Reductive Lie algebra of a Lie group - MathOverflow

Reductive Lie algebra of a Lie group

2010-05-06T13:14:05Z

In the answer of my question:

http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras

James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in characteristic 0 has no intrinsic interest, unless you study the Lie algebra of a Lie (or algebraic) group and relate their representations carefully."

Can please someone explain that to me or give to me any reference? thank you!

Answer by Ben Webster for Reductive Lie algebra of a Lie group

2010-05-06T14:15:56Z

What Jim means is that one naive definition of reductive Lie algebra

$\mathfrak{g}$ is reductive if all its finite-dimensional representations are semi-simple.

already has a name: semi-simple.

Another one

$\mathfrak{g}$ is reductive if all its representations are semi-simple.

is actually trivial; there are no Lie algebras that satisfy it.

Of course, there actually is a pretty good definition that matches better with reductive for groups:

$\mathfrak{g}$ is reductive if its adjoint representation is semi-simple.

but it's important to keep in mind that the properties above don't follow from that.

Answer by Pasha Zusmanovich for Reductive Lie algebra of a Lie group

2010-05-13T18:47:51Z

I wouldn't exclude the possibility that some nicely-looking ("intrinsic"?) characterization of reductive Lie algebras exist, say, in homological terms. How relevant such characterization might be to the questions discussed here (connection with Lie groups and representation theory) - that's another matter (probably it will not).