On orbits in $ \mathbb C^4 $ of 7–dimensional Lie algebras possessing two Abelian subalgebras

Abstract

The paper focuses on the problem on description of holomorphically homogeneous real hypersurfaces of multidimensional complex spaces based on the properties of the Lie algebras and their nilpotent and Abelian subalgebras corresponding to these manifolds. Using classifications of a large family of 7–dimensional solvable non–decomposable Lie algebras, earlier we studied the orbits of algebras with ``strong'' commutative properties. In particular, it was established that a 7–dimensional Lie algebra with an Abelian subalgebra of dimension 5 admits no Levi nondegenerate orbits in the space $\mathbb C^4.$

In the present paper we study all 82 types of solvable non–decomposable 7–dimensional Lie algebras, which have exactly two 4–dimensional Abelian subalgebras and a 6–dimensional nilradical. We prove that for 75 types of algebras, any 7–dimensional orbit in $ \mathbb C^4 $ is either Levi–degenerate or can be reduced to a tubular manifold by a holomorphic transformation. We provide all (up to local holomorphic coordinate transformations) realizations of 7 exceptional types of abstract Lie algebras as algebras of holomorphic vector fields in $ \mathbb C^4.$For most of these realizations, we give coordinate descriptions of orbits, which are holomorphically homogeneous nondegenerate real hypersurfaces in this space.