LOCAL STRUCTURES OF KV-ALGEBROIDS. LIE ALGEBROIDS

Michel Nguiffo Boyom

The term manifold is used for smooth or complex analytic manifolds. Manifolds considered are connected and paracompact. All geometric data on a manifold M are smooth or complex analytic according to M is smooth or complex analytic. Given a manifold MF(M,K) stands for the associatve K-algebra of real-valued smoth fonctions if K = R or the sheaf of complex analytic functions if K is C. Given a vector bundle over M sect (A) stands for the F(M,K) -module of sections ( resp. local sections if K = C ) of A.

A vector bundle A is a Koszul-Vinberg algebroid ( resp . Lie algebroid ) if (1) There is a vector bundle M-homomorphism from A to TM ; (2) sect(A) is endoved with a Koszul-Vinberg algebra structure { sect(A) ,. } (resp. Lie algebra structure { sect(A) , [ , ] ) such that for S and S’ in sect(A) and f in F(M,K) one gets (fS).S’ = f(S.S’) and S.(fS’) = f(S.S’) + (a(S)f)S’ ( resp. [ S , fS’ ] = f[ S , S’ ] + (a(S)f)S’.)

The purpose of the conference is to prove a decomposition theorem ( normal forms ) of KV-algebroids and Lie algebroids . Local classification of KV-algebroids (resp. Lie algebroids ) is deduced from the normal forms theorem .