On the automorphisms of quantum Weyl algebras

Kitchin, Andrew P.; Launois, Stéphane

doi:10.48550/arxiv.1511.01775

Cited by 4 publications

(3 citation statements)

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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: 73%

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: 73%

The automorphism groups for a family of generalized Weyl algebras

Tang

2018

J. Algebra Appl.

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“…The Dixmier conjecture has been proved to be stably equivalent to the Jacobian conjecture [21,7,34]. There have been several works studying a quantum analogue of the Dixmier conjecture for some quantum algebras [3,30,22,23,33]. In particular, it has recently been proved in [23] that each K−algebra endomorphism of a simple localization of…”

Section: Introductionmentioning

confidence: 99%

Automorphisms for Some "symmetric" Multiparameter Quantized Weyl Algebras and Their Localizations

Tang¹

2016

Preprint

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In this paper, we study the algebra automorphisms and isomorphisms for a family of "symmetric" multiparameter quantized Weyl algebras A q,Λ n (K) and some related algebras in the generic case. First, we compute the Nakayama automorphism for A q,Λ n (K) and give a necessary and sufficient condition for A q,Λ n (K) to be Calabi-Yau. We also prove that A q,Λ n (K) is cancellative. Then we determine the automorphism group for A q,Λ n (K) and its polynomial extension A q,Λ n (K[t]). As an application, we solve the isomorphism problem for {A q,Λ n (K)} and {A q,Λ n (K[t])}. Similar results will be established for the Maltisiniotis multiparameter quantized Weyl algebra A q,Γ n (K) and its polynomial extension A q,Γ n (K[t]). In addition, we prove a quantum analogue of the Dixmier conjecture for a simple localization (A q,Λ n (K)) Z of A q,Λ n (K). Moreover, we will completely determine the algebra automorphism group for (A q,Λ n (K)) Z .

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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).…”

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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 4 publications

References 20 publications

The automorphism groups for a family of generalized Weyl algebras

The automorphism groups for a family of generalized Weyl algebras

Automorphisms for Some "symmetric" Multiparameter Quantized Weyl Algebras and Their Localizations

Quantized Weyl algebras at roots of unity

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