nLab reductive group

Reductive groups

Reductive groups

Definition

Definition

A Lie group GG is called reductive if its Lie algebra 𝔤\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.