Zero divisors in reduction algebras

TL;DR

This paper proves that reduction algebras associated with Lie algebras and their reductive subalgebras have no zero divisors, extending known algebraic properties and conjectures to a broader class of algebras.

Contribution

It establishes the absence of zero divisors in reduction algebras and extends the Gelfand--Kirillov conjecture to these algebras, including diagonal reduction algebras of sl type.

Findings

No zero divisors in reduction algebras of Lie algebras.

Extension of Gelfand--Kirillov conjecture to reduction algebras.

Verification of the conjecture in a basic example.

Abstract

We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respect to its reductive Lie sub-algebra k. The class of reduction algebras include the Lie algebras (they arise when k is trivial) and the Gelfand--Kirillov conjecture extends naturally to the reduction algebras. We formulate the conjecture for the diagonal reduction algebras of sl type and verify it on a simplest example.