NON DEGENERACY OF AFFINELIKE LIE ALGEBRA

Michel Nguifo (1) , Stephane Puechmorel (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 ENAC - Ecole Nationale de l'Aviation Civile

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.

Domaines: Géométrie différentielle [math.DG]

Michel Nguifo, Stephane Puechmorel. NON DEGENERACY OF AFFINELIKE LIE ALGEBRA. 2022. ⟨hal-03655570⟩