Quantizations of multiplicative hypertoric varieties at a root of unity

TL;DR

This paper constructs a quantization framework for multiplicative hypertoric varieties at roots of unity, revealing their Azumaya algebra structure and establishing a localization theorem for associated quantum groups.

Contribution

It introduces a novel quantization method for multiplicative hypertoric varieties using q-difference operators at roots of unity, including explicit Azumaya algebra splitting and a localization theorem.

Findings

Construction of a matrix bundle (Azumaya algebra) over the variety.

Explicit finite étale splitting of the Azumaya algebra.

Establishment of a localization theorem for the hypertoric quantum group.

Abstract

We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite \'etale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.