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Simple Lie Algebra

A simple Lie algebra is a Lie algebra that 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, Lie Algebra, Semisimple Lie Algebra

This entry contributed by Skip Garibaldi

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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.

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Simple Lie Algebra

Cite this as:

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

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