Galois orders of symmetric differential operators

Author(s)

Futorny, Vyacheslav ; Schwarz, Joao

• Document type: Journal article
• Source: ALGEBRA & DISCRETE MATHEMATICS; v. 23, n. 1, p. 35-46, 2017.
• Web of Science Citations: 0

Abstract

In this survey we discuss the theory of Galois rings and orders developed in ({[}20], {[}22]) by Sergey Ovsienko and the first author. This concept allows to unify the representation theories of Generalized Wey1 Algebras ({[}4]) 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 ({[}24]) in the classical and the quarittini case for gl(n), and sl(n) in {[}18] and {[}21], 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. (AU)

• FAPESP's process: 14/09310-5 - Algebraic structures and their representations
• Grantee: Vyacheslav Futorny
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 14/25612-1 - Extensions of Noether's problem and Gelfand-Kirillov's conjecture to certain classes of noncommutative algebras
• Grantee: João Fernando Schwarz
• Support Opportunities: Scholarships in Brazil - Doctorate