2002/01/03 by Michael Mueger · 5 citations
Mathematics · #math.CT #msc:18D10 #msc:57M27
published as Proc. Lond. Math. Soc. 87 (2003) 291-308 · 19 pages, LaTeX2e
arxiv created 2002/01/03 · arxiv updated 2009/11/30
For a braided tensor category C and a subcategory K there is a notion of 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 iff the center Z2(C):=CC(C) (not to be confused with the center Z1 of a tensor category, related to the quantum double) is trivial, i.e. equivalent to Vect, and dim(C)<>0. Here dim(C)=sumi d(Xi)2, the Xi being the simple objects. We prove the following double centralizer theorem: Let C be a modular category and K a full tensor subcategory closed w.r.t. direct sums, subobjects and duals. Then CC(CC(K))=K and dim(K)dim(CC(K))=dim(C). We give several applications, the most important being the following. If C is modular and K is a full modular subcategory, then also L=CC(K) is modular and C is equivalent as a ribbon category to the direct product of K and L. Thus every modular category factorizes (non-uniquely, in general) into prime ones. We study the prime factorizations of the categories D(G)-Mod, where G is a finite abelian group.