View source for Semi-simple algebraic group

{{MSC|20G15|14L10}}

A semi-simple group is a connected
[[linear algebraic group]] of positive
dimension which contains only trivial solvable (or, equivalently,
Abelian) connected closed normal subgroups. The quotient group of a
connected non-solvable linear group by its radical is semi-simple.

A connected linear algebraic group $G$ of positive dimension is called
simple (or quasi-simple) if it does not contain proper connected
closed normal subgroups. The centre $\def\Z{\mathrm{Z}}\Z(G)$ of a simple group $G$ is
finite, and $G/\Z(G)$ is simple as an abstract group. An algebraic group $G$
is semi-simple if and only if $G$ is a product of simple connected
closed normal subgroups.

If the ground field is the field $\C$ of complex numbers, a semi-simple
algebraic group is nothing but a semi-simple Lie group over $\C$ (cf.
[[Lie group, semi-simple|Lie group, semi-simple]]). It turns out that
the classification of semi-simple algebraic groups over an arbitrary
algebraically closed field $K$ is analogous to the case $K=\C$, that is,
a semi-simple algebraic group is determined up to isomorphism by its
root system and a certain sublattice in the weight lattice that
contains all the roots. More precisely, let $T$ be a
[[Maximal torus|maximal torus]] in the semi-simple algebraic group $G$
and let $X(T)$ be the character group of $T$, regarded as a lattice in
the space $E=X(T)\otimes\R$. For a rational
[[Linear representation|linear representation]] $\rho$ of $G$, the group
$\rho(T)$ is diagonalizable. Its eigenvalues, which are elements of $X(T)$, are
called the weights of the representation $\rho$. The non-zero weights of
the adjoint representation $\mathrm{Ad}$ are called the roots of $G$. It turns
out that the system $\Sigma\subset X(T)$ of all roots of $E$ is a
[[Root system|root system]] in the space $E$, and that the irreducible
components of the system $\Sigma$ are the root systems for the simple
closed normal subgroups of $G$. Furthermore, $Q(\Sigma)\subseteq X(T)\subseteq P(\Sigma)$, where $Q(\Sigma)$ is the
lattice spanned by all roots and $P(\Sigma) = \{\lambda\in E \;|\;
\alpha^*(\lambda) \in \mathbb{Z} \textrm{ for all } \alpha\in\Sigma\}$ is the weight lattice in the root
system $\Sigma$. In the case $K=\C$ the space $E$ can be naturally identified
with a real subspace $\def\t{\mathfrak{t}}\t_\R^* \subset \t^*$, where $\t$ is the Lie algebra of the torus
$T$, spanned by the differentials of all characters, while the
lattices in $\t$ dual to $Q(\Sigma)\subseteq X(T)\subseteq P(\Sigma)$ coincide (up to a factor $2\pi i$) with $\Gamma_1\supseteq \Gamma(G)\supseteq \Gamma_0$
(see
[[Lie group, semi-simple|Lie group, semi-simple]]).

The main classification theorem states that if $G'$ is another
semi-simple algebraic group, $T'$ its maximal torus, $\Sigma'\subset E'$ a root system
of $G'$, and if there is a linear mapping $E\to E'$ giving an isomorphism
between the root systems $\Sigma$ and $\Sigma'$ and mapping $X(T)$ onto $X(T')$, then
$G\cong G'$ (local isomorphism). Moreover, for any reduced root system $\Sigma$ and
any lattice $\Lambda$ satisfying the condition $Q(\Sigma)\subseteq \Lambda\subseteq P(\Sigma)$ there exists a
semi-simple algebraic group $G$ such that $\Sigma$ is its root system with
respect to the maximal torus $T$, and $\Lambda = X(T)$.

The isogenies (in particular, all automorphisms, cf.
[[Isogeny]]) of a semi-simple algebraic group have also been
classified.

====References====
<table>
<TR><TD valign="top">[1]</TD>
 <TD valign="top"> R.G. Steinberg, "Lectures on Chevalley groups", Yale Univ. Press (1968) | {{MR|0466335}} | {{ZBL|1196.22001}}</TD>
</TR><TR><TD valign="top">[2]</TD>
 <TD valign="top"> J.E. Humphreys, "Linear algebraic groups", Springer (1975) | {{MR|0396773}} | {{ZBL|0471.20029}}</TD>
</TR>
<TR><TD valign="top">[a1]</TD>
 <TD valign="top"> T.A. Springer, "Linear algebraic groups", Birkhäuser (1981) | {{MR|1642713}} | {{ZBL|0927.20024}}</TD>
</TR>
</table>