pdf file

✷
Definition 5.1.11
A spherical category is a pivotal category whose left and right traces are equal,
for all endomorphisms f.
Tr l (f) = Tr r (f)
Remarks 5.1.12.
1. In a spherical category, we write Tr(f) and call it the trace of the endomorphism f.
2. In particular, left and right dimensions of all objects are equal, dim l X = dim r X. We call
this element of End C (I) the dimension dim X of X.
3. For spherical categories, the graphical calculus has the following additional property:
morphisms represented by diagrams are invariants under isotopies of the diagrams in the
two-sphere S 2 = R 2 ∪ {∞}. They are thus preserved under pushing arcs through the
point ∞. Left and right traces are related by such an isotopy. This explains the name
“spherical”.
4. Suppose that C = H−mod. Then the traces are given by
Tr l (ϑ) = Tr V (ϑρ V (ω −1 )), Tr r (ϑ) = Tr V (ϑρ V (ω)) for ϑ ∈ End H (V ).
Thus, H−mod is a spherical category, whenever H is a spherical Hopf algebra. One can
show [AAITC, Proposition 2.1] that it is sufficient to verify the trace condition on simple
H-modules to show that a pivotal Hopf algebra is spherical.
We also consider analogous additional structure on braided tensor categories. Recall from
remark 5.1.9 that for a braided category, the trace is always multiplicative.
Definition 5.1.13
Let C be a braided pivotal category.
1. For any object X of C, define the endomorphism
θ X = (id X ⊗ ˜d X ) ◦ (c X,X ⊗ id X ∗) ◦ (id X ⊗ b X ) .
This endomorphism is called the twist on the object X.
2. A ribbon category is a braided pivotal category where all twists are selfdual, i.e.
(θ X ) ∗ = θ X ∗ for all X ∈ C .
Lemma 5.1.14.
Let C be a braided pivotal category.
1. The twist is invertible with inverse
θ −1
X
= (d X ⊗ id X ) ◦ (id X ∗ ⊗ c −1
X,X ) ◦ (˜b X ⊗ id X ) .
121