Talk:Lie group

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated B-class, High-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
B Class
High Importance
 Field: Geometry


simply connected Lie groups & algebras[edit]

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)

"analytical structure"[edit]

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)

Request for technical explanation[edit]

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)

The article looks ok to me now, so I'm removing the tag. --C S (Talk) 05:09, 19 February 2007 (UTC)


Article states:

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

Anyone care to change this to \mathfrak{g}? 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)

is this useful?[edit]

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 (talkcontribs) .

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)

One whether this is useful... YES Mathchem271828 15:33, 16 January 2007 (UTC)

I have to say I sympathize with Birge. I also often feel quite frustrated trying to understand many of the mathematics articles in Wikipedia. I won't go as far as saying they are useless, which I feel is obviously false, but they probably are useless to most people not already familiar with the subject. (They have been to me so far, and I don't consider myself having any particular problem understanding mathematics in other contexts). I doubt it has to be this way, the manifold and group articles (that I just read) for example, are very easy to understand even with no previous knowledge of these concepts. Linking to other articles that explain more fundamental concepts such as group is great, but it's not very helpful if that article is equally dense and hard to understand which is often the case. Simply adding a more accessible introduction that doesn't try to be so rigorous, stringent and formal would help a lot I suspect. There is no need to "dumb down" all the math articles to a level we stupid people can understand, as suggested by lethe. --Apis O-tang (talk) 03:44, 28 March 2008 (UTC)

Maybe I was a bit too quick at commenting here, the quality of the math articles seems to have improved significantly compared to how I remember them. Thanks to anyone who's helping to address this issue. =) --Apis O-tang (talk) 05:28, 28 March 2008 (UTC)

yes yes it is useful, I mean, here I am reading it because I want to know what lie groups are. I find it helpful. IF you dont why not go read something else? If you find that you don't understand something, maybe you need to do more research and get the necessary prerequisites. For me, an intermediate student of math, I find this information pretty good. Somethings are over my head and some under it. Its good-gives a person perspective, don't get nervous.Jorge —Preceding unsigned comment added by (talkcontribs) 2009-03-01T14:50:53Z

Hi all, so, I needed quick brush up on what Lie groups are, since I'm having a go at symplectic maps, got to the Lie group page, found it disappontingly unhelpful. To the point that I got into this discussion to complain. Not surprisingly I find this = is this useful? = page where beside a precise echo of my impression I also learn that "IF you dont [find it useful] why not go read something else?". OK thanks. will do. So much for a "comprehensive written compendium" (talk) 12:53, 23 October 2009 (UTC)anonymouscontributor

Incorrect statement?[edit]

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)

Lie Groups in the news![edit]

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. --C S (Talk) 15:38, 19 March 2007 (UTC)


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)

Trouth grup // In Vandalism[edit]

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)


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)

Motivation section is contradictory[edit]

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. --C S (Talk) 09:00, 22 March 2007 (UTC)

early history[edit]

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)

Dear friend, please help me and others similarly affected to figure out this sentence, which I cut from the section: "Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: idea of symmetry, as explicated by Galois through the algebraic notion of a group; geometric theory and explicit solution of differential equations of mechanics, worked out by Poisson and Jacobi; and new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others and culminated in Riemann's revolutionary vision of the subject of geometry." To me, this is totally incomprehensible, like an art critique rather than technical history. —The preceding unsigned comment was added by Smylei (talkcontribs) 22:58, 3 April 2007 (UTC).
It was probably written by a Slav, and requires some thes to be idiomatic English, but it seems to me straightforward - and obviously correct. Septentrionalis PMAnderson 23:28, 3 April 2007 (UTC)

The concept of Lie group[edit]

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)

Dumbing it down?[edit]

Section 8, "The Lie algebra associated to a Lie group" contains the following, in its third bullet point:

"If G is a closed subgroup of GLn(R) then the Lie algebra of G can be thought of informally as the matrices m of Mn(R) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε2 = 0 (of course no such real number ε exists...). For example, the orthogonal group On(R) consists of matrices A with AAT = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)T = 1, which is equivalent to m + mT = 0 because ε2 = 0."

I'm no expert, but it seems to me like this is just a dumbing down of (a special case of) the concept of a limit. Why do we need to say "ε2 = 0"? Shouldn't this be phrased in terms of limits? R.e.b.--you wrote it last year, am I totally off base? --Tcamps42 02:56, 3 April 2007 (UTC)

This must be the first time that anyone has told me that my edits are insufficiently technical. The definition using infinitesimals is actually (more or less) the standard definition of a Lie algebra of an algebraic group. The advantage over the definition in terms of limits is that it works for groups defined over fields other than the reals. Also it's easier to work with; try calculating Lie aglebras using both definitions. R.e.b. 02:17, 4 April 2007 (UTC)

Link spam[edit]

I have removed the links that follow, for a few reasons: this is not article on Sigurdur Helgason, Sophus Lie or Wilhelm Killing; links are somewhat irrelevant; they clutter a reasonably balanced list of references. Arcfrk 17:59, 11 June 2007 (UTC)

  • Sigurdur Helgason , MIT Course on Introduction to Lie Groups[[2]]
  • Sigurdur Helgason , Sophus Lie, the mathematician [[3]]
  • Sigurdur Helgason , Sophus Lie and the Role of Lie Groups in Mathematics [[4]]
  • Sigurdur Helgason ,A Centennial: Wilhelm Killing and the Exceptional Groups[[5]]

p-adic lie groups[edit]

so what's that? anybody up to writing a quick explanation?

Missing definition of left-invariance[edit]

I have not found any definition of left-invariance in the whole contemporary Wikipedia. Amusingly, there is one in the nostalgia-wikipedia from 2001: Pavel Bazant 21:54, 21 September 2007 (UTC)

Interesting. Left-invariant vector fields are mentioned in multiple places, but the definition only appears at Maurer-Cartan form#Construction of the Maurer-Cartan form. Arcfrk 22:08, 21 September 2007 (UTC)
I've added the definition of left invariant vector fields and left invariant vector fields. I'm inexperienced at editing wikipedia articles so feel free to edit or delete my changes. (talk) 20:34, 17 April 2008 (UTC) GJS


LiE - computer algebra package for Lie group computations —Preceding unsigned comment added by Spoon! (talkcontribs) 01:09, 29 November 2007 (UTC)

Multiple forks[edit]

This article has grown too long and somewhat unwieldy. For example, the section "Examples" has expanded so much that it contains most of the list of Lie groups. On the other hand, certain important topics that deserve an article of their own, such as p-adic Lie groups or Lie correspondence cannot be easily found and/or treated systematically due to the sheer scope and the volume of material. As a first step, I propose to move out most of material dealing with the Lie algebra of a Lie group into a separate article (and use the summary style). Further, structure theory and types of Lie groups can probably stand on their own. I am also in favour of either trimming the examples considerably, or moving them into a separate article. Opinions? Arcfrk (talk) 02:32, 6 February 2008 (UTC)

I've long thought about doing this, but have never gotten around to it. Comments:
  • I agree the major material to fork off is the Lie algebra stuff. I would rather title it the Lie group-Lie algebra correspondence though; the idea being that we can discuss both the Lie algebra of a Lie group as well as the simply connected Lie group associated to every finite-dimensional Lie algebra.
  • We already have an article on the exponential map. It would probably be best to merge most of that content there and keep a summary style here.
  • We could also fork off an article on Lie group homomorphisms. I started a draft for this some time ago, but never finished it.
  • The examples should be merged with Table of Lie groups and then maybe renamed to Examples of Lie groups.
-- Fropuff (talk) 04:48, 6 February 2008 (UTC)
I really like one or two paragraph summaries with {{main}} used to indicate the more detailed article. I'd like to be able to read through the Lie groups article in a few minutes and have a basic idea of what they are, why they were studied, and what other articles I'd need to read to learn more. I think such a standard is hard to meet, but the shorter sections linking out to more detailed articles should help for sure.
I've been doing orphan and dead-end patrol lately, and I've noticed that sometimes these stub sections were written first, and then someone wrote the full article, but never linked it in. I don't remember any Lie group examples, but I did link in Darboux derivative as an orphan. Perhaps someone here has a better place to link it than "See also". JackSchmidt (talk) 04:51, 15 February 2008 (UTC)

Simple merge from analytic subgroup[edit]

The orphaned article analytic subgroup (and its dictionary definition friend analytic group) should be "merged" into this article. By that I mean that the terminology "analytic" used as a synonym for "connected" should be mentioned in this article, then the analytic articles turned into redirects. As far as I could tell, the terminology is unused on wikipedia, and there is no new content in the articles. This should not make the current article more unwieldy, as the merger should add a sentence or less to this article. JackSchmidt (talk) 04:44, 15 February 2008 (UTC)

Lie Brackets[edit]

I have to sympathize with the thread above about accessibility. I am trying to teach myself this material and am getting hung up on the bracket. This seems to be very important as it is emphasized on this and other pages, and in books, but I don't understand how it is useful.

I think I mostly understand the definition. I am a little confused if in the definition

[A,B] = AB-BA

A and B are matrices (coming from the matrix logarithm that mapped them into the Lie algebra—the tangent space) or are vectors. Either way, it is unclear if in AB-BA that's the elementwise (Hadamard) product or the matrix product.

Even assuming I understand the definition, I don't understand conceptually what [A,B] means. I think it is a measure of how much the group fails to commute. That's all well and good but (a) why is that so important? and (b) what's an example of a problem the solution of which is made easier by using Lie brackets? —Ben FrantzDale (talk) 05:05, 5 November 2009 (UTC)

If you ask your question at Wikipedia:Reference desk/Mathematics, you will probably get a response from a variety of people (I do not think that many people watch this page but I may be wrong). --PST 05:51, 5 November 2009 (UTC)

Error in display equation?[edit]

Is there an error in the third display equation in the section "The exponential map"? That is, shouldn't it read exp(v) = c(v) (rather than c(1) on the right-hand side)? Thanks, Fcy (talk) 05:16, 5 July 2010 (UTC)


The first statement in the section Properties is misleading although true. The fact that the diffeomorphism group acts transitively holds for every connected real manifold and is by no means specific for Lie groups. (talk) 16:56, 2 October 2012 (UTC)

I don't think it's misleading... the section makes no claim that the property is specific to Lie groups. Lots of properties are not specific to things in which they occur. Under "properties of squares" you could put "has four right angles", but that is not misleading. Rschwieb (talk) 17:23, 2 October 2012 (UTC)

Needs a rewrite and review.[edit]

I am tempted to rewrite major sections of this article. Not because I know much about it, but because as it is written it is such a mess. It claims that Lie groups must be real and yet it also mentions Complex Lie Groups. Come On! You can't have it both ways! It says only 0,1 and 3-spheres are Lie groups then says ALL groups are Lie Groups in a 'trivial' way. The phrase:" using an alternate metric completion of Q" is totally opaque to me (Q is not explained, defined or ever used again). What a mess. I also want to go on record as complaining that the structure of the article is inconsistent with wikipedia's normal stucture. Examples should come AFTER history and general discussion. This article appears to have been cobbled together from various patches and is poorly connected. The examples here are a bit puzzling (to me - a novice). The wikipedia article on General Covariance states:"the Lie group GL4(R) is a fundamental "external" symmetry of the world." If so, it gets my strong vote for inclusion on a short list of examples. Couldn't it replace GL(2,R)? It defies my present understanding that complex Lie groups aren't also physically fundamental, but that is a different subject. Quantum mechanics deals only with Real groups??? wrong.Abitslow (talk) 17:51, 10 May 2014 (UTC)

Under related notions it says, "Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds." This is not to confused as a claim that "Lie groups must be real". Note how a complex number can be expressed as "a+bi", where a and b are real numbers, which you can imagine as being "underlying". As for Q, it refers to the rational number domain. Also, beginning with history is usually not how math articles are done. See for example Addition, Subtraction, Multiplication, or Division (mathematics). Finally, the article on general covariance says, "A more modern interpretation of the physical content of the original principle of general covariance is that the Lie group GL4(R) is a fundamental 'external' symmetry of the world." In other words, some people believe it's fundamental. That's not a good case for it being a "better example" of a Lie group than GL(2,R) is.siNkarma86—Expert Sectioneer of Wikipedia
86 = 19+9+14 + karma = 19+9+14 + talk
13:05, 12 May 2014 (UTC)
When ignoring all fields except R and C, it is in one sense true that Lie groups are real because then they are all real analytic manifolds. But also, such a Lie group may be regarded as complex according to whether its Lie algebra is complex. If so, then (I believe) it can be endowed with a complex analytic manifold structure. YohanN7 (talk) 18:48, 11 October 2014 (UTC)

Smooth (differentiable) or analytic?[edit]

Most sources and this article defines Lie groups as smooth but not necessarily analytic manifolds. Rossmann, in Lie groups: An introduction through linear groups, defines them with the analytic structure. He argues that it is the natural way to define them, and it is clear that Lie groups allow an analytic structure. One thing is clear to me; the theory of flows of an analytic vector field becomes much more elegant in this setting. See page 152 and on in the referenced book. The reason that flows can be called exp (in this context) is extremely transparent. The power series of the usual exponential occurs as something that actually characterizes flows on Lie groups completely. I don't think this is possible on smooth manifolds with "merely" smooth vector fields and functions. YohanN7 (talk) 21:50, 11 October 2014 (UTC)

The formula, due to Sophus Lie himself (1888), I'm thinking about is

\varphi(\mathrm{exp}(\tau X)p) = \sum_{k = 0}^{\infty}\frac{\tau^k}{k!}X^k\varphi(p).

Here X is an analytic vector field, φ an analytic function, and exp is the flow of X, p is a point on the manifold and τ is the "time" of the flow. Its proof depends on φ having a Taylor series. In a smooth manifold setting, the exponential mapping is simply defined to be the time one flow of a (left invariant) smooth vector field without the strong motivational point of the power series for its name. YohanN7 (talk) 06:09, 12 October 2014 (UTC)

On the left hand side,

\mathrm{exp}(\tau X)

is, for each τR, a bi-analytic bijection (the analytic counterpart of diffeomorphism) from the analytic manifold (the Lie group) to itself. Thus

\mathrm{exp}(\tau X)p

is a new point on the manifold. When τ varies (p fixed), an integral curve of X is obtained. With p = the identity,

\mathrm{exp}(\tau X)

can be considered as a one-parameter subgroup as τ varies over all R (provided X is in the Lie algebra). All one-parameter subgroups of the Lie group are expressible as such for some X in the Lie algebra (the left-invariant vector fields ≃ tangent space at the identity). On the right hand side, X is to be thought of as a first order differential operator operating on φ. Every X in the Lie algebra can canonically be associated with such a differential operator on the whole Lie group. This extends the concept of a Taylor series globally and in a coordinate-independent way.

I think this sort of stuff should go into the present (convoluted) section Lie group#The exponential map (with a renaming of the section). I also think the Baker-Campbell-Hausdorff formula generalizes almost verbatim too in the analytic manifold setting. That is, begin with analytic Lie groups, consider vector fields, and the whole thing falls out naturally. YohanN7 (talk) 07:51, 12 October 2014 (UTC)

Here is my position:
(1) We should have a section regarding Hilbert's fifth problem-type fact, like any smooth group or group homomorphism is analytic (category of smooth group is the same as the category of analytic group). This is very important if it is not treated in depth in elementary textbooks.
(2) I'm not sure if we need to have too much stuff on exponential map; after all, the separate article is the best place for in-depth discussion. Perhaps more discussion exponential coordinate may make sense. In my opinion, there is too much emphasis on examples and structure theory and not enough abstract theory.
-- Taku (talk) 21:35, 29 November 2014 (UTC)
(2) Agreed, exponential coordinates should definitely be here, as well as the associated topology, the group topology. In relation to this a discussion on the equivalence of this topology with the subspace topology for Lie subgroups, relegating the bulk to Closed subgroup theorem. Also agreed that the exponential map need not be overly present here. But Lie's formula of above must get in somewhere. It's pretty and explains the name of exp in all generality without any a priory reference to the usual matrix exponential. But, it works only under the assumption of analyticity.
Actually, Lie's formula works for all analytic manifolds, not only Lie groups. YohanN7 (talk) 11:22, 1 December 2014 (UTC)
Yes, there should be a section on "topology and differentiable structure". As for Lie's formula, the obvious thing to do seems to create Lie's formula. It's also known as Taylor series, but it probably makes sense to have a separate article (and have obvious links between the two). As you said, the formula itself has nothing to do with Lie groups. -- Taku (talk) 22:19, 1 December 2014 (UTC)