On the gauge equivalence of twisted quantum doubles of elementary Abelian and extra-special 2-groups.(English)Zbl 1171.16021
Authors’ summary: We establish braided tensor equivalences among module categories over the twisted quantum double of a finite group defined by an extension of a group \(\overline G\) by an Abelian group, with 3-cocycle inflated from a 3-cocycle on \(\overline G\). 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 \(\omega\) is gauge equivalent to a twisted quantum double of an elementary Abelian 2-group if, and only if, \(\omega^2\) 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.
Reviewer: Sorin Dascalescu (Bucureşti)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
twisted quantum doubles;group cohomology;Frobenius-Schur indicators;Frobenius-Schur exponents;quasi-Hopf algebras;braided tensor categories;braided tensor equivalencesCite
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