On holomorphic realizations of 5-dimensional Lie algebras

An approach to the problem of describing holomorphically homogeneous real hypersurfaces of 3-dimensional complex spaces is discussed, connected with the Lie algebras of vector fields on such manifolds. After a series of papers by author, Fels-Kaup, Dubrov-Medvedev-The, only Levi-nondegenerate hypersurfaces are of interest in this problem for which the corresponding algebras are 5-dimensional and have full rank.
The report realizes a scheme for constructing homogeneous surfaces starting from abstract algebras, proposed by E.Cartan and developed by Beloshapka-Kossovskiy. It is shown that most of the known complete list containing 67 types of 5-dimensional algebras can be realized only on degenerate surfaces or on affine-homogeneous tubes. New examples (not reducible to known ones) of holomorphically homogeneous real hypersurfaces are obtained.