Skew group algebras in the representation theory of artin algebras

Abstract

Then the corresponding crossed product algebra /i * ?G, or /i x G for short, has an elements CgiEG ,gi A.-; li ~/i. Addition is componentwise, and multiplication is given by & = g(A)g and g, g2 = y(g, , g2) m. In this paper we assume that the values of y lie in the center Z(A) of/i. Hence the action of G on /i is given by a group homomorphism G-P Aut(/i), and (3) can be left out. In the special case that y is the trivial map we write /1G instead of/i * G, and the elements as C,,,c li gi. AG is then called a skew group ring. There is a lot of literature on skew group algebras and crossed product algebras, and on the relationship with the ring AC whose elements are those elements of A left fixed by G. Much work has been done on which properties of li are inherited by n * G or AC. Some of the work on the relationship between these rings has its roots in trying to develop a Galois theory for noncommutative rings. We refer to [3, 7, 13-15, 19, 21, 23-25, 27, 281 and their references. In this paper we study these constructions when /i is an artin algebra and G a finite group of order n such that n is invertible in /1. Under these assumptions the construction preserves central properties of interest in the