Homogeneous models for Levi degenerate CR manifolds

We extend the notion of a fundamental negatively Z-graded Lie algebra m_x=+_{p\leq 1}m_x^p associated to any point of a Levi-nondegenerate CR manifold to the class of k-nondegenerate CR manifolds (M;D;J) for all k\geq 2 and call this invariant the core at x\in M. It consists of a Z-graded vector space m_x=_{p\leq k-2} m_x^p of height k-2 endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of (M;D;J) and the Levi forms of higher order. In the case of CR manifolds of hypersurface type we propose a definition of a homogeneous model of type m, that is, a homogeneous k-nondegenerate CR manifold M=G/G_o with core m associated with an appropriate Z-graded Lie algebra Lie(G)=g=+g^p and subalgebra Lie(G_o)=g_o=+g^p_o of the nonnegative part +_{p\geq0} g^p. It generalizes the classical notion of Tanaka of homogeneous model for Levi-nondegenerate CR manifolds and the tube over the future light cone, the unique (up to local CR diffeomorphisms) maximally homogeneous 5-dimensional 2-nondegenerate CR manifold. We investigate the basic properties of cores and models and study the 7-dimensional CR manifolds of hypersurface type from this perspective. We first classify cores of 7-dimensional 2-nondegenerate CR manifolds up to isomorphism and then construct homogeneous models for seven of these classes. We finally show that there exists a unique core and homogeneous model in the 3-nondegenerate class.