On the structure of real transitive Lie algebras

Abstract

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