Algebras of infinitesimal CR automorphisms

This paper is devoted to the investigation of Lie algebras of local infinitesimal CR automorphisms. Such algebras are naturally associated to germs of homogeneous CR manifolds. We introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair (L,L_1), consisting of a real Lie algebra L and a subalgebra L_1 of its complexification, such that the factor space L/L∩L_1 is finite-dimensional. We investigate some formal properties of CR algebras and construct some "fibrations'' (i.e., L-equivariant submersions) of such algebras. We intend to extend the application of the É. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds. One of the main ideas of this paper is a decomposition of arbitrary CR algebras into three "parts'': totally real, totally complex and weakly nondegenerate CR algebras. There are some results about these three special classes of CR algebras. Some results about prolongations for transitive CR algebras are also obtained, in particular about maximality of parabolic CR algebras with respect to transitive prolongations.