Quantized Weyl algebras at roots of unity

TL;DR

This paper classifies the centers of quantized Weyl algebras at roots of unity, derives explicit discriminant formulas using Poisson geometry and quantum cluster algebra techniques, and applies these results to automorphism and isomorphism problems.

Contribution

It provides a comprehensive classification of PI quantized Weyl algebras at roots of unity, including explicit discriminant formulas and applications to automorphism and isomorphism problems.

Findings

Classified centers of PI quantized Weyl algebras at roots of unity.

Derived explicit discriminant formulas via two different methods.

Solved automorphism and isomorphism problems for these algebras.

Abstract

We classify the centers of the quantized Weyl algebras that are PI 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.