Group cohomology and gauge equivalence of some twisted quantum doubles

Mason, Geoffrey; Ng, Siu-Hung

doi:10.1090/S0002-9947-01-02771-4

Group cohomology and gauge equivalence of some twisted quantum doubles
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by Geoffrey Mason and Siu-Hung Ng PDF

Trans. Amer. Math. Soc. 353 (2001), 3465-3509 Request permission

Abstract:

We study the module category associated to the quantum double of a finite abelian group $G$ twisted by a 3-cocycle, which is known to be a braided monoidal category, and investigate the question of when two such categories are equivalent. We base our discussion on an exact sequence which interweaves the ordinary and Eilenberg-Mac Lane cohomology of $G$. Roughly speaking, this reveals that the data provided by such module categories is equivalent to (among other things) a finite quadratic space equipped with a metabolizer, and also a pair of rational lattices $L\subseteq M$ with $L$ self-dual and integral.

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