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2. We have θ I = id I and
θ V ⊗W = c W,V ◦ c V,W ◦ (θ V ⊗ θ W ) .
The reader should draw the graphical representation of this relation.
3. The twist is natural: for all morphisms f : X → Y , we have f ◦ θ X = θ Y ◦ f.
4. A braided pivotal category is a ribbon category, if and only if the identity
θ X = (d X ⊗ id X ) ◦ (id X ∗ ⊗ c X,X ) ◦ (˜b X ⊗ id X )
holds.
Proof.
1.,2. This is best seen graphically.
3. Using properties of the duality, one shows for f : U → V that
˜d V ◦ (f ⊗ id V ∗) = ˜d U ◦ (id U ⊗ f ∗ )
and similar relations for the other duality morphisms. The naturality of the twist now
follows from these relations and the naturality of the braiding and its inverse.
4. Follows from a graphical calculation that is left to the reader.
✷
Proposition 5.1.15.
A ribbon category C is spherical.
Proof.
To this end, one notes that
d V = d V ◦ c V,V ∗ ◦ (θ −1
V ⊗ id V ∗) = d V ◦ ◦c V,V ∗ ◦ (id V ⊗ θ −1
V ∗)
b V = (θ −1
V ⊗ id ∗ V ) ◦ c V,V ∗ ◦ b V = (id V ∗ ⊗ θ −1
V
) ◦ c V,V ∗ ◦ b V
form another left duality. The proof of this assertion can be found in [Kassel, p. 351-353],
with left and right duality interchanged as compared to our statement of the assertion. Since
all (left) dualities are equivalent, the rest of the proof can now be easily performed graphically. ✷
We finally express these structures on the level of Hopf algebras.
Definition 5.1.16
A ribbon Hopf algebra is a quasi-triangular Hopf algebra (H, R) together with an invertible
central element ν ∈ H such that
Δ(ν) = (R 21 ∙ R) −1 ∙ (ν ⊗ ν) , ɛ(ν) = 1 and S(ν) = ν .
The element ν is called a ribbon element.
A ribbon element is not unique, but only determined up to multiplication by an element in
{g ∈ G(H) ∩ Z(H) | g 2 = 1}, see [AAITC, Definition 2.13].
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