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A Darboux-Getzler theorem for scalar difference Hamiltonian operators

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