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Proposition 5.1.5.
Let H is a finite-dimensional Hopf algebra.
1. If ω ∈ G(H) is a pivot for H, then the action with ω endows the category H−mod fd of
finite-dimensional H-modules with a pivotal structure.
2. Conversely, if ω is a pivotal structure on the category H−mod fd , then ω := ω H (1 H ) is a
pivot for the Hopf algebra H.
Proof.
1. Assume that ω is a pivot. We already know that the category H−mod fd is rigid. The
bidual of the H-module (V, ρ V ) is the H-module (V, ρ V ◦ S 2 ). Use the pivot ω to define
the linear isomorphism
ω V : V → V
v ↦→ ω.v .
This is actually a morphism V → V ∨∨ of H-modules:
a.ω V (v) = S 2 (a) ∙ ω.v = ω ∙ a ∙ ω −1 ω.v = ω ∙ a.v = ω V (a.v) .
2. Suppose that H−mod fd is a pivotal category. We canonically identify H ∼ = H ∗∗ as a
K-vector space, on which h ∈ H acts by S 2 (h). We consider the endomorphism
All right translations by a ∈ A
ω H : H → H ∗∗ ∼ = H .
R a : H → H
h ↦→ h ∙ a
are H-linear. The naturality of ω thus implies ω(h ∙ a) = ω H (h) ∙ a for all h, a ∈ H. Since
ω H is a morphism of H-modules, we have
Altogether, we find
ω H (a ∙ b) = S 2 (a)ω H (b) .
S 2 (a)ω H (1) = ω H (a ∙ 1) = ω H (1 ∙ a) = ω H (1) ∙ a
which shows that ω H (1) is a pivot for the Hopf algebra H.
To understand the meaning of the notion of a spherical Hopf algebra, we need the notion
of a trace which is also central for applications to topological field theory.
Lemma 5.1.6.
In any monoidal category, the monoid End (I) is commutative.
✷
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