Доклады Академии наукДоклады Академии наук0869-5652The Russian Academy of Sciences1594910.31857/S0869-56524876607-610Research ArticleOn orbits of action of 5-dimensional non-solvable Lie algebras in three-dimensional complex spaceAtanovA. V.lobvgasu@yandex.ruKossovskiyI. G.lobvgasu@yandex.ruLobodaA. V.lobvgasu@yandex.ruVoronezh State UniversityMasaryk UniversityVoronezh State Technical University1009201948766076100409201904092019Copyright © 2019, Russian academy of sciences2019<p class="a"><span lang="EN-US">After the description by E. Cartan in 1932 holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, a similar study in the 3-dimensional case remains incomplete. In a series of works performed by several international teams of authors, the problem is reduced to describing homogeneous surfaces that are non-degenerate in Levi sense and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. A complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space, given in the paper, includes examples of new homogeneous hypersurfaces. The presented results bring to finish a large-scale scientific study of interest to various branches of mathematics.</span></p>homogeneous varietyholomorphic transformationnon-solvable Lie algebravector fieldreal hypersurfaceоднородное многообразиеголоморфное преобразованиенеразрешимая алгебра Ливекторное полевещественная гиперповерхность[Лобода А. В. // Тр. Мат. ин-та РАН. 2001. Т. 235. С. 114-142.][Fels G., Kaup W. // Acta Math. 2008. V. 201. P. 1-82.][Doubrov В., Medvedev А., The D. // arXiv (2017) 1711.02389v1. http://arxiv.org/abs/1711.02389v1.][Акопян Р. С., Лобода А. В. // Функц. анализ и его прил. 2019. Т. 53. № 2. С. 59-63.][Атанов А. В., Лобода А. В. // Материалы международной конференции ВЗМШ 2019. 2019. С. 135-138.][Мубаракзянов Г. М. // Изв. вузов. Матем. 1963. № 3. С. 99-106.][Beloshapka V. K., Kossovskiy I. G. // J. Geom. Anal. 2010. V. 20. № 3. P. 538-564.][Cartan E. // Ann. Math. Pura Appl. 1932. V. 11. № 4. P. 17-90.][Fels G., Kaup W. // J. Reine Angew. Math. 2007. V. 604. P. 47-71.][Chern S. S., Moser J. K. // Acta Math. 1974. V. 133. P. 219-271.][Исаев А. В., Мищенко М. А. // Изв. АН СССР. Сер. матем. 1988. Т. 52. № 6. С. 1123-1153.][Doubrov B., Komrakov B., Rabinovich M. // Geometry and Topology of Submanifolds, VIII. 1996. P. 168-178.]