Multiplicative Poisson vertex algebras and the classification of difference equations

Matteo Casati (University of Kent)

Tuesday 12th February 16:00-17:00 Maths 311B

Abstract

The theory of Poisson vertex algebras, introduced by Barakat, De Sole and Kac to provide an algebraic framework for Hamiltonian PDEs, can be adapted to characterise Hamiltonian difference operators and evolutionary difference equations. Local and nonlocal "Multiplicative Poisson vertex algebras" have been very recently defined for this purpose by De Sole, Kac, Valeri and Wakimoto [arXiv:1806.05536 and arXiv:1809.01735]. In particular, the authors provide the classification of local Hamiltonian difference structures up to the order (-5,5). A remark on the integrable systems arising from the biHamiltonian pairs present in such a classification, originally due to Carpentier and observing that all the corresponding hierarchies are always equivalent to the Volterra chain is explained as a consequence of the triviality of the Poisson-Lichnerowicz cohomology of the  constant Hamiltonian difference operator of order (-1,1) [arXiv:1810.08446].

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