What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple.

I wonder what beautiful properties does a simple Lie algebra have?

Well, I've only learned linear algebra, mathematical analysis and a bit of abstract algebra, so I might expect a more amateur answer~ I think there may be other readers of the book sharing the same question. And a good answer may help us gain more motivation to learn about Lie theory~

Much thanks!!!

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Simple Lie algebras can be completely classified (en.wikipedia.org/wiki/Semisimple_Lie_algebra#Classification), their representations can be completely classified, and they can be used as building blocks for general Lie algebras (en.wikipedia.org/wiki/Levi_decomposition). – Qiaochu Yuan May 7 '12 at 9:22
@QiaochuYuan Maybe you want to convert/extend this comment to an answer? – Julian Kuelshammer Jun 13 '13 at 15:50

1 Answer

First of all, we have a complete classification of the (complex) simple Lie algebras. Namely, we have that every simple Lie algebra is one of the following:

• $\mathfrak{sl}(n)$
• $\mathfrak{so}(n)$
• $\mathfrak{sp}(2n)$
• one of the five exceptional simple Lie algebras $\mathfrak{e}_6$, $\mathfrak{e}_7$, $\mathfrak{e}_8$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$

This gives us directly a classification of all semisimple Lie algebras (i.e. the "well behaved" Lie algebras). It also helps to study general Lie algebras through Levi's theorem, which says that every Lie algebra is the semi-direct product of its radical (i.e. its maximal solvable ideal) and a simple Lie algebra.

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You are missing a type. There are four non-exceptional types. – Tobias Kildetoft Aug 7 '13 at 12:03
@TobiasKildetoft: The list of classical types is complete. However the cases $\def\so{\mathfrak{so}}\so(2n+1)$ and $\so(2n)$ of odd and even orthogonal algebras have sufficiently different properties that they are considered to form different families (of types $B_n$ and $D_n$ respectively). – Marc van Leeuwen Aug 7 '13 at 12:27
@MarcvanLeeuwen Ahh, right. – Tobias Kildetoft Aug 7 '13 at 12:29
Calling semi-simple algebras the "well-behaved" Lie algebras is somewhat misleading; according to this description Abelian Lie algebras and the algebras $\mathfrak{gl}(n)$ would not be considered well-behaved, while they are in fact in many ways easier than the semisimple algebras. It is just that it sometimes simplifies reasoning to suppose that a Lie algebra is semisimple; with respect to handling reductive Lie algebras (as one should) it avoids drawing attention to the relatively uninteresting central part of the algebra. – Marc van Leeuwen Aug 7 '13 at 12:32