nLab
reductive group

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 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)