PreHamiltonian and Hamiltonian operators for differential-difference equations

In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo–difference Hamiltonian operator can be represented as a ratio AB−1 of two difference operators with coefficients from a difference field F, where A is preHamiltonian. A difference operator A is called preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields on F. The definition of a rational Hamiltonian operator can be reformulated in terms of its factors which simplifies the theory and makes it useful for applications. In particular we show that for a given rational Hamiltonian operator H in order to find a second Hamiltonian operator K compatible with H one only needs to find a preHamiltonian pair A and B such that K = AB−1H is skew-symmetric. We apply our theory to study multi Hamiltonian structures of Narita-Itoh-Bogayavlensky and AdlerPostnikov equations.