
Journal of Lie Theory 17 (2007), No. 2, 317335 Copyright Heldermann Verlag 2007 Cohomologie des Formes Divergences et Actions Propres d'Algèbres de Lie Abdelhak Abouqateb Faculté des Sciences et Techniques, Université CadiAyyad, B.P. 549 Gueliz, 40000 Marrakech, Morocco abouqateb@fstgmarrakech.ac.ma [Abstractpdf] For any action $\tau\colon{\cal G}\rightarrow{\cal V}(M)$ of a Lie algebra ${\cal G}$ on a manifold $M$, we introduce the notion of a cohomology $H^{\ast}_\tau(M)$ which we call the cohomology of $\tau$divergence forms. We show that this cohomology is invariant by a ${\cal G}$proper homotopy, and that there exists an analogue of the MayerVietoris lemma. We make the connection with the problem of integrability of a Lie algebra action to a proper Lie group action. The differentiable cohomology $H_d^{\ast}(G)$ of a unimodular Lie group $G$ is isomorphic to $H^{\ast+1}_\tau(G/K)$ (where $K$ a compact maximal subgroup of $G$ and $\tau\colon{\cal G}\rightarrow{\cal V}(G/K)$ is the natural homogeneous action of the Lie algebra ${\cal G}$ of $G$). Keywords: Gmanifolds, cohomology, noncompact Lie groups of transformations, compact Lie groups of differentiable transformations. MSC: 53B05, 57S15, 57S20, 17B56 [ Fulltextpdf (250 KB)] for subscribers only. 