On the semiprimitivity and the semiprimality problems for partial smash products

TL;DR

This paper investigates the conditions under which partial smash products of algebras with semisimple Hopf algebras are semiprimitve or semiprime, extending classical results to the setting of partial actions.

Contribution

It generalizes existing results on semiprimitivity and semiprimality to partial smash products involving Hopf algebras and partial actions.

Findings

If A is H-semiprimitive, then the partial smash product is semiprimitive under certain conditions.

The partial smash product is semiprime if A is H-semiprime and a PI-algebra.

Results extend classical theorems to the context of partial actions.

Abstract

In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let $H$ be a semisimple Hopf algebra over a field $\mathbb{k}$ and let $A$ be a left partial $H$-module algebra. We study the $H$-prime and the $H$-Jacobson radicals of $A$ and its relations with the prime and the Jacobson radicals of $\underline{A\#H}$, respectively. In particular, we prove that if $A$ is $H$-semiprimitive, then $\underline{A\#H}$ is semiprimitive provided that all irreducible representations of $A$ are finite-dimensional, or $A$ is an affine PI-algebra over $\mathbb{k}$ and $\mathbb{k}$ is a perfect field, or $A$ is locally finite. Moreover, we prove that $\underline{A\#H}$ is semiprime provided that $A$ is an $H$-semiprime PI-algebra, generalizing for the setting of partial actions, the main results of [20] and [19].