Chevalley group

Definition

Let $\mathfrak{g}$ be a simple Lie algebra over $\C$ with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$. Let $\set{h_1, \dots, h_\ell} \cup \set{e_\alpha : \alpha \in \Phi}$ be a Chevalley basis of $\mathfrak{g}$, and let $\mathfrak{g}_\Z$ be its $\Z$-span. Let $K$ be a field and let $\mathfrak{g}_K = \mathfrak{g}_\Z \otimes_\Z K$. For $\alpha \in \Phi$ and $t \in K$, let $x_\alpha(t) = \sum_{k \ge 0} t^k (\operatorname{ad} e_\alpha)^k / k!$, an automorphism of $\mathfrak{g}_K$. Here $(\operatorname{ad} e_\alpha)^k / k!$ is the reduction of an integer matrix on $\mathfrak{g}_\Z$ and is $0$ for $k \ge 4$. The (adjoint) Chevalley group of type $\Phi$ over $K$ is the subgroup of the automorphism group of $\mathfrak{g}_K$ generated by all these $x_\alpha(t)$.