Galois orders of symmetric differential operators

Abstract

In this survey we discuss the theory of Galois rings and orders developed in by Sergey Ovsienko and the first author. This concept allows to unify the representation theories of Generalized Weyl Algebras and of the universal enveloping algebras of Lie algebras. It also had an impact on the structure theory of algebras. In particular, this abstract framework has provided a new proof of the Gelfand-Kirillov Conjecture in the classical and the quantum case for gln and sln in and, respectively. We will give a detailed proof of the Gelfand-Kirillov Conjecture in the classical case and show that the algebra of symmetric differential operators has a structure of a Galois order.