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On degeneracy of orbits of nilpotent Lie algebras. (Russian. English summary) Zbl 1513.32037

Summary: In the paper we discuss 7-dimensional orbits in \(\mathbb{C}^4\) of two families of nilpotent 7-dimensional Lie algebras; this is motivated by the problem on describing holomorphically homogeneous real hypersurfaces. Similar to nilpotent 5-dimensional algebras of holomorphic vector fields in \(\mathbb{C}^3 \), the most part of algebras considered in the paper has no Levi non-degenerate orbits. In particular, we prove the absence of such orbits for a family of decomposable 7-dimensional nilpotent Lie algebra (31 algebra). At the same time, in the family of 12 non-decomposable 7-dimensional nilpotent Lie algebras, each containing at least three Abelian 4-dimensional ideals, four algebras has non-degenerate orbits. The hypersurfaces of two of these algebras are equivalent to quadrics, while non-spherical non-degenerate orbits of other two algebras are holomorphically non-equivalent generalization for the case of 4-dimensional complex space of a known Winkelmann surface in the space \(\mathbb{C}^3\). All orbits of the algebras in the second family admit tubular realizations.

MSC:

32M12 Almost homogeneous manifolds and spaces
32A10 Holomorphic functions of several complex variables
17B66 Lie algebras of vector fields and related (super) algebras
14H10 Families, moduli of curves (algebraic)
13A15 Ideals and multiplicative ideal theory in commutative rings
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References:

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