ∗ i ⊗ νe i
Evaluating this identity on any v ∈ V yields
∑
νe ∗ i (v) ⊗ e i = ∑ e ∗ i (v) ⊗ νe i
i
⇐⇒ S(ν) ∙ v = ν ∙ v
This shows that S(ν) = ν is a sufficient condition. Applying this to V = H and v = 1
shows that S(ν) = ν is necessary as well.
✷
Proposition 5.1.18.
Let V be any finite-dimensional module over a ribbon Hopf algebra H. Denote by u the Drinfeld
element of H. Then ν −1 u is a spherical element and we have for the trace
Tr(f) = Tr V ν −1 uf .
In particular, the dimension is the trace over the action of the element ν −1 u on V .
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