A note on the jordan decomposition
DOI:
https://doi.org/10.4067/10.4067/S0716-09172011000100011Abstract
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.References
[1] T. Ferraiol, M. Patrao and L. Seco: Jordan decomposition and dynamics on flag manifolds, Discrete Contin. Dyn. Syst. A, 26 No. 3, pp. 923-947, (2010).
[2] Helgason, S. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, (1978).
[3] Hoffman, K. and Kunze, R. Linear Algebra. Second Edition. PrenticeHall, (1971).
[4] Humphreys, J.E. Introduction to Lie Algebras and Representation Theory. Springer, (1972).
[5] Knapp, A. W. Lie Groups Beyond an Introduction, Progress in Mathematics, v. 140, Birkhäuser, (2004).
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[7] Varadarajan, V.S. Lie Groups, Lie Algebras and their Representations. Prentice-Hall Inc., (1974).
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[9] Warner, G. Harmonic Analysis on Semi-Simple Lie Groups I. Springer-Verlag, (1972).
Total 31 artículos.
[2] Helgason, S. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, (1978).
[3] Hoffman, K. and Kunze, R. Linear Algebra. Second Edition. PrenticeHall, (1971).
[4] Humphreys, J.E. Introduction to Lie Algebras and Representation Theory. Springer, (1972).
[5] Knapp, A. W. Lie Groups Beyond an Introduction, Progress in Mathematics, v. 140, Birkhäuser, (2004).
[6] Mostow, G. D.: Factor Spaces of Solvable Groups. Ann. of Math., 60, No. 1, pp. 1-27, (1954).
[7] Varadarajan, V.S. Lie Groups, Lie Algebras and their Representations. Prentice-Hall Inc., (1974).
[8] Varadarajan, V.S. Harmonic Analysis on Real Reductive Groups. Lecture Notes in Math. 576. Springer-Verlag, 1977.
[9] Warner, G. Harmonic Analysis on Semi-Simple Lie Groups I. Springer-Verlag, (1972).
Total 31 artículos.
Published
2011-05-25
How to Cite
[1]
M. Patrão, L. Santos, and L. Seco, “A note on the jordan decomposition”, Proyecciones (Antofagasta, On line), vol. 30, no. 1, pp. 123-136, May 2011.
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