MENU

You are here

Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type

TitleMultidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type
Publication TypeThesis
Year of Publication2015
AuthorsCasati, M
UniversitySISSA
KeywordsPoisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems
Abstract

The Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical
Poisson brackets, defined on an infinite dimensional space of maps Σ → M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson–Lichnerowicz cohomology. From this computations it
follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.

Custom 1

34902

Custom 2

Mathematics

Custom 4

1

Custom 5

MAT/07

Custom 6

Submitted by Matteo Casati (mcasati@sissa.it) on 2015-10-22T07:27:52Z
No. of bitstreams: 1
PhDThesis_Casati.pdf: 1027291 bytes, checksum: 49f551db40603ca035f2515ccb6ec7a2 (MD5)

Sign in