Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras
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Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras. / Hartwig, J.T. ; Öinert, Per Johan.
I: Journal of Algebra, Bind 373, 2013, s. 312-339.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras
AU - Hartwig, J.T.
AU - Öinert, Per Johan
PY - 2013
Y1 - 2013
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
VL - 373
SP - 312
EP - 339
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
ER -
ID: 113989408