Flags of subalgebras in contracted Lie algebras

Authors

DOI:

https://doi.org/10.15407/dopovidi2021.04.009

Keywords:

contractions of Lie algebras, flags of subalgebras, flags of subspaces in Lie algebras, six-dimensional nilpotent Lie algebras

Abstract

We prove a theorem that describes the behavior of subalgebra flags of Lie algebras under contractions and can be applied as a new criterion for the non-existence of contractions. A weaker version of the theorem is ob tained for flags of subspaces. Using the theorem, we prove the non-existence of contractions for a number of pairs of sixdimensional nilpotent real Lie algebras, for which the earlier known criteria do not work.

References

Burde, D. (1999). Degenerations of nilpotent Lie algebras. J. Lie Theory, 9, No. 1, pp. 193-202.

Gorbatsevich, V. V., Onishchik, A. L. & Vinberg, E. B. (1997). Lie groups and Lie algebras. III. Structure of Lie groups and Lie algebras. Berlin: Springer.

Grunewald, F. & O’Halloran, J. (1988). Varieties of nilpotent Lie algebras of dimension less than six. J. Algebra, 112, No. 2, pp. 315-325. https://doi.org/10.1016/0021-8693(88)90093-2

Popovych, D. R. (2014). Contractions with necessarily unbounded matrices. Linear Algebra Appl., 458, pp. 689-698. https://doi.org/10.1016/j.laa.2014.06.017

Umlauf, K. A. (1891). Über die Zusammensetzung der endlichen continuierlichen Transformationsgruppen, insbesondere der Gruppen von Range Null. (Unpublished Ph.D. thesis). University of Leipzig, Leipzig, Germany.

Morozov, V. V. (1958). Classification of nilpotent Lie algebras of sixth order. Izvestiya VUZ. Matematika, No. 4, pp. 161-171 (in Russian).

Magnin, L. (1986). Sur les algebres de Lie nilpotentes de dimension < 7. J. Geom. Phys., 3, No. 1, pp. 119-144.

https://doi.org/10.1016/0393-0440(86)90005-7

Nesterenko, M. & Popovych, R. O. (2006). Contractions of low-dimensional Lie algebras. J. Math. Phys., 47, No. 12, 123515. https://doi.org/10.1063/1.2400834

Seeley, C. (1990). Degenerations of 6-dimensional nilpotent Lie algebras over C . Arch. Math., 56, No. 10, ttps://doi.org/10.1080/00927879008824088

Published

26.08.2021

How to Cite

Popovych Д. (2021). Flags of subalgebras in contracted Lie algebras. Reports of the National Academy of Sciences of Ukraine, (4), 9–17. https://doi.org/10.15407/dopovidi2021.04.009