Rational recursion operators for integrable differential-difference equations

In this paper we introduce preHamiltonian pairs of difference operators and study their connections with Nijenhuis operators and the existence of weakly non-local inverse recursion operators for differential-difference equations. We begin with a rigorous setup of the problem in terms of the skew field $Q$ of rational (pseudo--difference) operators over a difference field $F$ with a zero characteristic subfield of constants $k\subset F$ and the principal ideal ring $M_n(Q)$ of matrix rational (pseudo-difference) operators. In particular, we give a criteria for a rational operator to be weakly non--local. A difference operator $H$ is called preHamiltonian, if its image is a Lie $k$-subalgebra with respect the the Lie bracket on $F$. Two preHamiltonian operators form a preHamiltonian pair if any $k$-linear combination of them is preHamiltonian. Then we show that a preHamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a preHamiltonian pair. This provides a systematical method to check whether a rational operator is Nijenhuis. As an application, we construct a preHamiltonian pair and thus a Nijenhuis recursion operator for the differential-difference equation recently discovered by Adler \& Postnikov. The Nijenhuis operator obtained is not weakly non-local. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well known examples including the Toda, the Ablowitz-Ladik and the Kaup-Newell differential-difference equations.