Cleft Extensions and Quotients of Twisted Quantum Doubles

  • Geoffrey MasonEmail author
  • Siu-Hung Ng
Part of the Developments in Mathematics book series (DEVM, volume 38)


Given a pair of finite groups F, G and a normalized 3-cocycle \(\omega\) of G, where F acts on G as automorphisms, we consider quasi-Hopf algebras defined as a cleft extension \(\mathbb{k}_{\omega }^{G}\#_{c}\,\mathbb{k}F\) where c denotes some suitable cohomological data. When \(F \rightarrow \overline{F}:= F/A\) is a quotient of F by a central subgroup A acting trivially on G, we give necessary and sufficient conditions for the existence of a surjection of quasi-Hopf algebras and cleft extensions of the type \(\mathbb{k}_{\omega }^{G}\#_{c}\,\mathbb{k}F \rightarrow \mathbb{k}_{\omega }^{G}\#_{\overline{c}}\,\mathbb{k}\overline{F}\). Our construction is particularly natural when F = G acts on G by conjugation, and \(\mathbb{k}_{\omega }^{G}\#_{c}\mathbb{k}G\) is a twisted quantum double D ω (G). In this case, we give necessary and sufficient conditions that Rep(\(\mathbb{k}_{\omega }^{G}\#_{\overline{c}}\,\mathbb{k}\overline{G}\)) is a modular tensor category.

Key words

Twisted quantum double Quasi Hopf algebra Modular tensor category 

Mathematics Subject Classification (2010):

16T10 18D10. 



Research of the first author was partially supported by NSA and NSF. The second author was supported by NSF DMS1303253.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of CaliforniaSanta CruzUSA
  2. 2.Louisiana State UniversityBaton RougeUSA

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