Smash products, inner actions and quotient rings

Miriam Cohen

Vol. 125 (1986), No. 1, 45–66

DOI: 10.2140/pjm.1986.125.45

Abstract

Let H be a Hopf algebra over a field k, and A an H-module algebra over k. Let A^H = {a ∈ A|h⋅a = 𝜖(h)a, all h ∈ H}. This paper is mainly concerned with inner actions. We prove the existence of a “symmetric” quotient ring Q of A, which is also an H-module algebra, and consider Q-inner actions, an analogue of X-inner automorphisms. Under certain conditions on A and H we show that Q contains B, a finite-dimensional separable algebra over its center C, a field. Moreover, the centralizer of B in Q is Q^H. This is used to prove that if A^H is P.I. then so is A, and that A is fully integral over A^HC of bounded degree. We also consider connections between the A, A^H and A#H module structures.

Mathematical Subject Classification

Primary: 16A24, 16A24

Secondary: 16A08

Milestones

Received: 2 May 1985

Published: 1 November 1986