Lie algebra classification for the Chazy equation and further topics related with this algebra.

Palabras clave: Ecuación de Chazy, Simetrías de Lie, Clasificación del grupo de simetrías de Lie, Álgebra de Lie, grupo a un parámetro


It is known that the classification of the Lie algebras is a classical problem. Due to Levi’s Theorem the question can be reduced to the classification of semi-simple and solvable Lie algebras. This paper is devoted to classify the Lie algebra generated by the Lie symmetry group of the Chazy equation. We also present explicitly the one parame-ter subgroup related to the infinitesimal generators of the Chazy symmetry group. Moreover the classification of the Lie algebra associated to the optimal system is investigated.

La clasificación de las álgebras de Lie es un problema clásico. Acorde al teorema de Levi la cuestión puede reducirse a la clasificación de álgebras de Lie semi-simples y solubles. Este artículo está dedicado a clasificar el álgebra de Lie generada por el grupo de simetría de Lie para la ecuación de Chazy. También presentamos explícitamente los subgrupos a un parámetro  relacionados con los generadores de las simetrías del grupo de Chazy. Además, la clasificación de la álgebra de Lie asociada al sistema optimo es investigada.

Biografía del autor/a

Yeisson Alexis Acevedo-Agudelo, Magister en Matemáticas Aplicadas, Universidad EAFIT

Magister en Matemáticas Aplicadas, Universidad EAFIT


R. O. Popovych, V. M. Boyko, M. O. Nesterenko, M. W. Lutfullin, Realizations of real low-dimensional lie algebras, Journal of Physics A: Mathematical and General 36 (26) (2003) 7337. doi: 10.1088/0305-4470/36/26/309

J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer science & Business media, 2012.9

N. Jacobson, Lie algebras, no. 10, Courier Corporation, 1979.

D. Rand, P. Winternitz, H. Zassenhaus, On the identification of a lie algebra given by its structure constants. i. direct de-compositions, Levi decompositions, and nilradicals, Linear algebra and its applications 109 (1988) 197–246. Doi: 10.1016/0024-3795(88)90210-8

A. Bourlioux, C. Cyr-Gagnon, P. Winternitz, Difference schemes with point symmetries and their numerical tests, Journal of Physics A: Mathematical and General 39 (22) (2006) 6877. doi: 10.1088/0305-4470/39/22/006

S. Shen, Lie symmetry reductions and exact solutions of some differential–difference equations, Journal of Physics A: Mathematical and Theoretical 40 (8) (2007) 1775. doi: 10.1088/1751-8113/40/8/006

R. Floreanini, L. Vinet, Lie symmetries of finite-difference equations, Journal of Mathematical Physics 36 (12) (1995) 7024–7042. doi: 10.1063/1.531205

Y. D. Bozhkov, P. R. da Conceição, On the generalizations of the Kummer–schwarz equation, Nonlinear Analysis 192 (2020) 111691. doi: 10.1016/

G. Loaiza, Y. Acevedo, O. Duque, D. A. G. Hernández, Lie algebra classification, conservation laws, and invariant solu-tions for a generalization of the Levinson–smith equation, International Journal of Differential Equations 2021 (2021) 1–11. doi:10.1155/2021/6628243

J. Chazy, Sur les equations differentielles dont l’integrale generale est uniforme et admet des singelarities essentielles mobiles, C. R. Acad. Sc.Paris 149 (1909) 563–565.

Van Dyke, M. Laminar Boundary Layers, Clarendon Oxford Press, (1964). doi: 10.1017/S0022112064210350

G. Loaiza, O. Duque, Y. Acevedo, Álgebra óptima y soluciones invariantes para la ecuación de Chazy, Ingeniería y Ciencia (2021). doi: 10.17230/ingciencia.17.33.1

R. Naz., F. M. Mahomed, D. P. Mason, Symmetry solutions of a third order ordinary differential equation which arises from Prandtl boundary 10 layer equations, Journal of Nonlinear Mathematical Physics 15 (2008) 179–191. doi: 10.2991/jnmp.2008.15.s1.16

A. Bowers, Classification of three-dimensional real lie algebras, Personal

A. W. Knapp, Lie groups beyond an introduction, Springer Science 140. ISBN: 978-1-4757-2453-0.

A. L. Onishchik, E. Vinberg, Lie groups and lie algebras III, Encyclopaedia of Mathematical Sciences 41. Springer, (1994). ISBN 978-3-540-54683-2.

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, (1986). ISBN 978-1-4684-0274-2.

P. Hydon, D. Crighton, Symmetry methods for differential equations: A beginner’s guide, Cambridge Texts in Applied Mathematics, Cambridge University Press, (2000). ISBN-13: 978-0521497862

G. Bluman, S. Anco, Symmetry and integration methods for differential equations, Springer Science and Business Me-dia, (2008). ISBN 978-0-387-21649-2

G. Loaiza, Y. Acevedo-Agudelo, O. Londoño-Duque, Álgebra óptima y soluciones invariantes para la ecuación de Chazy, Ingeniería y Ciencia 17 (33) (2021) 7–21. doi:10.17230/ingciencia.17.33.1

Cómo citar
Acevedo-Agudelo, Y. A., García-Hernández, D. A., Londoño-Duque, O. M., & Loaiza-Ossa, G. I. (2021). Lie algebra classification for the Chazy equation and further topics related with this algebra . Revista Politécnica, 17(34), 101-109.

Métricas de Artículo

Resumen : 47   HTML : 10  PDF : 12