On the structure of real transitive Lie algebras

Conn, Jack F.

doi:10.1090/S0002-9947-1984-0756031-0

On the structure of real transitive Lie algebras
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by Jack F. Conn

Trans. Amer. Math. Soc. 286 (1984), 1-71

DOI: https://doi.org/10.1090/S0002-9947-1984-0756031-0

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Abstract:

In this paper, we examine some of the ways in which abstract algebraic objects in a transitive Lie algebra $L$ are expressed geometrically in the action of each transitive Lie pseudogroup $\Gamma$ associated to $L$. We relate those chain decompositions of $\Gamma$ which result from considering $\Gamma$-invariant foliations to Jordan-Hölder sequences (in the sense of Cartan and Guillemin) for $L$. Local coordinates are constructed which display the nature of the partial differential equations defining $\Gamma$; in particular, locally homogeneous pseudocomplex structures (also called ${\text {CR}}$-structures) are associated to the nonabelian quotients of complex type in a Jordan-Hölder sequence for $L$.

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