| Version 5 |
Version 4 |
| \PMlinkescapeword{abelian} |
\PMlinkescapeword{abelian} |
|
|
| A Lie algebra is called {\em simple} if it has no proper ideals and is not abelian. A Lie algebra |
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. |
is called {\em semisimple} if it has no proper solvable ideals and is not abelian. |
|
|
| Let $k=\R$ or $\C$. Examples of simple algebras are $\fr{sl}_nk$, the Lie algebra |
Let $k=\R$ or $\C$. 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 |
of the special linear group (traceless matrices), $\fr{so}_nk$, the Lie algebra of the special |
| orthogonal group (skew-symmetric matrices), and $\fr{sp}_{2n} k$ the Lie algebra of the symplectic group. Over $\R$, there are other simple Lie algebas, such as $\fr{su}_n$, the Lie algebra of the special unitary group |
orthogonal group (skew-symmetric matrices), and $\fr{sp}_{2n} k$ the Lie algebra of the symplectic group. Over $\R$, there are other simple Lie algebas, such as $\fr{su}_n$, the Lie algebra of the special unitary group |
| (skew-hermitian matrices). Any |
(skew-hermitian matrices). Any |
| semisimple Lie algebra is a direct product of simple Lie algebras. |
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 |
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 |
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 |
Lie algebras. Secondly, their representation theory is very well understood. Finally, there is |
| a beautiful classification of simple Lie algebras. |
a beautiful classification of simple Lie algebras. |
|
|
| Over $\C$, there are 3 infinite series of simple Lie algebras: $\fr{sl}_n$, $\fr {so}_n$ and |
Over $\C$, 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 $\fr e_8$. |
$\fr{sp}_{2n}$, and 5 exceptional simple Lie algebras $\fr g_2,\fr f_4,\fr e_6,\fr e_7$, and $\fr e_8$. |
| Over $\R$ the picture is more complicated, as several different Lie algebras can have the same complexification (for example, $\fr{su}_n$ and $\fr{sl}_n\R$ both have complexification $\fr{sl}_n\C$). |
Over $\R$ the picture is more complicated, as several different Lie algebras can have the same complexification (for example, $\fr{su}_n$ and $\fr{sl}_n\R$ both have complexification $\fr{sl}_n\C$). |
