Talk:Lie group

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Needs more on history/motivation. Also needs gentler introduction. Tompw 09:59, 15 October 2006 (UTC)

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[edit] simply connected Lie groups & algebras

Where is a proof of the statement, that simply connected Lie groups are determined by their Lie algebras?

Yes, I agree, if someone goes ahead with this, I think that the "direct" proof for finite-dimensional Lie groups, along the lines of Wulf Rossmann's construction of a simply connected Lie group from any finite dimensional lie algebra in his "Lie Groups: An introduction through Linear Groups" should be included, beginning with the Baker-Campbell-Hausdorff formula and the few attendant details to patch up the fact that the BCH formula converges in a neighbourhood of the identity that is smaller in general than the simply connected group. One can always include the more general, slicker discussions grounded on the Frobenius integrability theorem. I have never seen a proof of Frobenius integrability theorem, elegant though it may be, that I understand (I have sighted, but not understood proofs!) and suspect I am not alone in this lacking! Rod Vance 20th Oktober, 2005

Hmmm, sounds as if it might go better on the BCH page. Charles Matthews 08:52, 19 October 2005 (UTC)

[edit] "analytical structure"

This phrase is used in the first paragraph of the article, and appears elsewhere on WP in only 3 or 4 places, always in reference to Lie groups. Does it simply mean "any mathematical structure that can be represented by an analytical function"? Or does it have some more specific formal mathematical meaning? Hv 15:57, 9 October 2005 (UTC)

There used to be an article which spelled out what the various differential structures were, but I can't find it now. Anyway, a C^k structure is a maximal atlas of open sets homeomorphic to Rⁿ such that transition functions are C^k. If k=0, then the transition functions are continuous, and you have a topological manifold; for finite k, the transition functions are k-times differentiable and you have a differential manifold; k=∞, that denotes smooth transition functions, and your have a smooth manifold; and k=ω denotes transition functions that are real analytic, that is, that have a convergent Taylor series. One of Hilbert's problems was to prove that a Lie group with a C⁰ structure actually has a unique compatible analytic structure. -Lethe | Talk 20:22, 9 October 2005 (UTC)

I found the page. It's differentiability class. -Lethe | Talk 23:02, 9 October 2005 (UTC)

[edit] Request for technical explanation

I think this concept would be a lot clearer if a specific example were explained in detail. For example, what specific properties of R³ make it eligible to a Lie group? What similar system(s) would not be? It would also be a good idea to show the definition of this Lie group (the set plus the operation) both in words and in mathematical symbols. -- Beland 16:40, 18 December 2005 (UTC)

R³ is a terrible example. Most people's first encounter with a Lie group is SO(3) and its related cover SU(2), the rotation group. Lie groups are best learend by example, and the article should emphasize this. linas 21:39, 5 March 2006 (UTC)

The first Lie group that every one meets is R, followed probably by Rⁿ. Most people's first nonabelian Lie group may be SU(2). -lethe ^{talk +} 22:45, 5 March 2006 (UTC)

[edit] gothic

Article states:

... usually denoted by a gothic g ...

Anyone care to change this to ? linas 21:35, 5 March 2006 (UTC)

Bad idea: not all browsers can handle gothic. R.e.b. 17:48, 19 April 2006 (UTC)

<math>\mathfrak{g}</math> produces an image, which any graphical browser can render. -lethe ^{talk +} 02:54, 20 April 2006 (UTC)

That's what I thought too until I found a counterexample. Maybe the browser was just having a bad day. R.e.b. 03:57, 20 April 2006 (UTC)

[edit] is this useful?

It seems to me that anybody who can understand this article, doesn't need this article. And anybody who needs this article will not get a damn thing from it. As such, this is entirely useless as an encyclopedia entry. Writing an article explaining an element of group theory using dense notation and verbiage that requires intimate knowledge of group theory is a waste of time. —This unsigned comment was added by Birge (talk • contribs) .

Well thank you for that very helpful constructive criticism. With that comment, I can now rewrite all our advanced math articles so that they're immediately accessible to anyone who has Birge's mathematics background, regardless of prerequisites. Happy day! -lethe ^{talk +} 20:24, 17 March 2006 (UTC)

I have a use: when I'm studying, sometimes I forget things or want some verification. For example, I came to this page wanting to know what the Lie bracket of a Lie algebra might have to do with the Lie derivative on the same Lie group as a manifold (bracket = derivative evaluated at a fixed point?). I know basic things and learn when reading this stuff. On an unrelated note, the following appears in the article:

'...we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not obvious that the Lie algebra does not depend on which representation we use."

There is always the trivial representation, right? So should it be that we need to have a faithful representation of the Lie group? Orthografer 15:15, 19 October 2006 (UTC)

There is a case that this could be a good candiadate for making more accessable. It is a fairly important mathematical concept, with 257 incoming links and no. 110 on my list of top linked maths articles. I've been meaning to include some examples of the most common lie groups GL(2,R), SL(2,R), Orthogonal group, so people new to the concept can have a bit of a gentler introduction. Probably no 3 on my list of things to do after Algebra, and getting on with the real world. Birge, might wish to look at Manifold, table of Lie groups, and General linear group in the mean time.--Salix alba (talk) 21:09, 17 March 2006 (UTC)

Lethe: I understand that your ego as a mathematician is intimately tied to nobody understanding what the heck you're talking about. Having said that, if you're going to bother to have an encyclopedia entry explaining group theory to the great unwashed, maybe it could explain something, not simply serve as a way to impress people with how complicated group theory is. I stand by my assertion that as it stands, this article does nobody any good. Someone like you doesn't need it, and by the time I figure out what any of it means, I won't need it either. I'm not saying everything in the current article needs to be covered such that it is self contained. I'm suggesting you don't need to cover most of the stuff in here. Covering less, but actually explaining it, would be better. This is supposed to be an encyclopedia article on Group Theory, not a review sheet for a class on Group Theory. It should be the first thing you read on Group Theory, not the last. Ask yourself, is this more or less understandable than an actual textbook on GT? Shouldn't it be the other way around? Birge 02:37, 21 March 2006 (UTC)

Just to be pedantic, this is not the main Group theory page, its about a particular important class of continuous group which mave a manifold structure. --Salix alba (talk) 10:32, 21 March 2006 (UTC)

Heh, well I don't think it's pedantic to distinguish Lie groups from general groups. Salix is right, Lie groups are considerably more technical than discrete groups. Anyway, I've added a pretty low-level sentence to the intro. Maybe it'll help? -lethe ^{talk +} 14:52, 21 March 2006 (UTC)

Well, it would help if we defined smooth manifold and group (and it would also be nice if the link to smooth manifold didn't just dump the reader at the top of differentiable manifold and let him figure out that there is a difference.) Septentrionalis 20:14, 6 August 2006 (UTC)

Having tried to do this, a question arises. What level of information can we assume? Septentrionalis 20:28, 6 August 2006 (UTC)

Ah... the eternal question of WP mathematics article. I personallty feel that some basic understanding of group theory (what a group is and what is does) should be assumed. Links to the articles on groups and group theory should mean that anyone who doesn't understand abotu groups will be easily able to look them up. Tompw 15:39, 19 October 2006 (UTC)