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.
