Structure and Geometry of Lie Groups pp 133-166 | Cite as
Root Decomposition
Chapter
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.
References
- [Ca94]Cartan, É., “Sur la structure des groupes de transformations finis et continue”, thèse, Paris, Nony, 1894 Google Scholar
- [Kil89]Killing, W., Die Zusammensetzung der stetigen endlichen Transformationsgruppen II, Math. Ann. 33 (1889), 1–48 MathSciNetCrossRefGoogle Scholar
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