Endomorphisms of Quantum Generalized Weyl Algebras

Kitchin, Andrew P.; Launois, Stéphane

doi:10.1007/s11005-014-0691-4

Cited by 11 publications

(13 citation statements)

References 22 publications

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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 A…”

Section: Introductionmentioning

confidence: 99%

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

Tang

2017

Algebra Colloq.

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We study a family of “symmetric” multiparameter quantized Weyl algebras [Formula: see text] and some related algebras. We compute the Nakayama automorphism of [Formula: see text], give a necessary and sufficient condition for [Formula: see text] to be Calabi-Yau, and prove that [Formula: see text] is cancellative. We study the automorphisms and isomorphism problem for [Formula: see text] and [Formula: see text]. Similar results are established for the Maltsiniotis multiparameter quantized Weyl algebra [Formula: see text] and its polynomial extension. We prove a quantum analogue of the Dixmier conjecture for a simple localization [Formula: see text] and determine its automorphism group.

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Section: Introductionmentioning

confidence: 99%

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

Tang

2017

Algebra Colloq.

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“…For instance, Richard first proved that each algebra endomorphism of a simple quantum torus is actually an algebra automorphism, and gave a criterion for an algebra endomorphism to be an algebra automorphism for non-simple quantum tori in [18]. Most recently, this result has been generalized to simple generalized Weyl algebras defined over the base ring K[h ±1 ] in [16]. Additionally, the relationship between algebra endomorphisms and automorphisms of mixed quantum polynomial algebras was also studied in [2].…”

Section: Introductionmentioning

confidence: 99%

Algebra endomorphisms and derivations of some localized down-up algebras

Tang

2014

J. Algebra Appl.

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We study algebra endomorphisms and derivations of some localized down-up algebras A S (r + s, −rs). First, we determine all the algebra endomorphisms of A S (r + s, −rs) under some conditions on r and s. We show that each algebra endomorphism of A S (r + s, −rs) is an algebra automorphism if r m s n = 1 implies m = n = 0. When r = s −1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A S (r + s, −rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A S (r + s, −rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r +s, −rs) is an algebra automorphism in either case. Second, we determine all the derivations of A S (r + s, −rs) and calculate its first degree Hochschild cohomology group.

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“…where n and σ i are defined as they have been throughout. Based on the results of this article, and those obtained in [19] and [30], we believe it is possible to show that every endomorphism is an automorphism when q i 1 1 q i 2 2 · · · q in n = 1 implies i 1 = i 2 = · · · = i n = 0, or when q = (q, . .…”

Section: A Quantum Tame Generators Problemmentioning

confidence: 67%

On the automorphisms of quantum Weyl algebras

Kitchin

Launois

2019

Journal of Pure and Applied Algebra

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Motivated by Weyl algebra analogues of the Jacobian conjecture and the tame generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of U + q (so 5 ).Mathematics Subject Classification (2010). 16W35, 16S32, 16W20, 17B37.

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Endomorphisms of Quantum Generalized Weyl Algebras

Cited by 11 publications

References 22 publications

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

Algebra endomorphisms and derivations of some localized down-up algebras

On the automorphisms of quantum Weyl algebras

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