A Darboux-Getzler theorem for scalar difference Hamiltonian operators

TL;DR

This paper extends Poisson-Lichnerowicz cohomology to scalar difference Hamiltonian operators, proving triviality of higher cohomology and explicitly computing the lowest cohomology groups, advancing the understanding of their deformations.

Contribution

It introduces the Poisson-Lichnerowicz cohomology framework for scalar difference Hamiltonian operators and computes key cohomology groups, paralleling known differential case results.

Findings

Higher cohomology groups vanish for the operator studied.

Explicit descriptions of the zeroth and first cohomology groups.

Application to classification of scalar Hamiltonian operators.

Abstract

In this paper we extend to the difference case the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators. A local scalar difference Hamiltonian operator is a polynomial in the shift operator and its inverse, with coefficients in the algebra of difference functions, endowing the space of local functionals with the structure of a Lie algebra. Its Poisson-Lichnerowicz cohomology carries the information about the center, the symmetries and the admissible deformations of such algebra. The analogue notion for the differential case has been widely investigated: the first and most important result is the triviality of all but the lowest cohomology for first order Hamiltonian differential operators, due to Getzler arXiv:math/0002164 . We study the Poisson-Lichnerowicz cohomology for the operator $K_0 = \mathcal{S} - \mathcal{S}^{-1}$, which is the normal form for $(-1,1)$ order scalar difference Hamiltonian operators; we obtain the same result as Getzler did, namely $H^p(K_0)=0$ $\forall p > 1$, and explicitly compute $H^0(K_0)$ and $H^1(K_0)$. We then apply our main result to the classification of lower order scalar Hamiltonian operators recently obtained by De Sole, Kac, Valeri and Wakimoto arXiv:1806.05536