Talk:Borel subgroup
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This article is not helpful at all in explaining why "parabolic" subgroups are called as such. %0?è�á -Mike
I'm not found of the sentence "There is an ambiguity here, which turns out not to matter much." I was going to change it so say that the meaning of the definition depends if the context is 'algebraic group theory' or 'group theory'. But I don't understand the material enough to change it (that is why I was reading this article, a book used the term in passing, I still don't know what it is saying). Can someone clarify what is amgiguous and why it doesn't matter? RJFJR 00:53, Dec 30, 2004 (UTC)
As written this is kind of choppy. There are a couple of very short paragraphs that I'm not sure how go together. RJFJR 00:53, Dec 30, 2004 (UTC)
Solvable groups
[edit]I didn't put in anything about the structure of Borel subgroups, though it is a basic part of the theory. That belongs to the theory of (connected) solvable algebraic groups, which needs its own page, and the editing I did took considerably more than the couple of minutes I intended.
The issue of connectivity should not be lost sight of. Also, there are at least three parallel languages for dealing with algebraic groups, never entirely equivalent, and while we don't need to dwell on the scheme theoretic one here it needs to be acknowledged. Also if one does focus on the group of rational points over a particular field then a lot of arithmetical questions arise which we don't acknowledge, such as the existence of these things in an appropriate set (in particular the notion of Borel and minimal parabolic can diverge and the latter can take over the main role ...). Anyway this sort of thing would muddy all the pages, and if it's going to be dealt with I suspect it should be dealt with on an entirely separate page. This isn't really my line either, just wandering through.
I think these days people generally take varieties to be irreducible, and algebraic groups to be connected, but over time terminology has wobbled a lot. Best to pin things down explicitly. Abu Amaal
Borel subgroup and homogeneous space
[edit]I'm not happy with the following sentence.
- Thus B is a Borel subgroup precisely when G/B is a homogeneous space for G and a complete variety, which is "as large as possible".
As far as I understand this, G/B is a homogeneous space for any subgroup B of G, so this should not be implied to be a necessary condition for Borel'ness. Of course, I'm likely missing something. My suggestion would be:
- Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is "as large as possible".