Atanov A.V., Loboda A.V. On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

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https://doi.org/10.15688/mpcm.jvolsu.2019.2.1

Artem Viktorovich Atanov
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Digital Technology,
Voronezh State University
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Universitetskaya Sq., 1, 394018 Voronezh, Russian Federation

Alexander Vasilyevich Loboda
Doctor of Physical and Mathematical Sciences, Professor,
Department of Applied Mathematics and Mechanics,
Voronezh State Technical University
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20 let Oktyabrya St., 84, 394006 Voronezh, Russian Federation

Abstract. This paper studies holomorphic homogeneous real hypersurfaces in C³ associated with the unique non-solvable indecomposable 5-dimensional Lie algebra g₅ (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with "highly symmetric" orbits, non-degenerate orbits of g₅ are "simply homogeneous", i.e. their symmetry algebras are exactly
5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order.

The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of g₅ and significantly simplify the third field.

There is a very important assumption which is used in our considerations: we suppose that all orbits of g₅ are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with g₅. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of g₅, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations)

(v − x₂y₁)² + y²₁y²₂=y₁,

which is the equation of the algebraic surface of the fourth order with the indefinite Levi form.

Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

Key words: homogeneous manifold, holomorphic transformations, non-solvable Lie algebras, vector field, real hypersurfaces in C³.

On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra by Atanov A.V., Loboda A.V. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 22 No. 2 2019, pp. 5-20

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