Double Multiplicative Poisson Vertex Algebras
Maxime Fairon, Daniele Valeri

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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons
