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 + 𝑦2
1𝑦2
2 = 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.
On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra
