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Proof.
Using the definition of d V from 5.1.15, we compute with R = R (1) ⊗ R (2)
d V (v ⊗ α) = 〈R (2) .α, R (1) θ −1 v〉 = 〈α, S(R (2) ∙ R (1) θ −1 v〉 = 〈α, u ∙ ν −1 .v〉
Therefore
Trf = d(f ⊗ id V ∗)b V = ∑ i
〈v i , u ∙ ν −1 .v i 〉
with v i a basis of V .
✷
Remark 5.1.19.
One can show that the Drinfeld double of any finite-dimensional Hopf algebra is a ribbon
algebra with the Drinfeld element as the ribbon element. Thus for doubles the categorical trace
equals the trace in the usual sense of vector spaces. In particular, all categorical dimensions are
non-negative integers.
5.2 Tanaka-Krein reconstruction
Let K be a field. In this subsection, we explain under what conditions a K-linear ribbon category
can be described as the category of modules over a K-ribbon algebra. We assume that the field
K is algebraically closed of characteristic zero and that all categories are essentially small , i.e.
equivalent to a small category, a category in which the class of objects is a set.
Definition 5.2.1
Let C, D be abelian tensor categories. A fibre functor is an exact faithful tensor functor Φ :
C → D.
Examples 5.2.2.
1. Let H be a bialgebra over a field K. The forgetful functor
F : H−mod → vect(K) ,
is a strict tensor functor. It is faithful, since by definition Hom H−mod (V, W ) ⊂
Hom vect(K) (V, W ). It is exact, since the kernels and images in the categories H−mod
and vect(K) are the same. Thus the forgetful functor F is a fibre functor.
2. There are tensor categories that do not admit a fibre functor to vector spaces.
We are looking for an inverse of the construction: give an a ribbon category, together with
a fibre functor
Φ : C → vect(K) .
Can we find a ribbon Hopf algebra H such that C ∼ = H−mod as a monoidal category?
We only sketch the proof of a slightly different result:
Theorem 5.2.3.
Let K be a field and C a K-linear abelian tensor category and
Φ : C → vect fd (K)
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