Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

TL;DR

This paper investigates the structure of twisted generalized Weyl algebras (TGWAs), establishing conditions for their simplicity and maximal commutativity, and providing concrete examples and generalizations of previous rank one results.

Contribution

It introduces new criteria for simplicity and commutativity in TGWAs, extending prior work from the rank one case to higher ranks and exploring the role of the centralizer.

Findings

Non-zero ideals intersect the centralizer non-trivially

Conditions for the centralizer to be commutative are provided

Necessary and sufficient conditions for simplicity of TGWAs are established

Abstract

In this paper we show that each non-zero ideal of a twisted generalized Weyl algebra (TGWA) $A$ intersects the centralizer of the distinguished subalgebra $R$ in $A$ non-trivially. We also provide a necessary and sufficient condition for the centralizer of $R$ in $A$ to be commutative, and give examples of TGWAs associated to symmetric Cartan matrices satisfying this condition. By imposing a certain finiteness condition on $R$ (weaker than Noetherianity) we are able to make an Ore localization which turns out to be useful when investigating simplicity of the TGWA. Under this mild assumption we obtain necessary and sufficient conditions for the simplicity of TGWAs. We describe how this is related to maximal commutativity of $R$ in $A$ and the (non-) existence of non-trivial $\Z^n$-invariant ideals of $R$. Our result is a generalization of the rank one case, obtained by D. A. Jordan in 1993. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples.