×

Cartan algebras and involutions. (English) Zbl 0795.17004

Summary: We identify Cartan algebras in certain semidirect products and prove that every involution of a real Lie algebra leaves invariant some Cartan algebra. Moreover, we establish the adaptability of invariant Cartan algebras and invariant Levi decompositions and prove some conjugacy theorems for invariant Cartan algebras.

MSC:

17B05 Structure theory for Lie algebras and superalgebras
17B20 Simple, semisimple, reductive (super)algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). · Zbl 0329.17002
[2] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0451.53038
[3] Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie D. Lawson, Lie groups, convex cones, and semigroups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. · Zbl 0701.22001
[4] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. · Zbl 0091.34802
[5] Henrik Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, Progress in Mathematics, vol. 49, Birkhäuser Boston, Inc., Boston, MA, 1984. · Zbl 0555.43002
[6] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. · Zbl 0265.22020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.