Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices

TL;DR

This paper investigates the structure of characteristic Lie rings in 2+1 dimensional Toda lattices, demonstrating that integrable cases have finitely generated modules, and proposes a classification approach based on these properties.

Contribution

It introduces a new framework linking characteristic Lie rings and module properties to integrability in 2+1 dimensional lattices, providing a basis for classification.

Findings

Integrable lattices have finitely generated modules.

Characteristic Lie rings are defined for Toda type lattices.

A classification algorithm based on module properties is discussed.

Abstract

Characteristic Lie rings for Toda type 2+1 dimensional lattices are defined. Some properties of these rings are studied. Infinite sequence of special kind modules are introduced. It is proved that for known integrable lattices these modules are finitely generated. Classification algorithm based on this observation is briefly discussed.