Discrete integrable systems and deformations of associative algebras

TL;DR

This paper explores the deep connections between discrete integrable systems and deformations of associative algebras, presenting a unified theory that links various discrete equations to algebraic deformation processes.

Contribution

It introduces a comprehensive theory of associative algebra deformations driven by a central system, unifying many discrete integrable equations as specific realizations.

Findings

Discrete equations like Boussinesq, WDVV, Schwarzian KP, BKP, and Hirota-Miwa are particular cases of the central deformation system.

The theory reveals reciprocal relationships between algebraic deformations and integrable systems.

Associativity conditions and gauge equivalences are interpreted through discrete integrable equations.

Abstract

Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. A theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the Deformation Driving Algebra and governed by the central system of equations. It is demonstrated that many discrete equations like discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BKP equations, discrete Hirota-Miwa equations for KP and BKP hierarchies are particular realizations of the central system. An interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful.An interpretation of the Menelaus relation (discrete Schwarzian KP equation), discrete Hirota-Miwa equation for KP hierarchy, consistency around the cube as the associativity conditions and the concept of gauge equivalence, for instance, between the Menelaus and KP configurations are particular examples.