# Chevalley group

2010 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A Chevalley group is a linear algebraic group over some field, related to a semi-simple complex Lie algebra, in the following way: Let $\def\fg{ {\mathfrak g}}\fg$ be a semi-simple Lie algebra over $\C$, let $\def\fh{ {\mathfrak h}}\fh$ be its Cartan subalgebra, let $\def\S{\Sigma}\S$ be a root system of $\fg$ with respect to $\fh$, let $\def\a{\alpha}\{\a_1,\dots,\a_k\}\subset \S$ be a system of simple roots, let $\{H_{\a_i} (1\le i\le k\}); X_\a (\a\in\S)\}$ be a Chevalley basis of the algebra $\fg$, and let $\fg_\Z$ be its linear envelope over $\Z$. Let $\def\phi{\varphi}\phi$ be a faithful representation of the Lie algebra $\fg$ in a finite-dimensional vector space $V$. It turns out that there is a lattice in $V$ (i.e. a free Abelian subgroup a basis of which is the basis of the space $V$) which is invariant with respect to all operators $\phi(X_\a)^m/m!$ ($\a\in\S$, $m$ is a natural number). If $k$ is an arbitrary field and if $V^k=M\otimes k$, then, for $\a\in\S$, homomorphisms $x_\a : k^+ \to {\rm GL}(V^k)$ of the additive group $k^+$ of $k$ into ${\rm GL}(V^k)$ are defined and given by the formulas

$$x_\a(t) = \sum_{m=0}^\infty t^m\frac{\phi(X_\a)^m}{m!}.$$ The subgroups $\def\fX{ {\rm X}}\fX_\a = {\rm Im}\; x_\a$, $\a\in\S$, generate in ${\rm GL}(V^k)$ some subgroup $G_k$, which is called the Chevalley group related to the Lie algebra $\fg$, the representation $\phi$ and the field $k$. If $\phi = {\rm ad}$ (the adjoint representation), the Chevalley groups were defined by C. Chevalley in 1955 (see [Ch]).

If $K$ is an algebraically closed field containing $k$, then a Chevalley group $G_K$ is a connected semi-simple linear algebraic group over $K$, defined and split over the prime subfield $k_0\subseteq k$. Its Lie algebra is isomorphic to $\fg_\Z\otimes K$. The group $G_k$ is the commutator subgroup of the group $G_K(k)$ of points of $G_K$ that are rational over $k$. Any connected semi-simple linear algebraic group over $K$ is isomorphic to one of the Chevalley groups. The algebraic groups $G_K$ (and $G_k$ as abstract groups) depend only on the lattice $\def\G{\Gamma}\G_\phi\subset \fh^*$ generated by the weights of the representation $\phi$. If $\G_\phi$ coincides with the lattice of roots $\G_0$, then $G_K$ is called the adjoint group, and if $\G_\phi=\G_1$ (the lattice of weights, see Lie group, semi-simple), then $G_K$ is called a universal or simply-connected group. If $G_K$ is universal, then $G_k=G_K(k)$.

The Chevalley group $G_K$ always coincides with its commutator subgroup. The centre of $G_K$ is finite. For example, the centre $Z$ of the universal group $G_K$ is isomorphic to ${\rm Hom}(\G_1/\G_0, k^*)$, and the corresponding adjoint group is isomorphic to $G_k/Z$ and has trivial centre.

If the algebra $\fg$ is simple, then the adjoint Chevalley group $G_k$ is simple, except in the following cases: $|k| = 2$ and $\fg$ is a Lie algebra of type $A_1, B_2$ or $G_2$; or $|k| = 3$ and $\fg$ is a Lie algebra of type $A_1$. Other series of simple groups can be obtained when considering the subgroups of fixed points of certain automorphisms of finite order of Chevalley groups (so-called torsion groups).

If the field $k$ is finite, then the order of the universal group $G_k$ can be computed by the formula

$$|G_k| = q^N \prod_{l=1}^r(q^{d_i} -1),$$ where $q = |k|$, $d_i$ ($i=1,\dots,r$) are exponents of the Lie algebra $\fg$, i.e. the degrees of the free polynomials on $\fh$, generating the algebras, that are invariant with respect to the Weyl group, and $N=\sum_{i=1}^r(d_i - 1)$ is the number of positive roots.

There is a well-developed theory of rational linear representations of the Chevalley groups $G_k$ over an infinite field $k$. It is reduced to the case of an algebraically closed field, and in the latter case coincides with the theory of rational representations (cf. Rational representation) of semi-simple algebraic groups. If $\fg$ is simple, $G_k$ is the universal Chevalley group over the infinite field $k$, and $\def\s{\sigma}\s$ is a non-trivial irreducible finite-dimensional representation of $G_k$ (as an abstract group) over an algebraically closed field $K$, then there exists a finite set of imbeddings $\phi_i:k\to K$ and a set of rational representations $\rho_i$ of the groups $G_{\phi_i}(k)$ such that $\s = \otimes_i\rho_i\circ \phi_i$. Concerning representations of Chevalley groups, see also [St], [BoCaCuIwSpSt], [Hu2].

How to Cite This Entry:
Chevalley group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chevalley_group&oldid=30675
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article