Quasi-Hopf Algebras of Dimension 6

TL;DR

This paper completes the classification of 6-dimensional quasi-Hopf algebras by proving their semisimplicity and providing explicit examples and structures, enriching the understanding of their algebraic and categorical properties.

Contribution

It establishes that all 6-dimensional quasi-Hopf algebras are semisimple and details the explicit structures and examples, expanding the classification framework.

Findings

All 6-dimensional quasi-Hopf algebras are semisimple.

15 non-twist equivalent quasi-Hopf algebras in dimension 6.

Explicit structures for semisimple 6-dimensional quasi-Hopf algebras.

Abstract

We complete the classification of the 6-dimensional quasi-Hopf algebras, by proving that any such algebra is semisimple. As byproducts, we provide examples of 6-dimensional quasi-bialgebras that are not semisimple as algebras, as well as the concrete quasi-Hopf structures of the 6-dimensional semisimple quasi-Hopf algebras previously classified by Etingof and Gelaki in terms of their category of representations. In total there are 15 quasi-Hopf algebras in dimension 6 which are not pairwise twist equivalent.