simple and semi-simple Lie algebras

A Lie algebra is called simple if it has no proper ideals and is not abelian. A Lie algebra is 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 algebra of the special linear group (traceless matrices), $\mathfrak{so}_{n}k$ , the Lie algebra of the special orthogonal 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). Any semi-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 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 $\mathbb{C}$ , 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