Talk:Simple Lie group

Merge with root system
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)

Yes, Root system goes with Simple Lie algebra rather than with Simple Lie group. -- Toby Bartels 04:06, 24 Aug 2003 (UTC)

Which Cartan?
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)

Restructure?

 * 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)

I did some restructuring. The classification section actually classified simply connected compact groups. I made that clear, and fleshed out the classification section to make it (hopefully) mathematically correct. I also added a part on complex simple Lie groups (very important: maybe needs to be elaborated to mention algebraic groups?) and added the words "locally compact" to the definition (these are unfortunately necessary to preclude infinite-dimensional examples: maybe people here have a suggestion about formulating it in a more friendly way). Mvaintrob (talk) 02:51, 21 December 2015 (UTC)

definition
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)

Classification
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?

Jjw19 (talk) 12:59, 11 May 2010 (UTC)

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.

Jjw19 (talk) 16:48, 14 May 2010 (UTC)

Accessibility? Could we try for a little?
Hey there, I consider myself a math geek. I got as far as Laplace Transforms before my community college ran out of math courses and I've been programming computers since 1978. Accordingly, the fact that I can not understand _ANY_ of this article suggests that it might be written with too much jargon and too little effort to explain. — Preceding unsigned comment added by 24.22.62.189 (talk) 14:17, 28 April 2014 (UTC)

I added a paragraph that hopefully helps. Mvaintrob (talk) 03:16, 21 December 2015 (UTC)

Merge from List of simple Lie groups
In Dec 2019 Taku placed a mergefrom banner on List of simple Lie groups, but did not actually start any merge discussion. So here it is. I myself am neutral. I can see a case for merging, but I can also see a case for two articles; one of which is devoted to general properties and examples/counter-examples, (for example, how to go from algebra to group, discussion of disconnected components, discussion of centers and universal covers...) and another article devoted to classification only. Oh, but wait: classical groups already has this generic discussion. And also reductive group is an excellent article covering the more general case. So yes, merge. 67.198.37.16 (talk) 20:05, 10 November 2020 (UTC)


 * merge and rename Rename to classification of simple Lie groups, because in the end, that is what it will be about!? 67.198.37.16 (talk) 20:27, 10 November 2020 (UTC)


 * ✅ I wonder whether a rename is necessary; in general, I think that the simpler, broader names are better and the merged articles includes descriptions that are outside of the scope of just classification. Klbrain (talk) 07:01, 8 December 2020 (UTC)

Question about symmetric spaces section
In the section Symmetric spaces, this passage appears:

"The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup H, and the compact one is a cover of the quotient of the compact form of G by the same subgroup H. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry."

I know very little about symmetric spaces. Should readers be assuming that all the Lie groups mentioned in this passage are real Lie groups?

If so, or if not, I hope someone knowledgeable about this subject will clarify this issue in the article. 2601:200:C000:1A0:F41B:AE21:91:D096 (talk) 01:39, 22 December 2022 (UTC)


 * Complex Lie groups are also real Lie groups, and they should be included in this description. In the case where K is a compact Lie group and G its complexification, the symmetric space of non-compact type is G/K and its compact dual is (KxK)/K (where K is diagonally embedded in KxK). This corresponds to type B in the more explicit classification fiven in Symmetric space. I added a "Main" template to this section to link to this more detailed article. jraimbau (talk) 08:27, 22 December 2022 (UTC)

Mistakes in list
1. As far as I understand, the maximal compact subgroup of SL(n,R) is SO(n,R), thus the maximal compact subgroup of A_n I (split) should be D_{(n+1)/2} or B_{n/2} (D_1 of A_1, B_1 of A2 and so on).

2. There is something wrong about the non compact F4: A quaternion Kähler symmetric space should correspond to a maximal compact subgroup ?xA1, here probably C3xA1. I guess they are mixed up. But I‘m not quite sure about which is the split one (but I guess, its not the one with quaternion Kähler symmetric space, but with maximal compact subgroup B4 (Spin 9)?). MrKling (talk) 15:37, 9 September 2023 (UTC)