Talk:Lie group–Lie algebra correspondence

Inline LaTeX

The article would benefit from HTML instead of LaTeX for inline equations, Inline LaTeX looks pretty awful in PNG rendering. (Looks like MathJax isn't even available now? At any rate, MathJax is too slow.) YohanN7 (talk) 14:38, 19 July 2014 (UTC)

screenshot

Actually, png looks fine to me: this is what I'm seeing. Is that really bad on your computer screen? Since I cannot see what others see, I think I will start a thread at the project page regarding this issue. -- Taku (talk) 16:48, 19 July 2014 (UTC)

If you mean by awful that the LaTeX math renderings are not the same typographic size/style as the prose, yeah, that is a known issue. Some people cannot stand it and render the inline math with HTML. Personally, I dislike inline HTML in the presence of displayed equations because the math symbols inline and the math symbols in the displayed math often look completely different and it can be a puzzle figuring out the correspondence between the two. I don't think there is a definite stylistic rule about this. Agreed, MathJax is still too slow. --Mark viking (talk) 18:51, 20 July 2014 (UTC)

I prefer displayed LaTeX too. But having every instance of Lie(G) displayed would be ridiculous. There used to be something in the style-guides about this, but it is gone now. YohanN7 (talk) 15:19, 22 July 2014 (UTC)

Another, more mathematical, point;

• If the image of f is closed, then $\operatorname {Lie} (\operatorname {im} (f))=\operatorname {im} (df)$

should really be

• If $G$ has countably many connected components, then $\operatorname {Lie} (\operatorname {im} (f))=\operatorname {im} (df)$.

There is an example in Rossmann, Lie Groups, An introduction through linear groups, showing the necessity of the condition. ("Closed" is not needed given we have a Lie group homomorphism.) YohanN7 (talk) 15:19, 22 July 2014 (UTC)

About that. An implicit assumption here is that every Lie group has at most countably many components. (If I remember correctly, this follows from the fact that a topological (second-countable) manifold has at most countably many components.) I think this is a very common assumption nowadays. The "closed image" on the other hand ensures that the image is in fact a Lie group; it's possible maybe it can be weakened. (The reference for the statement is Bourbaki.) -- Taku (talk) 18:35, 22 July 2014 (UTC)

Yes, you are right about the implicit assumption. It is standard, or very close to standard, to assume second countability. (The point could still be made in the article because it is somewhat interesting.) But surely, given a Lie group homomorphism, both ends are Lie groups. Besides, you can have a Lie group homomorphism where the range is an immersed manifold, hence not closed. See the irrational winding of a torus. There is a homomorphism between the Lie algebra, viewed as an additive group, and a subgroup of the torus. YohanN7 (talk) 18:55, 22 July 2014 (UTC)

That is, the closedness condition is not only unnecessary, it is wrong. For another, related example, the inclusion map of the irrational line on the torus is a Lie group homomorphism onto a non-closed subset endowed with the group topology. YohanN7 (talk) 21:07, 22 July 2014 (UTC)

You may want to phrase it like this: If the image is a subgroup of a larger group, H ⊂ G, with the subspace topology, then it is, according to the closed subgroup theorem, necessarily closed in G. YohanN7 (talk) 21:24, 22 July 2014 (UTC)

Maybe I'm missing something but why is the image a manifold at all? For example, how do you know the image is of pure dimension? (Also, it is not wrong; if the image is not closed, the statement is vacuous.) Maybe you simply transport the structure of G/kernel to the image? That seems to work but we need a reference for that. (I'll look up in the books.) -- Taku (talk) 22:58, 22 July 2014 (UTC)

The way you have worded things,

If

$f:G\to H$

is a Lie group homomorphism...,

implies that both G and H are Lie groups (or else f wouldn't be a Lie group homomorphism). You can then infer (using the closed subgroup theorem) that the image is closed if it happens to be a subgroup with the subspace topology. YohanN7 (talk) 02:03, 23 July 2014 (UTC)

B t w, I just noticed the screen shot. I agree, it actually looks decent. Here, the LaTeX is slightly too small. But on a regular computer screen it is way too big, more than double the size it ought to be (with my Google settings). YohanN7 (talk) 02:10, 23 July 2014 (UTC)

But don't you need a "closed" to have a manifold structure at all? Every subgroup of a Lie group can be made a "topological" subgroup. But it's not closed if such a subgroup is not Lie group to begin with. The image is always a topological subgroup but there need some condition that makes it into a Lie group. (I agree we probably should explain this in the article.) -- Taku (talk) 13:51, 23 July 2014 (UTC)

No, there are immersed Lie subgroups that are Lie groups (hence analytic manifolds with analytic group operations) when endowed with the appropriate topology (the group topology). Therefore, the "closed" hypothesis, while not strictly speaking incorrect, is misleading. The range is an embedded Lie subgroup if and only if it is closed. And again, you have ensured Lie groups at both ends when speaking of Lie group homomorphisms in the first place, whether embedded or immersed or standalone. YohanN7 (talk) 14:36, 23 July 2014 (UTC)

I think this is a matter of convention or terminology. I think you're implicitly assuming the first isomorphism theorem; the image is isomorphic as Lie group to G/kernel. This depends on which topology you're using on the image: if you transport the structure, then the first iso is tautologically true; otherwise, it's true only if the image is closed. (I will check what references besides Bourbaki are doing.) -- Taku (talk) 16:55, 23 July 2014 (UTC)

I'm just trying to help here, and you can take my advice if you want to. I don't want to get into an argument, so this is probably my last post on this. (I bet you feel the same way . Best is for someone else to comment now.) Here's what I've got to say about the subject:

There are no different conventions as to what constitutes a Lie group homomorphism. The domain and range are Lie groups period, you don't need to transport structure. You seem to miss that subsets can be endowed with different topologies. The subspace topology is only one of them. A Lie subgroup is a subgroup of a Lie group such that there exists a topology in which it is a Lie group. It does not have to be the subspace topology. In case it is the subspace topology, it would be closed as a conclusion (of the closed subgroup theorem), not by hypothesis. I have given two examples in this thread of Lie group homomormhisms where the range isn't a closed subgroup.

The article, as of now, suggests that you can have Lie group homomorphisms where the range is not a Lie group, and it suggests that the first isomorphism theorem sometimes holds and sometimes doesn't. (It holds for abstract groups = all groups.) The first isomorphism theorem is a purely algebraic theorem that has noting to do with topology or smooth structures. If you skip all text except $\operatorname {Lie} (\operatorname {im} (f))=\operatorname {im} (df)$, then you'll get it right. Also, the remark about countable many components is assumed is confusing. Either say nothing, or say that countably many components follow from second countability, which is usually assumed in the definition of a topological space. In case you do say this, you should also mention the then necessary hypothesis that the domain has countable many components when second countability is not assumed in the definition of a topological space. YohanN7 (talk) 18:16, 23 July 2014 (UTC)

If I gave you an impression that I'm annoyed by having an argument with you, that was completely unintended; it is just my personality to sound terse. Since differential geometry isn't my strong area, I'm happy to hear about any errors I might have missed.

Now on the point of contentions, your above post contains several inconsistencies with the literature as I understand. In the definition of "[[Lie group homomorphism, no source that I know of requires a range to be a Lie group instead of a codomain. Maybe that's not a good definition, but we just can't, here in Wikipedia, adopt a definition that is inconsistent with th literature. As said, this is just a matter of convention. The current article also does not exclude any Lie group that is immersed into another Lie group; in such a case, the image is simply not a Lie group and there is nothing with that. As for the first isomorphism, no, it really depends on a category. The first isomorphism theorem of Lie group is not the same thing as the first isomorphism theorem of abstract groups, since the notion of isomorphism depends on a category. This article is basically shamelessly a verbatim from Bourbaki. If there is some other source that does things differently, then I'm happy to hear about it. -- Taku (talk) 23:02, 23 July 2014 (UTC)

It all seems to be a matter of definition. For my version of it, see Introduction to Smooth Manifolds by John M. Lee and Lie Groups - An introduction through linear groups by Wulf Rossmann. Rossmann's book has plenty of small errors, but received extremely good reviews by heavyweights in the area: A. W. Knapp review. Some results regarding the Lie correspondence actually seem to be entirely new. He also states and proves the contested statement without the closedness hypothesis (but with the countable number of components hypothesis). In their terminology, immersed Lie subgroups are Lie groups when endowed with the correct topology — and this is an important ingredient in their versions of the Lie group Lie algebra correspondence. From Lee:

"Suppose G is a Lie group with Lie algebra g. If h is any Lie subalgebra of g, then there is a unique connected Lie subgroup of G whose Lie algebra is h."

Here, "Lie subgroup" means that there exists a topology on it such that it is a Lie group in its own. (Rossmann calls this topology the group topology.) I hope this explains what I have meant. YohanN7 (talk) 00:05, 24 July 2014 (UTC)

First of all, thank you for referring me to a textbook by Rossmann. I just got one from the library and it's actually a very beautiful book. His development is certainly different from the standard one that is in the context of differential geometry. And I did find that Lie(im(f)) = im(df), without the "closed" assumption. The only problem I can think is that he works with linear groups, not Lie groups (they are not the same thing). In particular, whether the image is a manifold or not isn't an issue for him the count ability assumption is important (in order to appeal to Baire category theorem.) I will keep looking up this matter in some other matter. (And of course we should include a detailed careful explanation of this; if we're confused, imagine the horror on the readers.) -- Taku (talk) 13:13, 24 July 2014 (UTC)

The problem with the specialization to linear groups may or may not be an issue. From what I have read, all results provable from linear groups hold true in the general case, but not vice versa. (Passing to a universal cover or a quotient aren't closed operations when considering linear groups.) Also, other authors define linear Lie groups as closed subsets of GL(n, C) or GL(n, R), see Brian C. Hall, Lie groups, Lie algebras and Representations — an Elementary Introduction. Thus in Hall's world, the irrational line on the Torus is not a linear Lie group, but it is a Lie group (but not in the subspace topology). In Rossmann's world, it is both a linear Lie group and a Lie group. He doesn't care about what it may be a subset of, and uses exponential coordinates to define a suitable topology making it a Lie group. Caveate: Rossmann calls what we call embedded submanifolds for simply submanifolds. Our immersed submanifolds he calls embedded submanifolds. Confusing? Yes YohanN7 (talk) 14:06, 24 July 2014 (UTC)

I just looked at Lee's book and he doesn't discuss Lie groups at all, so it cannot be used as a reference.

In any rate, I think we are in agreement that this is a matter of definition. When I say a submanifold I assume it is a subspace and so it is an embedded submanifold. By an immersed submanifold, I mean a smooth inclusion whose differential is injective everywhere and I don't put any topology on the image of the inclusion, which is as a set the same thing as the domain. I think this is fairly standard nowadays (not sure Wikipedia, though). I'm not sure about the distinction between Lie groups and linear Lie groups being of not fundamental. For the purpose of the representations, it's probably enough to consider only linear Lie groups (after all, you work with subgroups of the general linear groups, linear groups.) But for the general article such as this one I don't think we have liberty to deviate from the standard theory. After all, we are not here to teach a course but to record the standard theory. The definition like insisting the image of a Lie group homomorphism being a Lie group doesn't appear to be standard. Of course, the cavet is I'm not an expert and I might very well have missed something. -- Taku (talk) 14:25, 24 July 2014 (UTC)

We seem to be on the same page now, except for when it comes to Lee's book. It does discuss Lie groups in two big chapters, enough to develop the Lie algebra — Lie group correspondence, and he does so in an uncontroversial way. At least, this is the case with my copy. I know there is a new edition around, and he may have dumped Lie groups in favor of something more basic. It is, generally speaking, perfectly suitable as a Wikipedia reference since it is popular (= easily accessed) and fairly basic.

I think we can leave this discussion for a while, and perhaps return to it later. The topic might serve as a basis for a separate section in the article in the future? YohanN7 (talk) 19:57, 24 July 2014 (UTC)

I think it was my simple mistake: I must have got a wrong book; I looked for a book with "Lee" and "manifolds", and didn't realize there are two books satisfying this condition. Since the library is closed over the weekend, I will get the book Monday. Thank you again for the input. I'm planning to add more elaboration on a countability assumption. -- Taku (talk) 11:25, 26 July 2014 (UTC)

Avtually, there are three of them, Topological, Smooth, and Riemannian manifolds, respecively. YohanN7 (talk) 13:33, 27 July 2014 (UTC)

("3" books made me smile :) Anyway, I've just finished skimming Lee's "smooth" manifold book and, if I didn't miss anything, this article and his book are consistent; he doesn't require, for example, the image of a Lie group homomorphism, not just the codomain, is a Lie group. -- Taku (talk) 18:35, 29 July 2014 (UTC)

I realize now the I have been uncareful in distinguishing between image and codomain in this discussion. We are on the same page. The article has a lot of substance. Nice work! YohanN7 (talk) 19:22, 29 July 2014 (UTC)

Displays badly in MathML

The display of the article in MathML is strange. It is as if double (or even triple) line spacing is employed. (In PNG it displays correctly (but not beautifully).) YohanN7 (talk) 10:48, 17 June 2015 (UTC)

MathJax displays correctly. YohanN7 (talk) 13:43, 17 June 2015 (UTC)

Mistake?

Ref. 19: "It's surjective because $\operatorname {exp} ({\mathfrak {g}})^{n}=\operatorname {exp} ({\mathfrak {g}})$ as ${\mathfrak {g}}$ is abelian" — Really? or rather, $\operatorname {exp} ({\mathfrak {g}})^{n}=\operatorname {exp} (n{\mathfrak {g}})$? Boris Tsirelson (talk) 17:44, 7 December 2016 (UTC)