Some properties of abstract Galois algebras and generalized Weyl algebras

Abstract

In this paper we develop certain structural and representation theoretical aspects of abstract generalized Weyl algebra and Galois algebras. The later class of algebras has been introduced by V. Futorny and S. Ovsienko and has lead to recent breakthroughs in representation theory. From the structure theory side, we show how this theory provides a simple criteria to show that certain algebras are not PI-algebras. Then we compare many classes of Galois algebras considered recently, and show that the Gelfand-Kirillov Conjecture is satisfied in a broad situation. Refining previous work, we show that the invariants of a broad class of generalized Weyl algebras under the action of complex reflection groups G(m, p, n) are Galois orders. Finally, we consider the lifting of maximal ideals in generalized Weyl algebras.