Annals of Global Analysis and Geometry 12: 65-78, 1994. Q 1994 Kluwer Academic Publishers. Printed in the Netherlands. Poisson-Nijenhuis Structures and the Vinogradov Bracket J.V. BELTRAN* AND J. MONTERDE t 65 Abstract: We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frblicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively. Key words: Schouten-Nijenhuis bracket, Frblicher-Nijenhuis bracket, graded Lie algebras, bihamillonian manifolds MSC 1991: 58F05,58F07,58A10 1. Introduction A Poisson-Nijenhuis structure on a differentiable manifold is a pair formed by a Poisson bivector and a Nijenhuis tensor that satisfy certain compatibility conditions. Such kind of structures has been studied in [5] and they have its origin in previous works by Magri in the theory of completely integrable Hamiltonian systems. The condition that a bivector must fulfil in order to be a Poisson bivector can be expressed by means of a suitable graded Lie bracket: the Schouten-Nijenhuis bracket. We have the same situation with the Nijenhuis tensor: there is a graded Lie bracket, the Frblicher-Nijenhuis bracket, that allows us to write the condition that a vector valued differential 1-form must fulfil in order to be a Nijenhuis tensor (see [9] and [10] for their definitions and properties). In this paper, we express the compatibility conditions between both tensor fields, the Poisson bivector and the Nijenhuis tensor, by means of a graded Lie bracket defined by A.M. Vinogradov ([3], [11]). This graded Lie bracket is not defined on tensor fields but on graded differential operators on the algebra of differential forms, and in a certain sense, it is a generalization of the other two. Let us denote by P and N the Poisson bivector and the Nijenhuis tensor, and let ip and iN denote their associated tensorial graded differential operators. We show that the compatibility conditions are precisely the vanishing conditions of the higher order part of the Vinogradov bracket of ip and iN. Moreover, we characterize the Poisson-Nijenhuis structures for which the bracket vanishes. In fact, we show that this bracket vanishes if and only if the trace of the recursion operator, i.e., the Nijenhuis tensor N, is a Casimir function for the Poisson bivector P. * Partially supported by Fundaci6 Calxa Castell6. t Partially supported by the Spanish DGICYT grant ~PB91 - 0324.