Hamiltonian structures for integrable nonabelian difference equations

TL;DR

This paper develops a comprehensive algebraic and geometric framework for Hamiltonian structures in nonabelian difference equations, introducing new algebraic objects and classifying key integrable lattice systems.

Contribution

It introduces multiplicative double Poisson vertex algebras and characterizes nonabelian Poisson structures, advancing the understanding of Hamiltonian systems in noncommutative settings.

Findings

Established a correspondence between multiplicative double PVAs and difference operator Poisson structures

Defined nonabelian polyvector fields and Schouten brackets for noncommutative algebras

Constructed Hamiltonian structures for nonabelian Kaup, Ablowitz-Ladik, and Chen-Lee-Liu lattices

Abstract

In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and geometric (in terms of nonabelian Poisson bivectors) definitions. We introduce multiplicative double Poisson vertex algebras (PVAs) as the suitable noncommutative counterpart to multiplicative PVAs, used to describe Hamiltonian differential-difference equations in the commutative setting, and prove that these algebras are in one-to-one correspondence with the Poisson structures defined by difference operators, providing a sufficient condition for the fulfilment of the Jacobi identity. Moreover, we define nonabelian polyvector fields and their Schouten brackets, for both finitely generated noncommutative algebras and infinitely generated difference ones: this allows us to provide a unified characterisation of Poisson bivectors and double quasi-Poisson algebra structures. Finally, as an application we obtain some results towards the classification of local scalar Hamiltonian difference structures and construct the Hamiltonian structures for the nonabelian Kaup, Ablowitz-Ladik and Chen-Lee-Liu integrable lattices.