Talk:Lie group
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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 Ck structure is a maximal atlas of open sets homeomorphic to Rn such that transition functions are Ck. 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 article looks ok to me now, so I'm removing the tag. --Chan-Ho (Talk) 05:09, 19 February 2007 (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)
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)
One whether this is useful... YES Mathchem271828 15:33, 16 January 2007 (UTC)
[edit] Incorrect statement?
Is the following statement correct? "The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL2(R) is not surjective."
I think that for nilpotent Lie groups we also need that it is simply connected. See Knapp: Lie groups beyond an introduction, second edition, Theorem 1.127, page 107. Pierreback 10:11, 27 February 2007 (UTC)
[edit] Lie Groups in the news!
248-dimension maths puzzle solved
Mentions E8 and, unlike this article, is written in a way that is intelligible to most readers.
- Thanks for the link! As for your critique, I'm baffled as to what your point is. The BBC article is intelligible by virtue of not explaining what a Lie group is! Unfortunately, we don't have that luxury here. --Chan-Ho (Talk) 15:38, 19 March 2007 (UTC)
Also:
Math team solves the unsolvable E8
My point is this: I saw Lie Groups mentioned in the news and I wanted to learn more about them. So I turn to Wikipedia, right? Well, maybe not. What I found was only partially intelligible to me, even though I took several math courses in college (I hit the wall at analysis). I also had to search to find any mention of E8, even though the articles say it is significant. (I did find one instance of E8 in a list.) I expect most Wikipedia readers will have a reaction similar to mine, or worse. --Joe Wiki 22:48, 19 March 2007 (UTC)
- The real deal is [1]. I think we don't even have an article on Kazhdan-Lusztig polynomial. We do have articles on both David Kazhdan and George Lusztig. Regrettably if you have only an eighteenth-century background in mathematics, this stuff is not going to be immediately accessible. Charles Matthews 22:54, 19 March 2007 (UTC)
[edit] Trouth grup // In Vandalism
I deleted the following as it is obvious the consequence of vandalism;
Oscar Baltazar discoved E8 on a trip to the montezuma pyramid in Iceland. This radical discovery took him four days to compute which would take a normal person about 30 minutes to complete. Oscar Baltazar is a little slow, however this retardation helped his discovery.
Now how much of it was true data before the distortion is beyond me (At least it could be true that OB discovered it, that it was discovered either in an acheological site showing great knowledge from ancient times or in a lab in Iceland, of course "Montezuma Pyramid in Island" makes no sense, even "Montezuma Pyramid" is in itself misleading part of vandalism, then what follows is an insult, making it seem OB is just a personal acquaintance of the vandal or else the vandal targets OB out of whims, envy or both, and something may have taken 4 days to compute, that "it would take a normal person about 30 minutes to complete" is, of course, the offense but, as I've shown, parts could be true). Said what has been said... A cleaned version could be added to the article.Herle King 10:03, 20 March 2007 (UTC)
[edit] Rendering
I have deleted the section titled "Rendering", as not only did it not make any sense as it were, but I could not even see a possible way of making sense of it at all. Please, refer to the page on E8 and the discussion there. Arcfrk 01:19, 22 March 2007 (UTC)
[edit] Motivation section is contradictory
The motivation section as of March 22 is extremely confusing to me, as it presents an example that is impossible (path from identity to reflection), describes the flaw in the example, then says "but if we ignore this complication, it is perhaps possible to visualize how one symmetry (thought of as a motion) can be continuously modified to obtain another one." Wouldn't it be better to have an example that doesn't have the complication of being flawed and impossible? --Billgordon1099 06:29, 22 March 2007 (UTC)
- I have to admit to being puzzled by the point of this "example" also. What is wrong with simply considering a rotation instead of a reflection? This is simple to visualize as being part of a smooth family; one can pick an order 2 rotation (another point made in the motivation section) to see that such members of a family can have finite order. I suggest that simply explaining S0(2) or SO(3) is good fodder for this kind of section. --Chan-Ho (Talk) 09:00, 22 March 2007 (UTC)
[edit] early history
I have added a section on early history of the theory of Lie groups. However, as it grew somewhat large, I will probably summarize the content here and move the bulk of it to a separate article. Arcfrk 08:38, 22 March 2007 (UTC)
[edit] The concept of Lie group
I have restored this motivational section, because I believe it is unwise to remove it entirely. The rest of the article hardly provides any intuitive explanation of what Lie groups are and how they are used. While this sections was not a model of clarity, there was a lot of work already done on improving it. Besides, what were the errors that it was "full of", as the summary of the edit stated? Arcfrk 21:41, 23 March 2007 (UTC)
- Some of the errors: a rotation is not a Lie group, plenty of noncompact groups have no translations, a Lie group of transformations does not usually contain all nearby transformations, the definition of the Lie algebra has little to do with path connectedness (it works fine for p-adic groups for example), and the description of simple Lie algebras is wrong: see the list of simple Lie groups. R.e.b. 22:26, 23 March 2007 (UTC)
- I think you're reaching a bit on some of these. Of course a rotation is not a Lie group, but it's a concrete example of an element of such a group. Nowhere do I see an assertion, or even a hint, that the element was the group. I plead guilty to introducing path connectedness into the discussion in the first place, but the context was trying to help a non-specialist reader to picture a 1-parameter subgroup. Sure, you can define algebraic groups over fields other than R or C, but then the tangent space is usually introduced as the maximal ideal associated with a point, modulo its square. Formally, that is not unlike one of several definitions that can e given in the real case. But, be that as it may, it seems to me that one of the most intuitively appealing ways of introducting the bracket is by starting out with the idea that v(f) is just the rate of change of f along v, and then use the usual commutator [v,w] = vw - wv "applied" to f. It's hard motivate the introuction of the Lie algebra, and harder still to provide the beginner with a mental image he or she can easily grasp. Compactness came in for different reason, and that has to do with formalizing the Lie algebra/Lie group correspondence. My first attempt (written very quickly!) mentioned semisimplicity, and that was rightly removed, as an unnecessarily abstract concept in this setting. Compactness works very nicely in the representation theory though, of course, it does not extend to the setting of algebraic groups. I don't get your point about the list of Lie groups unless you're worried about differing real forms. The classification of Lie algebras presupposes an algebraically closed field (of characteristic 0) such as C. Deriving results about real Lie groups involves complications, of course. Oh yes, and of course there are natural examples of non-compact groups (the most obvious probably being GL(n,R)), but again, rotations were introduced as an aid to visualization, not to imply that they were in any way special.Greg Woodhouse 23:02, 23 March 2007 (UTC)
I agree that removing the 'motivation' section was excessive. Mathchem271828 04:56, 24 March 2007 (UTC)