---
tag: 42ZCE14AD2
kind: concept
name: Reductive Lie group
msc: ["22E46", "22E15"]
specializes: [7JBRGCVJRB]
sources:
  - url: https://en.wikipedia.org/wiki/Reductive_group
---

## Summary

A reductive Lie group is a [[Lie group|7JBRGCVJRB]] whose Lie algebra is a reductive Lie algebra.
Such a Lie algebra is the direct sum of its centre and a semisimple Lie algebra.
The class contains the semisimple groups, the tori, and the general linear groups $GL_n(\R)$ and $GL_n(\C)$, as well as every compact Lie group.
It is the natural setting for much of representation theory and harmonic analysis.
One reason is that Levi factors of parabolic subgroups of reductive groups are again reductive, while those of semisimple groups need not be semisimple.
Authors differ on the extra conditions they impose, such as finitely many components or conditions on the adjoint action.

## Explanations

### L1 human

A reductive Lie group is a group that is semisimple up to a commutative part that sits in the centre.
The basic example is $GL_n(\R)$: every invertible matrix is a scalar times a matrix of determinant $\pm 1$, and the scalars commute with everything.
The Lie algebra of $GL_n(\R)$ splits as the scalar matrices plus the trace-zero matrices.
The group of upper triangular invertible matrices is not reductive: its strictly upper triangular part is a non-central normal piece that cannot be split off.