Crossed products and Galois extensions of Hopf algebras

Robert James Blattner and Susan Montgomery

Vol. 137 (1989), No. 1, 37–54

DOI: 10.2140/pjm.1989.137.37

Abstract

In this paper we explore further the subject of crossed products A#_σH of an arbitrary Hopf algebra H (weakly) acting on a non-commutative algebra A over a field k. These general crossed products, which play a fundamental role in the theory of extensions of Hopf algebras, were introduced independently by Y. Doi and M. Takeuchi and by the present authors and M. Cohen. Here we give several characterizations of crossed products A#_σH with invertible cocycle σ. These characterizations are then used to extend to such crossed products known results for smash products concerning duality and Maschke-type theorems. We also prove a Noether-Skolem theorem using these methods.

Mathematical Subject Classification

Primary: 16A24, 16A24

Milestones

Received: 4 March 1988

Published: 1 March 1989