Timeline for Representations of reductive Lie group
Current License: CC BY-SA 2.5
15 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| May 16, 2010 at 18:38 | comment | added | Victor Protsak | Yes, indeed! I had only scanned through the first 2 paragraphs and was puzzled by your "messes up as many things as possible" remark as I didn't see any. So he is wrong, too. | |
| May 14, 2010 at 6:47 | comment | added | Vladimir Dotsenko | Victor, does not one of the sentences in Popov's article read "A linear algebraic group over a field of characteristic 0 is reductive if and only if its Lie algebra is a reductive Lie algebra"? It is more than an bit misleading, isn't it? | |
| May 13, 2010 at 19:48 | comment | added | Victor Protsak | (continued) In representation theory of reductive Lie groups (real or p-adic), where these objects play a major role, there is no disagreement on the fundamental aspects of the definition; the sticky points are details about disconnected groups and infinite centers. | |
| May 13, 2010 at 19:47 | comment | added | Victor Protsak | I didn't see any problems with Popov's article, and it only discusses algebraic groups, but Onishchik is certainly wrong - thanks for the link, that's definitely folk etymology I was thinking of. Of course, it stands to reason to look up the definition of a reductive Lie group in a book "Real reductive groups", even if "Lie" is not part of the title! | |
| May 13, 2010 at 8:24 | comment | added | Vladimir Dotsenko | @Victor: I believe that at some point one of the existing versions of terminology might have been "reductive" for Lie groups with reductive Lie algebras and "linearly reductive" for what now is more commonly called reductive. Some traces of that mess remained till today, see, e.g. eom.springer.de/R/r080440.htm (which messes up as many things as possible) and eom.springer.de/l/l058500.htm. But it's probably true that this terminology is mostly dormant, and is more or less non-existent in reliable textbooks. | |
| May 12, 2010 at 21:40 | comment | added | Victor Protsak | I wonder where a reductive Lie group is defined solely in terms of its Lie algebra: this sounds like folk etymology to me. For connected linear Lie groups, the following definition is common: a subgroup G of GL(n,R) which is fixed under the transpose map is a reductive Lie group, see e.g. Knapp or Wallach, vol 1. Then a reductive Lie group is algebraic and it is also reductive as an algebraic group. | |
| May 12, 2010 at 11:01 | comment | added | Vladimir Dotsenko | @Michele: that's fine, a good lesson for me to read things carefully. A reductive algebraic group is reductive as a Lie group, sure, but I fail to see how it is relevant: for a reductive Lie group, the statement you are interested in is false, while for a reductive algebraic group it is true, so you'd better be careful too :) | |
| May 12, 2010 at 10:58 | comment | added | Michele Torielli | Sorry for the misunderstanding. The fact is that a reductive algebraic group is reductive also as Lie group. | |
| May 12, 2010 at 10:51 | comment | added | Vladimir Dotsenko | @Victor: thanks! yes, I guess I read the title of this question better than the body :( I edited the answer accordingly. | |
| May 12, 2010 at 10:49 | history | undeleted | Vladimir Dotsenko | ||
| May 12, 2010 at 10:49 | history | edited | Vladimir Dotsenko | CC BY-SA 2.5 |
added 659 characters in body
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| May 12, 2010 at 10:37 | history | deleted | Vladimir Dotsenko | ||
| May 12, 2010 at 10:35 | comment | added | Victor Protsak | The additive group is unipotent, which is the opposite of reductive. | |
| May 12, 2010 at 10:31 | history | answered | Vladimir Dotsenko | CC BY-SA 2.5 |