Cohomologie des Formes Divergences et Actions Propres d'Algèbres de Lie
Journal of Lie Theory, Volume 17 (2007) no. 2, pp. 317-335
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 $G$).
DOI:
10.5802/jolt.449
Classification:
53B05, 57S15, 57S20, 17B56
Keywords: G-manifolds, cohomology, noncompact Lie groups of transformations, compact Lie groups of differentiable transformations
Keywords: G-manifolds, cohomology, noncompact Lie groups of transformations, compact Lie groups of differentiable transformations
@article{JOLT_2007_17_2_a5,
author = {A. Abouqateb},
title = {Cohomologie des {Formes} {Divergences} et {Actions} {Propres} {d'Alg\`ebres} de {Lie}},
journal = {Journal of Lie Theory},
pages = {317--335},
year = {2007},
volume = {17},
number = {2},
doi = {10.5802/jolt.449},
zbl = {1120.17009},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.449/}
}
A. Abouqateb. Cohomologie des Formes Divergences et Actions Propres d'Algèbres de Lie. Journal of Lie Theory, Volume 17 (2007) no. 2, pp. 317-335. doi: 10.5802/jolt.449
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