# Contents

## Reductive Lie groups

### Definition

A Lie group $G$ is called reductive if its Lie algebra $𝔤$ is reductive , which is the case if it is the the direct sum of an abelian and a semisimple Lie algebra.

### Properties

A Lie algebra is reductive if and only if its adjoint representation is completely reducible?, but this does not imply that all of its finite dimensional representations are completely reducible.

The concept of reductive is not quite the same for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of a unipotent algebraic group.

Revised on September 2, 2010 23:24:46 by Toby Bartels (173.190.149.107)