Talk:Simple Lie group
|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
I wouldn't say this page should be merged with root system. The duplicated material on simple Lie algebras should be edited out, and a proper account given (of the compact simple Lie groups first).
Charles Matthews 10:34, 21 Aug 2003 (UTC)
From the article:
- Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology), by computing the fundamental group of G. This was done by Cartan.
Which one? Father or son? -- Anon.
Well, you could follow the link, and see.
Charles Matthews 19:12, 3 Dec 2003 (UTC)
- I think this article needs a restructure to make it more accessible to the non-specialist. Diving straight into the method of classification makes it sound as though simple Lie groups only exist to be classified, and begs the question "what's the point?". We need some explanation of how they arise, why people study them and why a classification is useful. Also, the pictures/list describing the classification are incomprehensible without further description of what they mean. If they need to be here at all, they need a great deal more explanation.
- I've tidied up a little. I could try to do something more radical but it would probably be better done by somebody who really knows this stuff (I'm a discrete group theorist, for my sins....) Cambyses 04:20, 29 Apr 2004 (UTC)
I've hacked it about a bit, with some extra intro and organisation, and a bit of amplification. It's still not that great, of course.
Charles Matthews 07:15, 29 Apr 2004 (UTC)
- You do yourself a disservice: it is a great improvement already! Best wishes, Cambyses 03:15, 30 Apr 2004 (UTC)
Every book I look in defines a simple Lie group as a Lie group with a simple Lie algebra. This means that in general a simple Lie group G is *not* simple in the group sense, since it may have discrete normal subgroups corresponding to other Lie groups covered by G. See e.g. p147 Fuchs, "Symmetries, Lie Algebras and Representations : A Graduate Course for Physicists"
Indeed the first line of the article is wrong. The definition of simple Lie group does not imply that the underlying group is simple, not even topologically simple.
- The definition may well be given as a connected Lie group with simple Lie algebra, as at . Charles Matthews 12:34, 13 October 2006 (UTC)
I have changed the definition to make it more verifiable, and used this opportunity to trim down the lead. It remains too long and technical, in my opinion. At least, we don't have to go into too many details about discrete vs connected normal subgroups! Arcfrk 02:07, 26 March 2007 (UTC)
This article, quite rightly, talks about the classification of Lie groups. However, it doesn't seem to show the importance of the simple Lie groups. Am I right in saying that any compact, connected Lie group is isomorphic to a product of tori and non-abelian simple groups? If this is right, then shouldn't it be included, it seems quite important?
Ok, I was nearly right, every compact, connected Lie group is the quotient of a product of a torus and simple 1-connected Lie groups by a discrete subgroup.