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A connected [[Linear algebraic group|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.
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{{MSC|20G15|14L10}}
  
A connected linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843401.png" /> of positive dimension is called simple (or quasi-simple) if it does not contain proper connected closed normal subgroups. The centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843402.png" /> of a simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843403.png" /> is finite, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843404.png" /> is simple as an abstract group. An algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843405.png" /> is semi-simple if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843406.png" /> is a product of simple connected closed normal subgroups.
+
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.
  
If the ground field is the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843407.png" /> of complex numbers, a semi-simple algebraic group is nothing but a semi-simple Lie group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843408.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s0843409.png" /> is analogous to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434010.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434011.png" /> be a [[Maximal torus|maximal torus]] in the semi-simple algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434012.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434013.png" /> be the character group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434014.png" />, regarded as a lattice in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434015.png" />. For a rational [[Linear representation|linear representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434017.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434018.png" /> is diagonalizable. Its eigenvalues, which are elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434019.png" />, are called the weights of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434020.png" />. The non-zero weights of the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434021.png" /> are called the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434022.png" />. It turns out that the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434023.png" /> of all roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434024.png" /> is a [[Root system|root system]] in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434025.png" />, and that the irreducible components of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434026.png" /> are the root systems for the simple closed normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434027.png" />. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434029.png" /> is the lattice spanned by all roots and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434030.png" /> is the weight lattice in the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434031.png" />. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434032.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434033.png" /> can be naturally identified with a real subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434035.png" /> is the Lie algebra of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434036.png" />, spanned by the differentials of all characters, while the lattices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434037.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434038.png" /> coincide (up to a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434039.png" />) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434040.png" /> (see [[Lie group, semi-simple|Lie group, 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.
  
The main classification theorem states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434041.png" /> is another semi-simple algebraic group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434042.png" /> its maximal torus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434043.png" /> a root system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434044.png" />, and if there is a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434045.png" /> giving an isomorphism between the root systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434047.png" /> and mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434048.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434049.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434050.png" /> (local isomorphism). Moreover, for any reduced root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434051.png" /> and any lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434052.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434053.png" /> there exists a semi-simple algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434054.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434055.png" /> is its root system with respect to the maximal torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434056.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084340/s08434057.png" />.
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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 isogenies (in particular, all automorphisms, cf. [[Isogeny|Isogeny]]) of a semi-simple algebraic group have also been classified.
+
The main classification theorem states that if $G'$ is another
 
+
semi-simple algebraic group, $T'$ its maximal torus, $\Sigma'\subset E'$ a root system
====References====
+
of $G'$, and if there is a linear mapping $E\to E'$ giving an isomorphism
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.G. Steinberg,   "Lectures on Chevalley groups" , Yale Univ. Press  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.E. Humphreys,  "Linear algebraic groups" , Springer  (1975)</TD></TR></table>
+
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
====Comments====
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.A. Springer,   "Linear algebraic groups" , Birkhäuser (1981)</TD></TR></table>
+
<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>

Latest revision as of 19:22, 12 December 2023

2020 Mathematics Subject Classification: Primary: 20G15 Secondary: 14L10 [MSN][ZBL]

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). 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 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 $\rho$ 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 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).

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

[1] R.G. Steinberg, "Lectures on Chevalley groups", Yale Univ. Press (1968) | MR0466335 | Zbl 1196.22001
[2] J.E. Humphreys, "Linear algebraic groups", Springer (1975) | MR0396773 | Zbl 0471.20029
[a1] T.A. Springer, "Linear algebraic groups", Birkhäuser (1981) | MR1642713 | Zbl 0927.20024
How to Cite This Entry:
Semi-simple algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-simple_algebraic_group&oldid=18012
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article