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 ⊂ 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.

Abstract. First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield K((∂ −1 )) of pseudodifferential operators over K by the subalgebra K[∂] of all differential operators. Second, we show that the Dieudonnè determinant of a matrix pseudodifferential operator with coefficients in a differential subring A of K lies in the integral closure of A in K, and we give an example of a 2 × 2 matrix with entries in A[∂] whose Dieudonnè determiant does not lie in A.

The skewfield {Mathematical expression} of rational pseudodifferential operators over a differential field {Mathematical expression} is the skewfield of fractions of the algebra of differential operators {Mathematical expression}. In our previous paper, we showed that any {Mathematical expression} has a minimal fractional decomposition {Mathematical expression}, where {Mathematical expression}, and any common right divisor of {Mathematical expression} and {Mathematical expression} is a non-zero element of {Mathematical expression}. Moreover, any right fractional decomposition of {Mathematical expression} is obtained by multiplying {Mathematical expression} and {Mathematical expression} on the right by the same non-zero element of {Mathematical expression}. In the present paper, we study the ring {Mathematical expression} of {Mathematical expression} matrices over the skewfield {Mathematical expression}. We show that similarly, any {Mathematical expression} has a minimal fractional decomposition {Mathematical expression}, where {Mathematical expression} is non-degenerate, and any common right divisor of {Mathematical expression} and {Mathematical expression} is an invertible element of the ring {Mathematical expression}. Moreover, any right fractional decomposition of {Mathematical expression} is obtained by multiplying {Mathematical expression} and {Mathematical expression} on the right by the same non-degenerate element of {Mathematical expression}. We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures. © 2013 Springer Basel

For a rational differential operator L = AB −1 , the Lenard-Magri scheme of integrability is a sequence of functions F n , n ≥ 0, such that (1) B(F n+1 ) = A(F n ) for all n ≥ 0 and (2) the functions B(F n ) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of B(F n ) is unbounded, property (2) holds if and only if L belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator L is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence (F n ) starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures. Résumé: Nous démontrons quelques résultats algébriques sur l'anneau des matrices opérateurs différentiels sur un corp différentiel dans la généralité des anneaux non-commutatives principaux. Ces résultats sont utilisés dans la théorie des structures de Poisson non-locales.