Cleft Extensions and Quotients of Twisted Quantum Doubles

Abstract

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 $F\to \overline{F}:=F/A$ 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 ${\mathbb{k}}_{\omega }^G{\#}_c\mathbb{k}F\to {\mathbb{k}}_{\omega }^G{\#}_{\overline{c}}\mathbb{k}\overline{F}$. 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 $D^{\omega }(G)$. 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.

LaTex; 14 pages

References in corpus (1)

• Central Invariants and Frobenius-Schur Indicators for Semisimple Quasi-Hopf Algebras

Cited by in corpus (3)

• Fermionic Modular Categories and the 16-fold Way
• Morita equivalence of pointed fusion categories of small rank
• Orbifolds and minimal modular extensions