On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups

By Christopher D. Goff, Geoffrey Mason, Siu-Hung Ng. 2006

57 citations recorded by the source corpus.

Abstract We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group G ¯ by an abelian group, with 3-cocycle inflated from a 3-cocycle on G ¯ . We also prove that the canonical ribbon structure of the module category of any twisted quantum double of a finite group is preserved by braided tensor equivalences. We give two main applications: first, if G is an extra-special 2-group of width at least 2, we show that the quantum double of G twisted by a 3-cocycle ω is gauge equivalent to a twisted quantum double of an elementary abelian 2-group if, and only if, ω 2 is trivial; second, we discuss the gauge equivalence classes of twisted quantum doubles of groups of order 8, and classify the braided tensor equivalence classes of these quasi-triangular quasi-bialgebras. It turns out that there are exactly 20 such equivalence classes.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/af1d501831a10bd33cab55cc919efb30a844a1e1

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Citations in the corpus

The Picard crossed module of a braided tensor category (2012, background): “Note that in general the category Z(VecωA), where A is an abelian group and ω ∈ Z3(A, k), is not pointed, see [Goff et al. 2007].”
Categorical Morita Equivalence for Group-Theoretical Categories (2006, background): “We refer the reader to Coste et al. (2000) and Goff et al. (2006) for similar results.”