group theory

# Reductive groups

## Definition

###### Definition

A Lie group $G$ is called reductive if its Lie algebra $\mathfrak{g}$ is reductive, i.e., a direct sum of an abelian and a semisimple Lie algebra.

###### Remark

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.

###### Remark

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.

## References

Revised on October 30, 2013 03:12:20 by Todd Trimble (67.81.95.215)