A Darboux-Getzler theorem for scalar difference Hamiltonian operators

In this paper we extend the notion of Poisson–Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators, to the difference case. 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 information about the center, the symmetries and the admissible deformations of such an 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. We study the Poisson–Lichnerowicz cohomology for the operator \(K_0=S−S^{−1}\), which is the normal form for (−1,1) order scalar difference Hamiltonian operators; we obtain the same result as Getzler did, namely \(H^p(K_0)=0∀p>1\), 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.