On rational twisted generalized Weyl algebra

TL;DR

This paper explores the structure of rational twisted generalized Weyl algebras, providing explicit formulas and constructions based on Lie algebra representations, with applications to special unitary and orthogonal algebras of rank three.

Contribution

It introduces a new construction method for rational twisted generalized Weyl algebras using Gelfand-Zeitlin realization, expanding understanding of their structure and involution-invariant subalgebras.

Findings

Explicit formulas for algebra constructions

Description of involution-symmetric subalgebras

Applications to rank three unitary and orthogonal algebras

Abstract

The aim of this work is to investigate the structure of some skew twisted algebras, when the coefficient ring is a localization of the polynomial ring over the field of characteristic zero, and an involution is provided. A parallel construction of the rational twisted generalized Weyl algebras is given. We propose a method and explicit formulas for a constructive description of these algebras and their involution-symmetric invariant subalgebras based on the Gelfand-Zeitlin realization of the universal enveloping algebra of some complex Lie algebras. As concrete examples we discuss special unitary and orthogonal algebras of rank three.