On actions of Drinfel'd doubles on finite dimensional algebras

TL;DR

This paper investigates how Drinfel'd doubles of certain finite-dimensional Hopf algebras, including Taft algebras and others, act on finite-dimensional algebras, extending previous classifications and exploring the uniqueness of such actions.

Contribution

It analyzes actions of Drinfel'd doubles of Hopf algebras close to Taft algebras on finite-dimensional algebras, highlighting conditions for unique module algebra structures.

Findings

Actions of Drinfel'd doubles extend uniquely in specific cases

Identifies properties of Hopf algebras that lead to unique module algebra structures

Examines a variety of Hopf algebras including Sweedler and quantum groups

Abstract

Let $q$ be an $n^{th}$ root of unity for $n > 2$ and let $T_n(q)$ be the Taft (Hopf) algebra of dimension $n^2$. In 2001, Susan Montgomery and Hans-J\"urgen Schneider classified all non-trivial $T_n(q)$-module algebra structures on an $n$-dimensional associative algebra $A$. They further showed that each such module structure extends uniquely to make $A$ a module algebra over the Drinfel'd double of $T_n(q)$. We explore what it is about the Taft algebras that leads to this uniqueness, by examining actions of (the Drinfel'd double of) Hopf algebras $H$ "close" to the Taft algebras on finite-dimensional algebras analogous to $A$ above. Such Hopf algebras $H$ include the Sweedler (Hopf) algebra of dimension 4, bosonizations of quantum linear spaces, and the Frobenius-Lusztig kernel $u_q(\mathfrak{sl}_2)$.