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

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 $(\A)_{\mathcal{Z}}$.