X ⊗ Y is commutative for all X, Y ∈ C. In particular, one has r 1 = l 1 = ι.Proof.This follows by applying the pentagon axiom for the quadruple of objects X, 1, 1, Y .More specifically, we have the following diagram:(1.2.2)
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Etingof et al. (2004) studied this question.
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