Discretization of Baker-Akhiezer Modules and Commuting Difference Operators in Several Discrete Variables

TL;DR

This paper introduces discrete Baker-Akhiezer modules as a discretization of continuous modules, enabling the construction of commuting difference operators with matrix coefficients across multiple discrete variables.

Contribution

It presents a novel framework for discretizing Baker-Akhiezer modules and constructs new commuting difference operators in several discrete variables.

Findings

Defined discrete Baker-Akhiezer modules as modules over difference operators

Constructed commuting difference operators with matrix coefficients

Extended the theory to multiple discrete variables

Abstract

We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a certain discretization of Baker-Akhiezer modules which are modules over the ring of differential operators. We use it to construct commuting difference operators with matrix coefficients in several discrete variables.