Noncommutative differential calculi and the unifying zero curvature representation of integrable systems

TL;DR

This paper introduces a unified framework for representing various integrable systems using noncommutative differential calculus, emphasizing the role of zero-curvature representations across different types of systems.

Contribution

It develops a connection and curvature within a deformed derivation-based calculus, unifying the zero-curvature representation for continuum and discrete integrable systems.

Findings

Unified zero-curvature representation for different integrable systems

Application of noncommutative differential calculus in integrable systems

Framework applicable to continuum and discrete cases

Abstract

Derivation-based differential calculi are of great importance in noncommutative geometry, noncommutative gauge theory and integrable systems. In this paper, we propose the connection and curvature from a class of deformed derivation-based differential calculus. By means of this theory, we give out the zero-curvature representation of the continuum-continuum, discrete-continuum and discrete-discrete integrable systems in an unifying manner.