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Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

På svenska

Author

  • Jonas T. Hartwig
  • Johan Öinert

Summary, in English

In this paper we prove three theorems about twisted generalized Weyl algebras (TGWAs). First, we show that each non-zero ideal of a TGWA has non-zero intersection with the centralizer of the distinguished subalgebra R. This is analogous to earlier results known to hold for crystalline graded rings. Second, we give necessary and sufficient conditions for the centralizer of R to be commutative (hence maximal commutative), generalizing a result by V. Mazorchuk and L. Turowska. And third, we generalize results by D.A. Jordan and V. Bavula on generalized Weyl algebras by giving necessary and sufficient conditions for certain TGWAs to be simple, in the case when R is commutative. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples. We also discuss how simplicity of a TGWA is related to the maximal commutativity of R and the (non-)existence of non-trivial Z^n-invariant ideals of R.

Publishing year

2013

Language

English

Pages

312-339

Publication/Series

Journal of Algebra

Volume

373

Document type

Journal article

Publisher

Elsevier

Topic

  • Mathematics

Keywords

  • Twisted generalized Weyl algebra
  • Simple ring
  • Maximal commutative subalgebra

Status

Published

Research group

  • Non-commutative Geometry

ISBN/ISSN/Other

  • ISSN: 0021-8693