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

TL;DR

This paper investigates automorphisms, isomorphisms, and structural properties of symmetric multiparameter quantized Weyl algebras and their localizations, including automorphism groups, Calabi-Yau conditions, and a quantum Dixmier conjecture analogue.

Contribution

It provides explicit automorphism groups, solves the isomorphism problem, and establishes a quantum analogue of the Dixmier conjecture for these algebras.

Findings

Computed Nakayama automorphism and Calabi-Yau conditions.

Determined automorphism groups for algebras and their extensions.

Proved a quantum analogue of the Dixmier conjecture.

Abstract

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