On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there is a notion of a centralizer CC K, which is a full tensor subcategory of C. A pre‐modular tensor category is known to be modular in the sense of Turaev if and only if the center Z2C≡ CCC (not to be confused with the center Z1 of a tensor category, related to the quantum double) is trivial, that is, consists only of multiples of the tensor unit, and dimC ≠ 0. Here dimC=∑id(Xi)2 , the Xi being the simple objects.

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