Modular quasi-Hopf algebras and groups with one involution

TL;DR

This paper constructs new modular quasi-Hopf algebras from groups with a single involution, extending previous work on twisted quantum doubles and exploring conditions for modularity and super-modularity.

Contribution

It verifies cohomological conditions for groups with one involution, leading to explicit constructions of new modular quasi-Hopf algebras and explores their categorical properties.

Findings

Identified conditions under which $Rep(D^{ ext{ω}}(G, A))$ is modular for groups with one involution.

Constructed explicit examples involving binary polyhedral groups and sporadic simple groups.

Developed a theory distinguishing when the representation category is super-modular.

Abstract

In a previous paper the authors constructed a class of quasi-Hopf algebras $D^{\omega}(G, A)$ associated to a finite group $G$, generalizing the twisted quantum double construction. We gave necessary and sufficient conditions, cohomological in nature, that the corresponding module category $Rep(D^{\omega}(G, A))$ is a modular tensor category.\ In the present paper we verify the cohomological conditions for the class of groups $G$ which \emph{contain a unique involution}, and in this way we obtain an explicit construction of a new class of modular quasi-Hopf algebras.\ We develop the basic theory for general finite groups $G$, and also a parallel theory concerned with the question of when $Rep(D^{\omega}(G, A))$ is super-modular rather than modular. We give some explicit examples involving binary polyhedral groups and some sporadic simple groups.