---
tag: 2XS84PWE3S
kind: concept
name: Reductive group
msc: [20G15, 20G07]
specializes: [GA81VVQHEP]
sources:
  - url: https://en.wikipedia.org/wiki/Reductive_group
same_as:
  wikidata: Q1006450
  wikipedia: Reductive group
---

## Summary

A reductive group is a connected [[linear algebraic group|GA81VVQHEP]] with no nontrivial connected normal unipotent subgroup.
Examples are $\mathrm{GL}_n$, $\mathrm{SL}_n$, the special orthogonal groups $\mathrm{SO}_n$, the symplectic groups, and the [[algebraic tori|66FQFAJNS4]].
Over an algebraically closed field, reductive groups are classified by root data.
In characteristic $0$, they are the connected linear groups whose representations are all semisimple.

## Explanations

### L1 human

Inside the group of invertible upper triangular matrices sits the subgroup of those with all diagonal entries $1$.
That subgroup is unipotent and normal, and it makes the group hard to analyze.
A reductive group is one with no such part: nothing unipotent hides in it as a normal subgroup.
This is the class of groups with a clean classification, much like compact Lie groups.