In the answer of my question:

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!

  • $\begingroup$ I should let the experts answer, but my understanding is that all this means is that there is very little you can say (except structurally) about a reductive Lie algebra in characteristic zero. For example, you cannot infer much about its representation theory simply from the fact that a Lie algebra is reductive. On the other hand, if you know that it the Lie algebra of a Lie group, then at least for those representations which integrate to representations of the group you can say more. $\endgroup$ – José Figueroa-O'Farrill May 6 '10 at 13:32

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 (EDIT: nonzero) 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.

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  • 2
    $\begingroup$ To reinforce what Ben says, the standard definition of "reductive" Lie algebra in characteristic 0 just tells you that it is the direct sum of a semisimple and an abelian Lie algebra. The semisimple ones are characterized by having a nondegenerate Killing form (and resulting rich structure), while the abelian ones are just boring vector spaces. Even for algebraic groups, "reductive" has strong representation-theoretic implications only in characteristic 0; but the structure is interesting in any characteristic and over various ground fields. $\endgroup$ – Jim Humphreys May 6 '10 at 14:33
  • $\begingroup$ Thank you. The fact is that I'm dealing with a Lie algebra of a connected algebraic group and I'm trying to understand their connection, also because I'd like to prove that a certainly representation is semisimple. $\endgroup$ – Michele Torielli May 6 '10 at 14:55
  • $\begingroup$ Your callous disregard of the $0$ Lie algebra as the exemplar of your second definition of reductivity surely wounds it. $\endgroup$ – LSpice Jan 11 '19 at 16:41
  • $\begingroup$ If one wishes to be precise, it's most likely that $\mathfrak{g}$ is implicitly assumed finite-dimensional, and possibly there's some assumption on the field too. (There are infinite-dimensional Lie algebras with no nonzero finite-dim rep, and hence all their finite-dim reps are semisimple.) Anyway the question was not very precise either. $\endgroup$ – YCor Jan 11 '19 at 17:48

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).

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