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:

Abstract

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.

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References

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Mauro Patrão
Departamento de Matemática
Universidade de Brasília-DF
Brazil
e-mail : mpatrao@mat.unb.br

Laércio Santos
Universidade Federal de São Carlos
Campus de Sorocaba
Sorocaba - SP
Brazil
e-mail : lsantos@ufscar.br

Lucas Seco
Departamento de Matemática
Universidade de Brasília-DF
Brazil
e-mail : lseco@mat.unb.br

Received : November 2010. Accepted : December 2010