TOPICS
Search

Simple Lie Algebra


A Lie algebra is said to be simple if it is not Abelian and has no nonzero proper ideals.

Over an algebraically closed field of field characteristic 0, every simple Lie algebra is constructed from a simple reduced root system by the Chevalley construction, as described by Humphreys (1977).

Over an algebraically closed field of field characteristic >7, every simple Lie algebra is constructed from a simple reduced root system (as in the characteristic 0 case) or is a Cartan algebra.

There also exist simple Lie algebras over algebraically closed fields of field characteristic 2, 3, and 5 that are not constructed from a simple reduced root system and are not Cartan algebras.


See also

Cartan Algebra, Semisimple Lie Algebra, Lie Algebra

This entry contributed by Skip Garibaldi

Explore with Wolfram|Alpha

References

Huang, J.-S. "Simple Lie Algebras." Part II in Lectures on Representation Theory. Singapore: World Scientific, pp. 27-70, 1999.Humphreys, J. E. §25 in Introduction to Lie Algebras and Representation Theory, 3rd ed. New York: Springer-Verlag, 1977.Mathieu, O. "Classification des algèbres de Lie simples." Astérisque, No. 266, 245-286, 2000.Strade, H. and Wilson, R. L. "Classification of Simple Lie Algebras Over Algebraically Closed Fields of Prime Characteristic." Bull. Amer. Math. Soc. 24, 357-362, 1991.

Referenced on Wolfram|Alpha

Simple Lie Algebra

Cite this as:

Garibaldi, Skip. "Simple Lie Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SimpleLieAlgebra.html

Subject classifications