On the Structure of Modular Categories

By Michael Mueger. 2002

255 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/971f09b2269cd811d485359c202885972fc02889

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Citations in the corpus

A Finiteness Property for Braided Fusion Categories (2009, background): “By (a generalized version of) a theorem of Müger [25] this is equivalent to s̃X,Y = dim(X) dim(Y ) for simple X and Y where s̃ is the normalized modular S-matrix (see Section 3).”
Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups (2007, background): “G� are weakly Morita equivalent in the sense of M. Muger [ Mu2 ].”
WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES (2008, background): “2.9. M¨uger’s theorem. ([ M4 ]) Theorem 2.4. Let C be a braided category, and D a nondegenerate subcategory in C. Then C is naturally equivalent, as a braided category, to D ⊠ D′.”
Fusion subcategories of representation categories of twisted quantum doubles of finite groups (2008, methodology): “Assuming that categorical dimensions in C are positive, simple objects X and Y in C centralize each other if and only if S(X, Y ) = d(X)d(Y ) [ M1 , Prop.”
On braided fusion categories I (2009, background): “The next Proposition was proved by Altschuler and Bruguieres in [1] for pre-modular categories (see [ 31 ] for the proof in the special case of pseudo-unitary braided fusion categories).”
Comments on one-form global symmetries and their gauging in 3d and 4d (2018, methodology): “conditions of a consistent TQFT are satised separately for T0and AN;p and hence we have the factorization (2.21). The factorization (2.21) also follows from a theorem in modular tensor category (see [16] and Theorem 3.13 in [17]). In physics language, the theorem states that if a 3d TQFT T 15We use ”
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES (2007, background): “(16) b(X,Y1)b(X,Y2) = b(X,Z), as explained in [ Mu2 ].”
Tensor categories: a selective guided tour (2008, background): “) • There is a “Double commutant theorem” for modular categories (Müger [193], inspired by Ocneanu [219]): Let M a modular category and a C ⊂ M a replete full tensor subcategory.”