Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

Abstract

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.

The automorphism groups for a family of generalized Weyl algebras

“…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]. Since the generalized Weyl algebra A p (λ, µ, K q [s, t]) has a simple localization, it is natural to ask whether a quantum analogue of the Dixmier conjecture holds for the simple localization.…”

Zariski cancellation problem for non-domain noncommutative algebras

“…This paper can be considered as a sequel to [BZ1], where Bell-Zhang studied Zariski Cancellation Problem for noncommutative domains (in particular, for several families of Artin-Schelter regular algebras [AS]). Many mathematicians have been making significant contributions to this research direction and related topics, see Brown-Yakimov [BY], Ceken-Palmieri-Wang-Zhang [CPWZ1,CPWZ2], Chan-Young-Zhang [CYZ1,CYZ2], Gaddis [Ga], Gaddis-Kirkman-Moore [GKM], Gaddis-Won-Yee [GWY], Levitt-Yakimov [LY], Lü-Mao-Zhang [LMZ], Nguyen-Trampel-Yakimov [NTY], Tang [Ta1,Ta2] and others.…”

On the automorphisms of quantum Weyl algebras

“…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, . .…”

“…Following the submission to the arXiv of the preprint to this article, a Dixmier type problem for a quantized Weyl algebra was solved in [30] by Tang. The algebra studied, denoted (Aq ,Λ n (K)) Z , is isomorphic to A(n, 1, q), when q = (q 1 , .…”