Double Multiplicative Poisson Vertex Algebras

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.

v3: 46 pages, 1 figure. Minor corrections following the referee's suggestions

References in corpus (2)

• Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems
• Hamiltonian structures for integrable nonabelian difference equations

Cited by in corpus (1)

• Around Van den Bergh's double brackets for different bimodule structures