SciELO - Scientific Electronic Library Online

vol.30 número1Some mathieu-type series for the I-function occuring in the fokker-planck equation índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados




Links relacionados


Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.30 no.1 Antofagasta  2011 

Proyecciones Journal of Mathematics
Vol. 30, N° 1, pp. 123-136, May 2011.
Universidad Católica del Norte
Antofagasta - Chile

A note on the jordan decomposition

Mauro Patrão1
Laércio Santos2
Lucas Seco1

1Universidade de Brasília-DF, Brazil
2Universidade Federal de São Carlos, Brazil

Correspondencia a:


The multiplicative Jordan decomposition of a linear isomorphism of Rn into its elliptic, hyperbolic and unipotent components is well know. One can define an abstract Jordan decomposition of an element of a Lie group by taking the Jordan decomposition of its adjoint map. For real algebraic Lie groups, some results of Mostow implies that the usual multiplicative Jordan decomposition coincides with the abstract Jordan decomposition. Here, for a semisimple linear Lie group, we obtain this fact by elementary methods. We also obtain the corresponding results for semisimple linear Lie algebras. Complete and simple proofs of these facts are lacking in the literature, so that the main purpose of this article is to fill this gap.

Texto completo sólo en formato PDF



[1] T. Ferraiol, M. Patr˜ao and L. Seco: Jordan decomposition and dynamics on flag manifolds, Discrete Contin. Dyn. Syst. A, 26 No. 3, pp. 923-947, (2010).         [ Links ]

[2] Helgason, S. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, (1978).         [ Links ]

[3] Hoffman, K. and Kunze, R. Linear Algebra. Second Edition. Prentice-Hall, (1971).         [ Links ]

[4] Humphreys, J.E. Introduction to Lie Algebras and Representation Theory. Springer, (1972).         [ Links ]

[5] Knapp, A. W. Lie Groups Beyond an Introduction, Progress in Mathematics, v. 140, Birkh¨auser, (2004).         [ Links ]

[6] Mostow, G. D.: Factor Spaces of Solvable Groups. Ann. of Math., 60, No. 1, pp. 1-27, (1954).         [ Links ]

[7] Varadarajan, V.S. Lie Groups, Lie Algebras and their Representations. Prentice-Hall Inc., (1974).         [ Links ]

[8] Varadarajan, V.S. Harmonic Analysis on Real Reductive Groups. Lecture Notes in Math. 576. Springer-Verlag, 1977.         [ Links ]

[9] Warner, G. Harmonic Analysis on Semi-Simple Lie Groups I. Springer-Verlag, (1972).         [ Links ]

Mauro Patrão
Departamento de Matemática
Universidade de Brasília-DF
e-mail :

Laércio Santos
Universidade Federal de São Carlos
Campus de Sorocaba
Sorocaba - SP
e-mail :

Lucas Seco
Departamento de Matemática
Universidade de Brasília-DF
e-mail :

Received : November 2010. Accepted : December 2010

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons