THE SCHOUTEN-NIJENHUIS BRACKET, COHOMOLOGY AND GENERALIZED POISSON STRUCTURES

Newly introduced generalized Poisson structures based on suitable skew -symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated 'Jacobi identities' are expressed as conditions on these tensors, the cohomological conten ts of which is given. In particular, we determine the linear generaliz ed Poisson structures which can be constructed on the dual spaces of s imple Lie algebras.