On the Consistency of Twisted Generalized Weyl Algebras

TL;DR

This paper revises the conditions for the nontriviality of twisted generalized Weyl algebras, correcting previous claims and establishing new necessary and sufficient consistency conditions for the base algebra's injective embedding.

Contribution

It identifies errors in prior sufficiency claims, introduces revised consistency conditions, and proves their necessity and sufficiency for algebra nontriviality and injectivity.

Findings

Counterexample disproves previous sufficiency claim.

New set of consistency conditions established.

Conditions ensure the algebra's nontriviality and injective base algebra embedding.

Abstract

A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms s_i of R, n central elements t_i of R and on some additional scalar parameters. In a paper by V.Mazorchuk and L.Turowska (1999) it is claimed that certain consistency conditions for s_i and t_i are sufficient for the algebra to be nontrivial. However, in this paper we give an example which shows that this is false. We also correct the statement by finding a new set of consistency conditions and prove that the old and new conditions together are necessary and sufficient for the base algebra R to map injectively into A. In particular they are sufficient for the algebra A to be nontrivial. We speculate that these consistency relations may play a role in other areas of mathematics, analogous to the role played by the Yang-Baxter equation in the theory of integrable systems.