On the consistency of twisted generalized Weyl algebras

Futorny, Vyacheslav; Hartwig, Jonas

doi:10.1090/S0002-9939-2012-11184-0

On the consistency of twisted generalized Weyl algebras
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by Vyacheslav Futorny and Jonas T. Hartwig PDF

Proc. Amer. Math. Soc. 140 (2012), 3349-3363 Request permission

Abstract:

A twisted generalized Weyl algebra $A$ of degree $n$ depends on a base algebra $R$, $n$ commuting automorphisms $\sigma _i$ of $R$, $n$ central elements $t_i$ of $R$ and on some additional scalar parameters.

In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for $\sigma _i$ 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.

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