From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors

By Michael Mueger. 2001

362 citations recorded by the source corpus.

Abstract For every tensor category C there is a braided tensor category Z ( C ) , the ‘center’ of C . It is well known to be related to Drinfel'd's notion of the quantum double of a finite dimensional Hopf algebra H by an equivalence Z (H -mod ) ≃ ⊗ br D(H) -mod of braided tensor categories. In the Hopf algebra situation, whenever D(H)-mod is semisimple (which is the case iff D(H) is semisimple iff H is semisimple and cosemisimple iff S2=id and char F ∤ dim H ) it is modular in the sense of Turaev, i.e. its S-matrix is invertible. (This was proven by Etingof and Gelaki in characteristic zero. We give a fairly general proof in the appendix.) The present paper is concerned with a generalization of this and other results to the quantum double (center) of more general tensor categories. We consider F -linear tensor categories C with simple unit and finitely many isomorphism classes of simple objects. We assume that C is either a ∗ -category (i.e. F = C and there is a positive ∗ -operation on the morphisms) or semisimple and spherical over an algebraically closed field F . In the latter case we assume dim C ≡∑ i d(X i ) 2 ≠0 , where the summation runs over the isomorphism classes of simple objects. We prove that Z ( C ) (i) is a semisimple spherical (or ∗ -) category and (ii) is weakly monoidally Morita equivalent (in the sense of Muger (J. Pure Appl. Algebrea 180 (2003) 81–157)) to C ⊗ F C op . This implies dim Z ( C )=( dim C ) 2 . (iii) We analyze the simple objects of Z ( C ) in terms of certain finite dimensional algebras, of which Ocneanu's tube algebra is the smallest. We prove the conjecture of Gelfand and Kazhdan according to which the number of simple objects of Z ( C ) coincides with the dimension of the state space H S 1 ×S 1 of the torus in the triangulation TQFT built from C . (iv) We prove that Z ( C ) is modular and we compute Δ ± ( Z ( C ))≡∑ i θ(X i ) ±1 d(X i ) 2 = dim C . (v) Finally, if C is already modular then Z ( C ) ≃ ⊗ br C ⊗ F C ≃ ⊗ C ⊗ F C op , where C is the tensor category C with the braiding c X,Y =c Y,X −1 .

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

View Paper: https://www.semanticscholar.org/paper/1f47c1e135ddc462f69d5fe01e75b668b3bbd073

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Citations in the corpus

Central invariants and Frobenius–Schur indicators for semisimple quasi-Hopf algebras (2003, background): “Following [24], for any simple object V in C and an isomorphism f :V- V ; we define jV j(2) 1⁄4 trV ðf Þtr V ð ðf (1)ÞÞ: ð7:2Þ Clearly, jV j(2) is independent of the choice of f : By Etingof et al.”
GROUP-THEORETICAL PROPERTIES OF NILPOTENT MODULAR CATEGORIES (2007, background): “The central charge of Z(Vec ! G) equals 1 by [ Mu4 , Theorem 1.2].”
Torus bundles not distinguished by TQFT invariants (2011, methodology): “According to Müger’s results (see [40]) the Drinfeld double D(C) of a spherical fusion category C is a modular tensor category.”
Module categories over the Drinfeld double of a finite group (2002, background): “But I got an idea that this problem can be solved (and actually is trivial) only after reading papers [14, 15].”
On fusion categories (2002, neutral): “(X, Y ) ⊗ Z = X ⊗ Z ⊗ Y . Then the dual category is the Drinfeld center Z(C) of C (see [O1, Mu2 ]; for the basic theory of the Drinfeld center see [K]).”
The Ponzano–Regge model (2008, background): “As noted in [24, 37], there is also a problem with constructing a vector space of quantum states for a boundary surface in this case.”
From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories (2001, background): “A less obvious example is provided in the companion paper [47].”
Tannaka–Kreĭn duality for compact quantum homogeneous spaces. I. General theory (2012, background): “Now ergodic actions on finite-dimensional C-algebras can be characterized abstractly inside of ReppGq as (irreducible) abstract Q-systems ([20], [22]).”
Annular representation theory for rigid $C^{*}$-tensor categories (2015, methodology): “enter of a fusion categories, with equivalence classes of irreducible representations of the tube algebra being in 1 1 correspondence with simple objects in the center in the case jIrr(C)j<1, [12],[22]. Thus the tube algebra of a category can be viewed as a \substitute" for the Drinfeld doub”