Algebras of infinitesimal CR automorphisms

Abstract

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. The authors introduce a corresponding abstract notion of CR algebra. A CR algebra is a pair $(L,L_1)$(L,L1), consisting of a real Lie algebra $L$L and a subalgebra $L_1$L1 of the complexification $\bold C\otimes_{\bold R} L$C⊗RL, such that the factor space $L/L\cap L_1$L/L∩L1 is finite-dimensional.\ud The authors investigate some formal properties of CR algebras and construct some "fibrations'' (i.e., $L$L-equivariant submersions) of such algebras. They introduce three new notions of nondegeneracy of CR algebras---strict, weak and ideal nondegeneracy. These three concepts are weaker than those used previously by some other authors. The authors intend to extend the application of the É. Cartan method of investigating the equivalence of CR structures to some larger classes of CR manifolds.\ud 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 (Theorems 5.3 and 5.4). 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

Aspects of the Levi form

“…Let g be the complex Lie algebra obtained by complexifying g 0 and set (7.5) q = {Z ∈ g | dπ e (Z) ∈ T 0,1 p 0 M 0 }. For the following statements we refer to [AMN1,AMN2,MN].…”

Classification of Levi degenerate homogeneous CR-manifolds in dimension 5

“…Before we briefly outline the main steps of our proof, we recall the notion of a CR-algebra, taken from [28], and introduce some notation.…”

“…(i) In [28] also the case is allowed where g has infinite dimension, but q has to have finite codimension in l. In this part of the paper, however, only finitedimensional Lie algebras occur.…”

Locally homogeneous finitely nondegenerate CR-manifolds

“…• In [13], certain purely algebraic nondegeneracy conditions of CR-algebras have been introduced. However, their geometric interpretation, in particular the characterization of holomorphic (non)degeneracy as given in the remark following Prop.…”

“…involves a manipulation of the defining equations, which, in concrete cases, can be quite hard. A (locally) homogeneous CR-manifold can also be described by a purely algebraic datum, for instance by a CR-algebra in the sense of [13]. In fact, one can show that there is a natural equivalence between the category of germs of locally homogeneous CR-manifolds on the complex geometric side and the category of CR-algebras on the algebraic side, see Section 4 for further details.…”