PlanetMath: simple and semi-simple Lie algebras

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

simple and semi-simple Lie algebras

(Definition)

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 $\mathfrak{sl}_nk$ , the Lie algebra of the special linear group (traceless matrices), $\mathfrak{so}_nk$ , 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 $\mathfrak{sl}_n\mathbb{C}$ ).