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On the consistency of twisted generalized Weyl algebras

Authors: Vyacheslav Futorny and Jonas T. Hartwig
Journal: Proc. Amer. Math. Soc. 140 (2012), 3349-3363
MSC (2010): Primary 16D30, 16S35; Secondary 16S85
Published electronically: February 17, 2012
MathSciNet review: 2929005
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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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Additional Information

Vyacheslav Futorny
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo SP, 05315-970, Brazil

Jonas T. Hartwig
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Received by editor(s): March 22, 2011
Received by editor(s) in revised form: April 7, 2011
Published electronically: February 17, 2012
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2012 American Mathematical Society

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