Noncommutative Inverse Scattering Method for the Kontsevich system

TL;DR

This paper develops a noncommutative inverse scattering method for integrable systems on associative algebras, unifying classical and quantum cases, and constructs a noncommutative Lax matrix for Kontsevich's differential equations.

Contribution

It introduces a universal noncommutative inverse scattering framework, including Hamilton flows, Casimir elements, and a noncommutative Lax matrix, extending classical integrability concepts.

Findings

Existence of an infinite family of commuting flows

Construction of a noncommutative Lax matrix for Kontsevich's system

Unification of classical and quantum integrable systems

Abstract

We formulate an analog of Inverse Scattering Method for integrable systems on noncommutative associative algebras. In particular we define Hamilton flows, Casimir elements and noncommutative analog of the Lax matrix. The noncommutative Lax element generates infinite family of commuting Hamilton flows on an associative algebra. The proposed approach to integrable systems on associative algebras satisfy certain universal property, in particular it incorporates both classical and quantum integrable systems as well as provides a basis for further generalization. We motivate our definition by explicit construction of noncommutative analog of Lax matrix for a system of differential equations on associative algebra recently proposed by Kontsevich. First we present these equations in the Hamilton form by defining a bracket of Loday type on the group algebra of the free group with two generators. To make the definition more constructive we utilize (with certain generalizations) the Van den Bergh approach to Loday brackets via double Poisson brackets. We show that there exists an infinite family of commuting flows generated by the noncommutative Lax element.