A Darboux–Getzler Theorem for Scalar Difference 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. We study the Poisson-Lichnerowicz cohomology for the operator K0 = 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 (K0) = 0 ∀p > 1, and explicitly compute H 0 (K0) and H 1 (K0). We then apply our main result to the classification of lower order scalar Hamiltonian operators recently obtained by

Hamiltonian structures for integrable nonabelian difference equations

“…This is the basic idea leading to the definition of double Poisson algebras [59], double Poisson vertex algebras [17], and multiplicative double Poisson vertex algebra (see Section 3.3). In this paper, we show how the classical (and somehow geometric, in the sense that it exploits the language and machinery widely used in Poisson geometry) notion of Poisson bivector, and of Schouten brackets between polyvector fields, can be tailored to the functional nonabelian case (functional polyvector fields for Abelian differential systems are very well known and long-established, see for instance [49], and we have introduced them for Abelian differentialdifference systems in [13]). The Schouten brackets we defined in Section 4 unify the two aforementioned languages, as well as the standard language of Poisson geometry.…”

“…The definition of local p-vector fields (see [13] for the difference Abelian case) must be postponed to Section 3; however, for computational reasons we anticipate the main practical result, namely that it is possible to adopt a tailored version of the so-called θ formalism following Olver and Sokolov's treatment [50].…”

“…4.1. θ formalism and functional polyvector fields. The complex of functional polyvector fields, that is presented in [32,13] for the difference Abelian case, generalises naturally to the nonabelian case we are dealing with; the θ formalism for the nonabelian differential case is presented in [50] and the one for the difference case in [14].…”

“…We have defined the Poisson cohomology of F using a Poisson bivector and the notion of Schouten bracket. However, the computation of such cohomology even in the commutative case is a challenging task (see for instance [35,11,13] for the Abelian differential and difference case). An investigation of the Poisson cohomology in the nonabelian case will be discussed in a forthcoming work.…”

Recursion and Hamiltonian operators for integrable nonabelian difference equations

“…The notion of functional vector field was introduced in the context of Hamiltonian (commutative) PDEs [23,36,21]. It was then generalised to the nonabelian case [38] and to the commutative difference one [24,11].…”