Biproduct Quasi-Hopf Algebras of Rank 2

TL;DR

This paper classifies biproduct quasi-Hopf algebras of rank 2 derived from arbitrary quasi-Hopf algebras, revealing new classes of such algebras and associated tensor categories, especially from group-based structures.

Contribution

It provides a comprehensive description of all rank 2 biproduct quasi-Hopf algebras over any quasi-Hopf algebra, including new classes from group cocycles and radical structures.

Findings

Classified all Hopf algebras of dimension 2 in Yetter-Drinfeld categories.

Constructed new biproduct quasi-Hopf algebras from group cocycles.

Discovered new tensor categories from these algebraic structures.

Abstract

Inspired by the work of Radford, for $H$ an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension $2$ within the braided category of left Yetter-Drinfeld modules over $H$ and determine the biproduct quasi-Hopf algebras defined by them. Classes of such biproduct quasi-Hopf algebras are obtained by taking $H$ as the Hopf algebra of functions on a group $G$, endowed with the quasi-Hopf algebra structure provided by a non-trivial $3$-cocycle on $G$ (especially when $G$ is a finite cyclic group or the double dihedral group), or as being a quasi-Hopf algebra with radical of codimension two. In this way we uncover new classes of basic quasi-Hopf algebras of even dimension, as well as new classes of tensor categories.