Classification of realizations of Lie algebras of vector fields on circle

  • S. V. Spichak Institute of Mathematics of National Academy of Sciences of Ukraine, Kyiv
Keywords: realizations of Lie algebras, global equivalence transformations

Abstract

UDC 517.986.5

The realizations of finite-dimensional Lie algebras of smooth tangent vector fields on circle are described.
The ``canonical'' realizations of two-dimensional noncommutative algebra, as well as the algebra $\mathfrak{sl}(2,\mathbb R)$ are constructed. It is shown that any realization of these algebras by smooth vector fields is reduced to one of a ``canonical'' realization by piecewise-smooth global transformations of circle onto itself.
Formulas for calculating the number of non-equivalent realizations are obtained.

References

S. Lie, Theorie der Transformationsgruppen, 3, Teubner, Leipzig (1893).

Gonzalez-López A., N. Kamran, P. J. Olver, Lie algebras of vector fields in the real plane, Proc. London Math. Soc., 64, no. 2, 339 – 368 (1992), https://doi.org/10.1112/plms/s3-64.2.339

I. A. Yehorchenko, Nonlinear representation of the Poincare algebra and invariant equations, Symmetry Anal. Equat. Math. Phys., Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv, 62 – 66 (1992).

R. Z. Zhdanov, V. I. Lahno, W. I. Fushchych, On covariant realizations of the Euclid group, Comm. Math. Phys., 212, no. 3, 535 – 556 (2000).

M. Nesterenko, S. Posta, O. Vaneeva, Realizations of Galilei algebras, J. Phys. A: Math. Theor., 49, 115203, 26 pp. (2016), https://doi.org/10.1088/1751-8113/49/11/115203

R. O. Popovych, V. M. Boyko, M. O. Nesterenko, M. W. Lutfullin, Realizations of real low-dimensional Lie algebras, J. Phys. A: Math. Gen., 36, no. 26, 7337 – 7360; (2003) arXiv:math-ph/0301029, https://doi.org/10.1088/0305-4470/36/26/309

A. G. Sergeev, Geometricheskoe kvantovanie prostranstv petel', Sovr. probl. matematiki, 13, MIAN, Moskva, 3 – 294 (2009).

M. S. Strigunova, Konechnomernye podalgebry v algebre Li vektornyh polej na okruzhnosti, Tr. MIAN, 236, 338 – 342 (2002).

V. A. Zajceva, V. V. Kruglov, A. G. Sergeev, M. S. Strigunova, K. A. Trushkin, Konechnomernye podalgebry v algebre Li vektornyh polej na okruzhnosti, Tr. MIAN,224, 139 – 151 (1999).

S. V. Spichak, Preliminary classification of realizations of two-dimensional Lie algebras of vector fields on a circle, Group analysis of differential equations and integrable systems, 212 – 218, Department of Mathematics and Statistics, University of Cyprus, Nicosia, 2013.

E. Pressli, G. Sigal, Gruppy petel', Mir, Moskva (1990).

B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Sovremennaya geometriya. Metody i prilozheniya, Nauka, Moskva (1986).

V. I. Lagno, S. V. Spichak, V. I. Stognij, Simetrijnij analiz rivnyan' evolyucijnogo tipu, Praci Institutu matematiki NAN Ukrayni. Matematika ta ii zastosuvannya, Kiyv (2002).

V. B. Stepanov, Kurs differencial'nyh uravnenij, Moskva: Gos. izd-vo tekhn.-teoret. lit.,1950.

G. M. Mubarakzyanov, O razreshimyh algebrah Li, Izv. vyssh. uchebn. zaved. Matematika, № 1, 114 – 123 (1963).

G. M. Mubarakzyanov, Klassifikaciya veshchestvennyh struktur algebr Li pyatogo poryadka, Izv. vyssh. uchebn. zaved. Matematika, № 3, 99 – 106 (1963).

G. M. Mubarakzyanov, Klassifikaciya razreshimyh algebr Li shestogo poryadka s odnim nenil'potentnym bazisnym elementom, Izv. vyssh. uchebn. zaved. Matematika, № 4, 104 – 116 (1963).

P. Turkowski, Solvable Lie algebras of dimension six, J. Math. Phys., 31, № 6, 1344 – 1350 (1990), https://doi.org/10.1063/1.528721

K. Shevalle, Teoriya grupp Li, T. 3. Obshchaya teoriya algebr Li, Izd-vo inostr. lit., Moskva (1958).

Published
26.04.2022
How to Cite
Spichak, S. V. “Classification of Realizations of Lie Algebras of Vector Fields on Circle”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 3, Apr. 2022, pp. 389-9, doi:10.37863/umzh.v74i3.6270.
Section
Research articles