On the Semiprimitivity and Semiprimality Problems for Partial Smash Products

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 𝕜 and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and their relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A#H is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over 𝕜 and 𝕜 is a perfect field, or A is locally finite. Moreover, we prove that A#H is semiprime provided that A is an H-semiprime PI-algebra, generalizing to the setting of partial actions the known results for global actions of Hopf algebras.