simple and semi-simple Lie algebras

A Lie algebraMathworldPlanetmath is called simple if it has no proper idealsMathworldPlanetmath and is not abelianMathworldPlanetmath. A Lie algebra is called semi-simplePlanetmathPlanetmath if it has no proper solvablePlanetmathPlanetmath ideals and is not abelian.

Let k= or . Examples of simple algebras are 𝔰𝔩nk, the Lie algebra of the special linear groupMathworldPlanetmath (traceless matrices), 𝔰𝔬nk, the Lie algebra of the special orthogonal groupMathworldPlanetmath (skew-symmetric matrices), and 𝔰𝔭2nk the Lie algebra of the symplectic group. Over , there are other simple Lie algebas, such as 𝔰𝔲n, the Lie algebra of the special unitary group (skew-Hermitian matrices). Any semi-simple Lie algebra is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of simple Lie algebras.

Simple and semi-simple Lie algebras are one of the most widely studied classes of algebras for a number of reasons. First of all, many of the most interesting Lie groups have semi-simple Lie algebras. Secondly, their representation theory is very well understood. Finally, there is a beautiful classification of simple Lie algebras.

Over , there are 3 infinite series of simple Lie algebras: 𝔰𝔩n, 𝔰𝔬n and 𝔰𝔭2n, and 5 exceptional simple Lie algebras 𝔤2,𝔣4,𝔢6,𝔢7, and 𝔢8. Over the picture is more complicated, as several different Lie algebras can have the same complexification (for example, 𝔰𝔲n and 𝔰𝔩n both have complexification 𝔰𝔩n).

Title simple and semi-simple Lie algebras
Canonical name SimpleAndSemisimpleLieAlgebras
Date of creation 2013-03-22 13:11:28
Last modified on 2013-03-22 13:11:28
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 17B20
Related topic LieAlgebra
Related topic LieGroup
Related topic RootSystem
Related topic RootSystemUnderlyingASemiSimpleLieAlgebra
Defines simple Lie algebra
Defines semi-simple Lie algebra
Defines semisimple Lie algebra
Defines simple
Defines semi-simple
Defines semisimplePlanetmathPlanetmathPlanetmath