Classifying bicrossed products of two Taft algebras

TL;DR

This paper classifies all Hopf algebras that factorize through two Taft algebras, providing explicit descriptions, counting isomorphism types, and analyzing automorphism groups.

Contribution

It offers a complete classification of bicrossed products of two Taft algebras, including explicit generators, relations, and automorphism group structures.

Findings

Classified all bicrossed products of two Taft algebras.

Explicitly described the generators and relations of these Hopf algebras.

Counted the number of isomorphism types and characterized their automorphism groups.

Abstract

We classify all Hopf algebras which factorize through two Taft algebras $\mathbb{T}_{n^{2}}(\bar{q})$ and respectively $T_{m^{2}}(q)$. To start with, all possible matched pairs between the two Taft algebras are described: if $\bar{q} \neq q^{n-1}$ then the matched pairs are in bijection with the group of $d$-th roots of unity in $k$, where $d = (m,\,n)$ while if $\bar{q} = q^{n-1}$ then besides the matched pairs above we obtain an additional family of matched pairs indexed by $k^{*}$. The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.