POISSON STRUCTURES ON LIE ALGEBROIDS LIVIU POPESCU University of Craiova, Dept. of Applied Mathematics in Economy 13 A.I.Cuza st., 200585, Craiova, Romania e-mail: liviupopescu@central.ucv.ro, liviunew@yahoo.com Abstract. In this paper the properties of Lie algebroids with Poisson structures are investigated. We generalize some results of Fernandes [1] regarding linear contravariant connections on Poisson manifolds at the level of Lie algebroids. In the last part, the notions of complete and horizontal lifts on the prolongation of Lie algebroid are studied and their compatibility conditions are pointed out. Key words: Poisson manifolds, Lie algebroids, contravariant connection, complete and horizontal lifts. Mathematics Subject Classification (2000): 53D17, 17B66, 53C05 1. Introduction Poisson manifolds were introduced by A. Lichnerowicz in his famous pa- per [8] and their properties were later investigated by A. Weinstein [15]. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. In the last years a lot of papers deal with the study of various aspects of this subject in the different directions of research [14], [12], [1]. The Lie algebroid [9] is a generalization of a Lie al- gebra and integrable distribution. In fact, a Lie algebroid is a vector bundle with a Lie bracket on his space of sections whose properties are very similar to those of a tangent bundle. We remark that the cotangent bundle of a Poisson manifold has a natural structure of Lie algebroid. The purpose of this paper is to study some aspects of Lie algebroids ge- ometry endowed with a Poisson structures, which generalize the Poisson manifolds. The paper is organized as follows. In the section 2 we recall the Cartan calculus and the Schouten-Nijenhuis bracket at the level of on Lie algebroids and introduce the Poisson structure on Lie algebroid. We investigate the properties of linear contravariant connection and its tensors of torsion and curvature. In the last part of this section we find a Poisson connection which depends only on the Poisson bivector and structural func- tions of Lie algebroid which generalize some results of Fernandes from [1]. The section 3 deals with the prolongation of Lie algebroid [5] over the vector Date : July 30, 2008. 1