On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there is a notion of a centralizer C C 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 Z 2 C≡ C C C (not to be confused with the center Z 1 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 dim C = ∑ i d ( X i ) 2 , the X i being the simple objects. We prove several structural properties of modular categories. Our main technical tool is the following double centralizer theorem. Let C be a modular category and K a full tensor subcategory closed with respect to direct sums, subobjects and duals. Then C C C C K = K and dim K·dim C C K = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory, then L=C C K is also modular and C is equivalent as a ribbon category to the direct product: C ≃ K ⊠ L . Thus every modular category factorizes (non‐uniquely, in general) into prime modular categories. We study the prime factorizations of the categories D ( G )‐Mod, where G is a finite abelian group. (2) If C is a modular *‐category and K is a full tensor subcategory then dim C ⩾ dim K · dim Z 2 K. We give examples where the bound is attained and conjecture that every pre‐modular K can be embedded fully into a modular category C with dim C=dim K·dim Z 2 K. (3) For every finite group G there is a braided tensor *‐category C such that Z 2 C≃Rep, G and the modular closure/modularization C is non‐trivial. 2000 Mathematics Subject Classification 18D10.