Abstract
The aim of this note is to prove some cohomological vanishing theorems for non solvable affinelike Lie algebras, say ALLA. There are some relevant consequences of our vanishing theorems: (1) Every real non solvable affinelike Lie algebra is formally nondegenerate in the sense of A. Weinstein, (Theorem 1.1). (2) Let G be a non solvable Lie group whose Lie algebra is an affinelike Lie algebra (g,e). If the radical of [ker(ad(e)),ker(ad(e))] is commutative, then G admits a left invariant symplectic structure if and only if it has an open coadjoint orbit, (Theorem 1.2). (a) In Section 6, we use our vanishing theorems to supply an algebraic proof of the normal form theorem for Lie non solvable a-algebroids.