Double Poisson vertex algebras and non-commutative Hamiltonian equations

TL;DR

This paper introduces double Poisson vertex algebras to analyze non-commutative Hamiltonian PDEs, extending Van den Bergh's double Poisson algebra theory to integrable hierarchies like KP and Gelfand-Dickey.

Contribution

It generalizes double Poisson algebra theory to vertex algebras, enabling the study of non-commutative Hamiltonian PDEs and hierarchies.

Findings

Developed formalism of double Poisson vertex algebras

Applied theory to non-commutative KP and Gelfand-Dickey hierarchies

Constructed non-commutative de Rham and variational complexes

Abstract

We develop the formalism of double Poisson vertex algebras (local and non-local) aimed at the study of non-commutative Hamiltionan PDEs. This is a generalization of the theory of double Poisson algebras, developed by Van den Bergh, which is used in the study of Hamiltonian ODEs. We apply our theory of double Poisson vertex algebras to non-commutative KP and Gelfand-Dickey hierarchies. We also construct the related non-commutative de Rham and variational complexes.