Double Multiplicative Poisson Vertex Algebras

TL;DR

This paper introduces double multiplicative Poisson vertex algebras, establishing their theoretical foundations, classification, and connection to non-abelian integrable differential-difference equations, expanding the algebraic framework for integrable systems.

Contribution

It develops the theory of double multiplicative Poisson vertex algebras, linking them to local lattice double Poisson algebras and non-abelian integrable equations.

Findings

Established a one-to-one correspondence with local lattice double Poisson algebras.

Derived classification results for these algebraic structures.

Connected the theory to non-abelian integrable differential-difference equations.

Abstract

We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on the corresponding representation spaces. Moreover, we prove that they are in one-to-one correspondence with local lattice double Poisson algebras, a new important class among Van den Bergh's double Poisson algebras. We derive several classification results, and we exhibit their relation to non-abelian integrable differential-difference equations. A rigorous definition of double multiplicative Poisson vertex algebras in the non-local and rational cases is also provided.