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Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

Hartwig, Jonas T. and Öinert, Johan LU (2013) In Journal of Algebra 373. p.312-339
Abstract
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... (More)
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. (Less)
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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Twisted generalized Weyl algebra, Simple ring, Maximal commutative subalgebra
in
Journal of Algebra
volume
373
pages
312 - 339
publisher
Elsevier
external identifiers
  • wos:000311918900016
  • scopus:84868133065
ISSN
0021-8693
DOI
10.1016/j.jalgebra.2012.10.009
language
English
LU publication?
yes
id
81deeeb4-d466-4a06-aa66-b737c99963ef (old id 3409319)
date added to LUP
2016-04-01 10:57:26
date last changed
2022-01-26 04:09:04
@article{81deeeb4-d466-4a06-aa66-b737c99963ef,
  abstract     = {{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.}},
  author       = {{Hartwig, Jonas T. and Öinert, Johan}},
  issn         = {{0021-8693}},
  keywords     = {{Twisted generalized Weyl algebra; Simple ring; Maximal commutative subalgebra}},
  language     = {{eng}},
  pages        = {{312--339}},
  publisher    = {{Elsevier}},
  series       = {{Journal of Algebra}},
  title        = {{Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras}},
  url          = {{http://dx.doi.org/10.1016/j.jalgebra.2012.10.009}},
  doi          = {{10.1016/j.jalgebra.2012.10.009}},
  volume       = {{373}},
  year         = {{2013}},
}