---
tag: ZMDAEW6DHN
kind: definition
name: Chevalley group
msc: [20G40, 20D06, 20G15]
defines: YPCH4Q0V34
uses: [0K1SCPBNKJ]
sources:
  - url: https://en.wikipedia.org/wiki/Group_of_Lie_type
    locator: Chevalley groups
---

## 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|0K1SCPBNKJ]] of $\mathfrak{g}_K$ generated by all these $x_\alpha(t)$.

## Remarks

It is written $\Phi(K)$, for example $A_n(K)$, $E_8(K)$ or $G_2(q)$ when $K = \F_q$.
The sum defining $x_\alpha(t)$ is finite because $\operatorname{ad} e_\alpha$ is nilpotent, and the Chevalley basis makes its divided powers integral, so the formula makes sense in every characteristic.
Example: for $\Phi$ of type $A_1$, the group $A_1(K)$ is isomorphic to $\mathrm{PSL}_2(K)$.
Non-example: let $K$ be a field of characteristic not $2$.
Then $\mathrm{SL}_2(K)$ is not the adjoint Chevalley group of type $A_1$.
Its centre $\set{\pm I}$ has order $2$, while $A_1(K) \cong \mathrm{PSL}_2(K)$ has trivial centre.
Chevalley proved that $\Phi(K)$ is a [[simple group|0YMYMNQ0QZ]] except for $A_1(\F_2)$, $A_1(\F_3)$, $B_2(\F_2)$ and $G_2(\F_2)$.

## Explanations

### L1 human

A Chevalley group is the group generated by exponentials of root vectors of a simple Lie algebra, rebuilt over an arbitrary field.