Talk:Reductive group

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Why does this page restrict the definition to the category of algebraic groups? One certainly speaks/writes of reductive Lie groups as well. Michael Kinyon 19:20, 13 September 2006 (UTC)

According to this [1] it would be a small addition: namely define reductive Lie algebra here and then (connected, I guess) reductive Lie group also. In other words local isomorphisms allowed. Charles Matthews 13:45, 14 September 2006 (UTC)

Having finitely many connected components is OK. Since every Lie group is the semidirect product of a real reductive group and a solvable group, it is certainly worth saying something about the Lie case here. We could also mention various special cases that help give some intuition about reductive groups, e.g., a matrix Lie group is reductive iff it is closed under conjugate transpose. Michael Kinyon 16:47, 14 September 2006 (UTC)

Nice work so far, Charles. Sorry I haven't contributed more to this particular page. Hopefully soon. Michael Kinyon 21:22, 23 September 2006 (UTC)

[edit] Reductive Lie algebras

Currently Reductive Lie algebra redirects to Lie algebra, and the word "reductive" occurs nowhere on that page. I think it might be a good idea to create the former so that this page can point to it. The page could include the various equivalent forms (semisimple plus center, completely reducible adjoint rep, etc.) and an example or two, including fleshing out R.e.b.'s reminder in a recent edit that "reductive" in the Lie algebra context doesn't mean that all reps are completely reducible. This page could then focus more on the group level. If I get some time in the next couple of weeks, I'll make such a page, but for now, I'll put something at the request page in case someone wants to get right to it. Michael Kinyon 17:54, 24 September 2006 (UTC)