Semi-simple algebraic group

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.