# 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 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