pdf file

Remark 5.1.9.
Equation (13) always holds for K-linear categories for which End C (I) ∼ = Kid I and thus in
particular for categories of modules over Hopf algebras. It also holds for all braided pivotal
categories, since c X,I
∼ = cI,X
∼ = idX .
Proof.
We only show the assertions for one trace. The proof is best performed in graphical notation.
✷
Corollary 5.1.10.
From the properties of the traces, we immediately deduce the following properties of the left
and right dimensions:
1. Isomorphic objects have the same left and right dimension.
2. dim l X = dim r X ∗ = dim l X ∗∗ , and similarly with left and right dimension interchanged.
3. dim l I = dim r I = id I .
4. Suppose, relation (13) holds. Then the dimensions are multiplicative:
dim l (X ⊗ Y ) = dim l X ∙ dim l Y and dim r (X ⊗ Y ) = dim r X ∙ dim r Y
for all objects X, Y of C.
5. The dimension is additive for exact sequences: from
we conclude dim V = dim V ′ + dim V ′′ .
0 → V ′ → V → V ′′ → 0
Proof.
1. Choose f : X → Y and g : Y → X such that id X = g ◦ f and id Y = f ◦ g. Then by the
symmetry of the trace
dim l X = Tr l id X = Tr l g ◦ f = Tr l f ◦ g = Tr l id Y = dim l Y .
2. The axioms of a duality imply that id ∗ X = id X ∗. Now the claim follows from the second
identity of lemma 5.1.8.
3. Follows from the canonical identification I ∼ = I ⊗ I ∗ .
4. Follows from the identity id X⊗Y = id X ⊗ id Y which is part of the definition of a tensor
product.
5. We refer to [AAITC, 2.3.1].
120