96 11 22 1 v 2 2 7 N ov 1 99 6 Generalized Poisson structures ∗

New generalized Poisson structures are introduced by using skew-symmetric contravariant tensors of even order. The corresponding ‘Jacobi identities’ are given by the vanishing of the Schouten-Nijenhuis bracket. As an example, we provide the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras. 1 Standard Poisson structures We report here on the generalized Poisson structures (GPS) introduced in [1]. They are different from these of Nambu [2] and Takhtajan [3]. Let us start by recalling how the standard Poisson structures are introduced. Let M be a manifold and F(M) be the associative algebra of smooth functions on M . Definition 1.1 (PB) A Poisson bracket {·, ·} on F(M) is a bilinear mapping F(M) × F(M) → F(M), (f1, f2) 7→ {f1, f2} such that {·, ·} is (a) skew-symmetric, (b) satisfies the Leibniz rule (derivation property) and (c) the Jacobi identity (JI). M is then called a Poisson manifold. Because of (a), (c) the space F(M) endowed with a PB {·, ·} becomes an (infinite-dimensional) Lie algebra. Let x be local coordinates on U ⊂ M and consider a PB of the form {f(x), g(x)} = ω(x)∂jf∂kg , ∂j = ∂ ∂xj , j, k = 1, . . . , n = dimM (1) Then, ω(x) defines a PB if ω(x) = −ω(x) [(a)] and [(c)] ω∂kω lm + ω∂kω mj + ω∂kω jl = 0 . (2) Talk at XXI Int. Coll. on Group Theoretical Methods in Physics (Goslar, July 1996). To appear in the proceedings. St. John’s College Overseas Visiting Scholar. On leave of absence from Inst. for Theor. and Exper. Phys., 117259 Moscow, Russia. On sabbatical (J.A.) leave and on leave of absence (J.C.P.B.) from Departamento de F́ısica Teórica and IFIC (Centro Mixto Univ. de Valencia-CSIC), E–46100 Burjassot (Valencia), Spain.