Root Decomposition

  • Joachim HilgertEmail author
  • Karl-Hermann Neeb
Part of the Springer Monographs in Mathematics book series (SMM)


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.


  1. [Ca94]
    Cartan, É., “Sur la structure des groupes de transformations finis et continue”, thèse, Paris, Nony, 1894 Google Scholar
  2. [Kil89]
    Killing, W., Die Zusammensetzung der stetigen endlichen Transformationsgruppen II, Math. Ann. 33 (1889), 1–48 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of PaderbornPaderbornGermany
  2. 2.Department of MathematicsFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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