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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11184-0

Published electronically:
February 17, 2012

MathSciNet review:
2929005

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Abstract | References | Similar Articles | Additional Information

Abstract: A twisted generalized Weyl algebra of degree depends on a base algebra , commuting automorphisms of , central elements of and on some additional scalar parameters.

In a paper by Mazorchuk and Turowska, it is claimed that certain consistency conditions for and 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 to map injectively into . In particular they are sufficient for the algebra 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

Email:
futorny@ime.usp.br

**Jonas T. Hartwig**

Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305

Email:
jonas.hartwig@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-2012-11184-0

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