Reductive Group - Lie Group Case

Lie Group Case

For more details on this topic, see Reductive Lie algebra.

More generally, in the case of Lie groups, a reductive Lie group G can be defined in terms of its Lie algebra, namely a reductive Lie group is one whose Lie algebra g is a reductive Lie algebra; concretely, a Lie algebra that is the sum of an abelian and a semisimple Lie algebra. Sometimes the condition that identity component G0 of G is of finite index is added.

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.

For example, the one-dimensional, abelian Lie algebra R is obviously reductive, and is the Lie algebra of both a reductive algebraic group G_m (the multiplicative group of nonzero real numbers) and also a unipotent (non-reductive) algebraic group G_a (the additive group of real numbers). These are not isomorphic as algebraic groups; at the Lie algebra level we see the same structure, but this is not enough to make any stronger assertion (essentially because the exponential map is not an algebraic function).