APPLICATION OF SYSTEMS ANALYSIS AND COMPUTER ALGORITHMS IN STUDYING ORBITS OF 7-DIMENSIONAL LIE ALGEBRAS

We discuss a systematic approach to the problem of describing holomorphically homogeneous real hypersurfaces in the space C4, each of which is an orbit of some real Lie algebra. When studying the family of 7-dimensional Lie algebras, which plays an important role in the problem at hand and contains more than a thousand different types of algebras, it is natural to use computer algorithms. With the participation of the authors of this article, classification results on the orbits of several large blocks of algebras from this family were previously obtained. Relations are established between the presence and dimensions of nilpotent and Abelian subalgebras of the original Lie algebras and such properties of their orbits in C4 as Levi degeneracy and tubularity. In this article, the above ideas and computer algorithms are applied to a family of 18 types of 7-dimensional Lie algebras that have a common 6-dimensional nil-radical. Holomorphic realizations in C4 of these algebras are constructed and by integrating them, all holomorphically homogeneous (in the local sense) Levi-nondegenerate 7-dimensional orbits of this family are obtained. Using a quadratic change of variables, it is shown that all these orbits are holomorphically equivalent to tubular hypersurfaces.