Talk:Reductive group

Why restricted?

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)

• My impression on the usage is that most of the time it's either "reductive (linear) algebraic group" or "reductive Lie group", and almost never simply "reductive group". Of course, you can say that "reductive group" sounds less technical, but there is currently a fairly well-defined division between those two concepts, and in many contexts they are not interchangeable.
• On a slightly different note, it seems that the introduction is so technical as to be virtually useless to a non-expert. Was the intent here simply to give a short definition for reference purposes? Shouldn't there be some motivations explained first, with defintions following later in the main text? Arcfrk 08:25, 10 March 2007 (UTC)

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)

Reductive Lie group?

Do many people actually use the notion of "reductive Lie group" given here? For example for complex reductive groups (which are algebraic and Lie groups) surely one wants to take the class of groups that arise by complexifying compact groups (i.e. taking the affine group associated to the ring of representative functions on a comapact group). It seems to me to be mighty confusing to define reductive Lie groups in a way that is incompatible with this. —Preceding unsigned comment added by 78.239.179.184 (talk) 18:42, 24 January 2011 (UTC)

Complete reducibility

The article included the incorrect claim that representations of a reductive group are completely reducible. I have removed the offending assertion. A counterexample: the multiplicative group of a non-archimedean local field.

The conter example is irrelevant, since the context is algebraic representation (strictly speacing the multiplicative group of a non-archimedean local field.is a topological group of algebraic nature and not algebraic group).

However, the statement is only true on 0 characteristic. Rami (talk) 09:07, 14 July 2024 (UTC)

rename

I sujest to rename to Algebraic reductive groups. The Artical talking only on of them. the notion Reductive group exists in other contextses, like "real reductive groups". The do not have to be algebraic. Rami (talk) 09:09, 14 July 2024 (UTC)