On the automorphisms of quantum Weyl algebras

Kitchin, Andrew P.; Launois, Stéphane

doi:10.1016/j.jpaa.2018.06.016

Cited by 5 publications

(3 citation statements)

References 28 publications

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“…It was proved by Richard that each algebra endomorphism of a simple quantum torus is an algebra automorphism in [20]. It was also proved that a quantum analogue of the Dixmier conjecture holds for some quantum generalized Weyl algebras in [15,16]. Similar results were established for the simple localizations of some two-parameter down-up algebras in [25] and some simple localizations of the multiparameter quantized Weyl algebras in [26].…”

Section: Introductionmentioning

confidence: 70%

The automorphism groups for a family of generalized Weyl algebras

Tang

2018

J. Algebra Appl.

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Abstract. In this paper, we study a family of generalized Weyl algebras {A p (λ, µ, K q [s, t])} and their polynomial extensions. We will show that the algebra) S when none of p and q is a root of unity. As an application, we determine all the height-one prime ideals and the center foris cancellative. Then we will determine the automorphism group and solve the isomorphism problem for the generalized Weyl algebras A p (λ, µ, K q [s, t]) and their polynomial extensions in the case where none of p and q is a root of unity. We will establish a quantum analogue of the Dixmier conjecture and compute the automorphism group for the simple localization (A p (1, 1, K q [s, t])) S . Moreover, we will completely determine the automorphism group for the algebra A p (1, 1, K q [s, t]) and its polynomial extension when p = 1 and q = 1.

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Section: Introductionmentioning

confidence: 70%

The automorphism groups for a family of generalized Weyl algebras

Tang

2018

J. Algebra Appl.

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“…For example, to determine whether two presented groups are isomorphic is known to be an NP hard problem. The isomorphism problem for GWAs has been studied in [2,5,9,28,30,21].…”

Section: Introductionmentioning

confidence: 99%

Classification of twisted generalized Weyl algebras over polynomial rings

Hartwig

Rosso

2020

Journal of Algebra

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Let R be a polynomial ring in m variables over a field of characteristic zero. We classify all rank n twisted generalized Weyl algebras over R, up to Z n -graded isomorphisms, in terms of higher spin 6-vertex configurations. Examples of such algebras include infinite-dimensional primitive quotients of U (g) where g = gl n , sln, or sp 2n , algebras related to U ( sl2) and a finite W-algebra associated to sl4. To accomplish this classification we first show that the problem is equivalent to classifying solutions to the binary and ternary consistency equations. Secondly, we show that the latter problem can be reduced to the case n = 2, which can be solved using methods from previous work by the authors [17], [15]. As a consequence we obtain the surprising fact that (in the setting of the present paper) the ternary consistency relation follows from the binary consistency relation.

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“…Quantized Weyl algebras. The quantized Weyl algebras and their generalizations have been studied from many different points of view: quantum groups and Hecke type quantizations [20,30], structure of prime spectra and representations [6,21,22,27], automorphism and isomorphism problems [3,16,23,28,33,34], homological and ring theoretic dimensions [15], quantizations of multiplicative hypertoric varieties [12,18] and others. Most of these results concern the generic case when the algebras are not polynomial identity (PI).…”

Section: Introductionmentioning

confidence: 99%

Quantized Weyl algebras at roots of unity

Levitt

Yakimov

2018

Isr. J. Math.

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We classify the centers of the quantized Weyl algebras that are polynomial identity algebras and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.2000 Mathematics Subject Classification. Primary: 17B37; Secondary: 53D17, 16W20, 15A69. Key words and phrases. Quantized Weyl algebras, discriminants of PI algebras, Poisson prime elements, quantum cluster algebras, automorphism and isomorphism problems.

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On the automorphisms of quantum Weyl algebras

Cited by 5 publications

References 28 publications

The automorphism groups for a family of generalized Weyl algebras

The automorphism groups for a family of generalized Weyl algebras

Classification of twisted generalized Weyl algebras over polynomial rings

Quantized Weyl algebras at roots of unity

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