Basic quasi-Hopf algebras over cyclic groups

TL;DR

This paper extends the classification of basic quasi-Hopf algebras over cyclic groups to those of order m, introducing new non-semisimple examples that are not twist equivalent to Hopf algebras.

Contribution

It generalizes prior classifications to cyclic groups of order m and constructs new non-semisimple quasi-Hopf algebras not twist equivalent to Hopf algebras.

Findings

Classifies all basic quasi-Hopf algebras over cyclic groups of order m

Introduces a family of non-semisimple quasi-Hopf algebras $A(H,s)$

Shows these are not twist equivalent to Hopf algebras

Abstract

Let $m$ a positive integer, not divisible by 2,3,5,7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in \cite{EG3} to the case of cyclic groups of order $m$. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras $A(H,s)$, constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order $m$ is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type $A(H,s)$.