On the Semiprime Smash Product Question

Abstract

This is a survey article on a question, posed in 1986 by M.Cohen and D.Fishman, whether the smash product A#H of a semisimple Hopf algebra and a semiprime left H-module algebra A is itself semiprime.

Prime and Semiprime Quantum Linear Space Smash Products

“…More generally, many authors have considered the question of (semi)primeness for smash products A#H, where H is a Hopf algebra and A an H-module algebra [3, 8, 9, 12, 16, 17, 19, 23]. The interested reader is directed to the survey article by Lomp for a thorough account of the state of the art with regard to semiprime smash products [18]. This paper concerns the question of (semi)prime extensions for certain pointed Hopf algebras H. In particular, we build on the work of Bergen who studied this question when H is the Taft algebra [3].The Taft algebras were originally defined by Earl Taft [24] in the study of finitedimensional Hopf algebras whose antipode had arbitrarily large order.…”