Reductive groups

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 (see at reductive algebraic group) because a reductive Lie group can be the group of real points of a unipotent algebraic group.

## References

Review includes

Discussion of the “basic” multiplicative holomorphic line 2-bundle (Chern-Simons line 3-bundle) on complex reductive groups is in

Last revised on September 3, 2014 at 16:28:50. See the history of this page for a list of all contributions to it.