reductive group

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

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.

- Wikipedia,
*Reductive group*

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