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Journal of Lie Theory 17 (2007), No. 2, 317--335
Copyright Heldermann Verlag 2007

Cohomologie des Formes Divergences et Actions Propres d'Algèbres de Lie

Abdelhak Abouqateb
Faculté des Sciences et Techniques, Université Cadi-Ayyad, B.P. 549 Gueliz, 40000 Marrakech, Morocco


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 Mayer-Vietoris 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: G-manifolds, cohomology, noncompact Lie groups of transformations, compact Lie groups of differentiable transformations.

MSC: 53B05, 57S15, 57S20, 17B56

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