en>Michael Hardy at 16:51, 24 September 2013

2013-09-24T16:51:14Z

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In algebra, '''Weyl's theorem on complete reducibility''' is a fundamental result in the theory of [[Lie algebra representation]]s. Let <math>\mathfrak{g}</math> be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over <math>\mathfrak{g}</math> is semisimple as a module (i.e., a direct sum of simple modules.)

The theorem is a consequence of [[Whitehead's lemma (Lie algebras)|Whitehead's lemma]] (see Weibel's [[homological algebra]] book). Weyl's original proof was analytic in nature: it famously used the [[unitarian trick]].

A Lie algebra is called [[reductive Lie algebra|reductive]] if its [[adjoint representation of a Lie algebra|adjoint representation]] is semisimple. Thus, the theorem says that a [[semisimple Lie algebra]] is reductive. (But this can be seen more directly.)

== References ==
* [[Nathan Jacobson|Jacobson, Nathan]], ''Lie algebras'', Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
* {{cite book |last=Weibel |first=Charles A. |authorlink=Charles Weibel |title=An Introduction to Homological Algebra |url= |accessdate= |year=1995 |publisher=Cambridge University Press |location= |isbn= |page=}}

== External links ==
* A [http://amathew.wordpress.com/2010/01/31/weyls-theorem-on-complete-reducibility/ blog post] by Akhil Mathew

[[Category:Lie algebras]]

Iterated limit - Revision history

en>Michael Hardy at 16:51, 24 September 2013