Quantum Weyl algebras and reflection equation algebras at a root of unity

TL;DR

This paper analyzes quantum Weyl algebras and reflection equation algebras at roots of unity, computing their centers and Azumaya loci to understand their algebraic structure and representation theory.

Contribution

It provides explicit computations of centers and Azumaya loci for specific quantum algebras at roots of unity, illustrating broader phenomena in quantum algebra.

Findings

Computed centers of quantum Weyl algebras and quantum differential operators.

Identified Azumaya loci in these quantum algebras.

Demonstrated elementary examples of complex quantum phenomena.

Abstract

We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].