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'simple and semi-simple Lie algebras'
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| Title of object: |
simple and semi-simple Lie algebras |
| Canonical Name: |
SimpleAndSemiSimpleLieAlgebras2 |
| Type: |
Definition |
| Created on: |
2002-12-04 00:44:36 |
| Modified on: |
2002-12-04 12:30:36 |
| Classification: |
msc:17B20 |
| Defines: |
simple, semi-simple |
Revision comment (for changes between this and next version):
| Changes for correction #2898 ('su_n real'). |
Preamble:
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Content:
\newcommand{\fr}[1]{\mathfrak{#1}}
A Lie algebra is called {\em simple} if it has no proper ideals and is not abelian. A Lie algebra
is called {\em semisimple} if it has no proper solvable ideals and is not abelian.
Let $k=\mathbb{R}$ or $\mathbb{C}$. Examples of simple algebras are $\fr{sl}_nk$, the Lie algebra
of the special linear group (traceless matrices), $\fr{so}_nk$, the Lie algebra of the special
orthogonal group (skew-symmetric matrices), $\fr{su}_n$ the Lie algebra of the special unitary group
(skew-hermitian matrices), and $\fr{sp}_{2n} k$ the Lie algebra of the symplectic group. Any
semisimple 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: $\fr{sl}_n$, $\fr {so}_n$ and
$\fr{sp}_{2n}$, and 5 exceptional simple Lie algebras $\fr g_2,\fr f_4,\fr e_6,\fr e_7$, and $\fr e_8$.
Over $\mathbb{R}$ the picture is more complicated, but not much more so. |
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