ON THE GEOMETRY OF LIE-ALGEBRAS AND POISSON TENSORS

A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced and an algorithm to construct all solvable Lie algebras is presented. Poisson-Liouville structures are also introduced to discuss a new class of Lie algebras which include, as a subclass, semi-simple Lie algebras. A decomposition theorem for Poission tensors is proved for a class of Poisson manifolds including linear ones. Simple Lie algebras are also discussed from this viewpoint and lower-dimensional real Lie algebras are analysed.