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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Additional Information
- Vyacheslav Futorny
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo SP, 05315-970, Brazil
- MR Author ID: 238132
- Email: futorny@ime.usp.br
- Jonas T. Hartwig
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 776335
- Email: jonas.hartwig@gmail.com
- 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
- © Copyright 2012 American Mathematical Society
- 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
- MathSciNet review: 2929005