Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

J.T.  Hartwig; Per Johan Öinert

doi:10.1016/j.jalgebra.2012.10.009

Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

J.T. Hartwig, Per Johan Öinert

Department of Mathematical Sciences

6 Citations (Scopus)

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 Zn-invariant ideals of R.

Original language	English
Journal	Journal of Algebra
Volume	373
Pages (from-to)	312-339
ISSN	0021-8693
DOIs	https://doi.org/10.1016/j.jalgebra.2012.10.009
Publication status	Published - 1 Jan 2013

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10.1016/j.jalgebra.2012.10.009

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title = "Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras",

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 Zn-invariant ideals of R.",

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T1 - Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

AU - Hartwig, J.T.

AU - Öinert, Per Johan

PY - 2013/1/1

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N2 - 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 Zn-invariant ideals of R.

AB - 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 Zn-invariant ideals of R.

U2 - 10.1016/j.jalgebra.2012.10.009

DO - 10.1016/j.jalgebra.2012.10.009

M3 - Journal article

SN - 0021-8693

VL - 373

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EP - 339

JO - Journal of Algebra

JF - Journal of Algebra

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