Casati M. A Darboux-Getzler theorem for scalar difference Hamiltonian operators (abstract)

Speaker: Matteo Casati

Title: A Darboux-Getzler theorem for scalar difference Hamiltonian operators

Abstract:
The classification of Hamiltonian operators in the formal calculus of variations relies on their corresponding Poisson-Lichnerowicz cohomology. We consider the case of scalar difference Hamiltonian operators, such as the ones which constitute the biHamiltonian pair for the Volterra chain, and prove that 1) the normal form of a order 1 scalar difference operator $P$ is constant; 2) $H^{p}(P)=0$ $\forall p>1$, so that in particular there are not nontrivial infinitesimal deformations and any infinitesimal deformation can be extended to an Hamiltonian operator; 3) as a consequence, any higher order compatible Hamiltonian operator can be brought to the constant, order 1 form by a (Miura-like) change of coordinates.

Slides: CasatiTrieste2018slides.pdf

References:
arXiv:1810.08446

Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.