simple and semi-simple Lie algebras

A Lie algebra is called simple if it has no proper ideals and is not abelian. A Lie algebrais called semi-simple if it has no proper solvable ideals and is not abelian.

Let $k=\\mathbb{R}$ or $\\mathbb{C}$ . Examples of simple algebras are $\\mathfrak{sl}_{n}k$ , the Lie algebraof the special linear group (traceless matrices), $\\mathfrak{so}_{n}k$ , the Lie algebra of the specialorthogonal group (skew-symmetric matrices), and $\\mathfrak{sp}_{2n}k$ the Lie algebra of the symplectic group. Over $\\mathbb{R}$ , there are other simple Lie algebas, such as $\\mathfrak{su}_{n}$ , the Lie algebra of the special unitary group(skew-Hermitian matrices). Anysemi-simple Lie algebra is a direct product of simple Lie algebras.

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

Over $\\mathbb{C}$ , there are 3 infinite series of simple Lie algebras: $\\mathfrak{sl}_{n}$ , $\\mathfrak{so}_{n}$ and $\\mathfrak{sp}_{2n}$ , and 5 exceptional simple Lie algebras $\\mathfrak{g}_{2},\\mathfrak{f}_{4},\\mathfrak{e}_{6},\\mathfrak{e}_{7}$ , and $\\mathfrak{e}_{8}$ .Over $\\mathbb{R}$ the picture is more complicated, as several different Lie algebras can have the same complexification (for example, $\\mathfrak{su}_{n}$ and $\\mathfrak{sl}_{n}\\mathbb{R}$ both have complexification $\\mathfrak{sl}_{n}\\mathbb{C}$ ).

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	semisimple