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

## Summary

A Chevalley group is a group of automorphisms of a simple Lie algebra rebuilt over an arbitrary field.
The rebuilding uses the integral structure given by a Chevalley basis.
Claude Chevalley introduced this construction in 1955.
It gives, in a uniform way, a simple group for each simple complex Lie algebra and each field, apart from a few small exceptions.
Over finite fields these groups give most of the families of finite simple groups of Lie type, including new families of exceptional type.

## Explanations

### L1 human

A complex simple Lie algebra, such as the algebra of $2 \times 2$ matrices of trace $0$, has a basis in which all the structure constants are integers.
Because of this, one can reduce it modulo a prime, or more generally build a copy of it over any field.
The Chevalley group is generated by exponentials of certain nilpotent operators on this copy.
For the algebra of type $A_1$ and a field $K$, the result is the group $\mathrm{PSL}_2(K)$.