Abstract
Since a simple Lie algebra \({\mathfrak{g}}\) has no other ideals than \({\mathfrak{g}}\) and {0}, we cannot analyze its structure by breaking it up into an ideal \({\mathfrak{n}}\) and the corresponding quotient algebra \({\mathfrak{g}}/{\mathfrak{n}}\). We therefore need refined tools to look inside simple Lie algebras. It turns out that Cartan subalgebras and the corresponding root decompositions provide such a tool.
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References
Cartan, É., “Sur la structure des groupes de transformations finis et continue”, thèse, Paris, Nony, 1894
Killing, W., Die Zusammensetzung der stetigen endlichen Transformationsgruppen II, Math. Ann. 33 (1889), 1–48
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Hilgert, J., Neeb, KH. (2012). Root Decomposition. In: Structure and Geometry of Lie Groups. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-84794-8_6
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DOI: https://doi.org/10.1007/978-0-387-84794-8_6
Publisher Name: Springer, New York, NY
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