On orbits in $ \mathbb C^4 $ of $7$-dimensional Lie algebras possessing two Abelian subalgebras
A. V. Lobodaab,
R. S. Akopyanc a Voronezh State Technical University, Moskovsky av. 14, 394026, Voronezh, Russia
b Lomonosov Moscow State University, Leninskie Gory 1, 119991, Moscow, Russia
c MIREA — Russian Technological University, Vernadsky av. 78, 119454, Moscow, Russia
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.
Keywords:
Lie algebra, nilradical, Abelian ideal, homogeneous manifold, holomorphic transformation, vector field, orbit of algebra, tubular manifold, real hypersurface.
UDC:
517.55,
512.816,
514.7
MSC: 22F30,
57M60,
53C30 Received: 03.09.2024
Fulltext:
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