Reductive Lie algebra of a Lie group - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:19:03Z http://mathoverflow.net/feeds/question/23706 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23706/reductive-lie-algebra-of-a-lie-group Reductive Lie algebra of a Lie group Michele Torielli 2010-05-06T13:14:05Z 2010-05-13T18:47:51Z <p>In the answer of my question:</p> <p><a href="http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras" rel="nofollow">http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras</a></p> <p>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."</p> <p>Can please someone explain that to me or give to me any reference? thank you!</p> http://mathoverflow.net/questions/23706/reductive-lie-algebra-of-a-lie-group/23714#23714 Answer by Ben Webster for Reductive Lie algebra of a Lie group Ben Webster 2010-05-06T14:15:56Z 2010-05-06T14:15:56Z <p>What Jim means is that one naive definition of reductive Lie algebra </p> <ul> <li>\$\mathfrak{g}\$ is <strong>reductive</strong> if all its finite-dimensional representations are semi-simple.</li> </ul> <p>already has a name: <strong>semi-simple.</strong></p> <p>Another one</p> <ul> <li>\$\mathfrak{g}\$ is <strong>reductive</strong> if all its representations are semi-simple.</li> </ul> <p>is actually trivial; there are no Lie algebras that satisfy it.</p> <p>Of course, there actually is a pretty good definition that matches better with reductive for groups:</p> <ul> <li>\$\mathfrak{g}\$ is <strong>reductive</strong> if its adjoint representation is semi-simple.</li> </ul> <p>but it's important to keep in mind that the properties above don't follow from that.</p> http://mathoverflow.net/questions/23706/reductive-lie-algebra-of-a-lie-group/24531#24531 Answer by Pasha Zusmanovich for Reductive Lie algebra of a Lie group Pasha Zusmanovich 2010-05-13T18:47:51Z 2010-05-13T18:47:51Z <p>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).</p>