Orbits of decomposable $7$-dimensional Lie algebras with $\mathfrak{sl}(2)$ subalgebra

Abstract

The problem on complete classification of holomorphically homogeneous real hypersurfaces in two-dimensional complex spaces was resolved by E. Cartan in 1932. A similar description in the three-dimensional case was recently obtained by A. Loboda. In this work we discuss a part of classification of locally holomorphic homogeneous hypersurfaces in $4$-dimensional complex space being orbits in $\mathbb{C}^4$ by one family of $7$-dimensional Lie algebra. As it was shown in works by Beloshapka, Kossovskii, Loboda and other, the ideas by E. Cartan allow one to obtain rather simply the descriptions of the orbits for the algebras having Abelian ideals for rather large dimensions. In particular, the presence of a $4$-dimensional Abelian ideal in $7$-dimensional Lie algebra of holomorphic vector fields in $\mathbb{C}^4$ often gives rise to the tubularity property for all orbits of such algebra. The Lie algebras in the family we consider are direct sums of the algebra $\mathfrak{sl}(2)$ and several $4$-dimensional Lie algebras and they have at most $3$-dimensional Abelian subalgebras. By means of a technique of the simultaneous «flattening» of vector field we obtain a complete description of all Levi non-degenerate holomorphically homogeneous hypersurfaces being the orbits of the considered algebras in $\mathbb{C}^4$. Many of the obtained homogeneous hypersurfaces turn out to be tubular manifolds. At the same time, the issue on possible reduction of other hypersurfaces to tubes requires further studying. As an effective tool for such study, as well as for a detailed investigation of issues on holomorphic equivalent of the obtained orbits, the technique of Moser normal forms can serve. By means of this technique, we study the issue on the sphericity for representatives of one of the obtained family of hypersurfaces. However, the application of the method of normal forms for the hypersurfaces in complex spaces of dimension $4$ and higher requires a further developing of this technique.