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From Problem simplication.... 1.46.131.75 (talk) 11:56, 10 April 2022 (UTC)[reply]

In #Additional examples, The projective group is confusing. There are distinct types of projective groups, general (full) and special. ${\displaystyle \mathrm {PSL} (3,R)\cong \mathrm {SL} (3,\mathbb {R} )}$ preserves orientation while ${\displaystyle \mathrm {GSL} (3,R)}$ does not. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:14, 1 June 2022 (UTC)[reply]

As written it was pretty useless so i removed it. However it points to a deficiency of the current version of the article, that there the Lie groups ocurring as symmetry groups of differential equations are not mentioned outside of the history section and (not very clearly) the section on representations. This is treated in some detail in the page Lie point symmetry but there should probably be a short paragraph on that here as well; for the time being i added a "see also". jraimbau (talk) 13:33, 1 June 2022 (UTC)[reply]

Yes, I believe that this article is too technical for readers interested in this subject, because it doesn't begin with the most obvious starting point for Lie groups: the circle group.

I do not believe there is ever any reason to introduce anyone to Lie groups without beginning with the circle group.

The first paragraph states that the subgroups of the groups of ${\displaystyle n\times n}$ invertible matrices over ${\displaystyle \mathbb {R} }$ or ⁠${\displaystyle \mathbb {C} }$⁠ are now called classical groups.

Is this not quite misleading? By Ado's theorem, any Lie algebra has a faithful representation, and thus can be realized as a matrix Lie algebra. Going, via the exponential, from the Lie Algebra to the corresponding group, it should thus follow that any Lie group is locally isomorphic to a subgroup of the General Linear group. In any case, there are plenty of groups that are not classical, which are so represented.

I can't find this term anywhere, neither online or in any of the references. There's an in-line reference to Kobayashi and Oshima, but that book is in Japanese. Is this a Japanese term that has been translated? Sheddow (talk) 14:26, 27 July 2025 (UTC)[reply]

I've never seen it either, your guess that it is a translation makes a lot of sense to me given the definition. I think this paragraph could be shortened drastically without much loss to the article: it seems to me that it's only paraphrasing an immediate corollary to Ado's theorem and the Lie correspondence. jraimbau (talk) 16:54, 27 July 2025 (UTC)[reply]

It’s actually not that simple. Here, the point of the section (topological definition) is that the notion “Lie group” is actually a topological one, which is fairly nontrivial and doesn’t really follow only from algebraic considerations (roughly you need to use some repression theory). Taku (talk) 01:16, 26 August 2025 (UTC)[reply]

Sorry for late reply. The term was introduced to the article by me. The problem I had was the term “linear Lie group” can mean either a closed subgroup of a GL_n(R) or a (not-necessarily-closed) Lie subgroup of a GL_n(R). I added “immensely” to indicate the second meaning. I don’t know any better term; but simply using the term “linear Lie group” seems problematic. —- Taku (talk) 01:01, 26 August 2025 (UTC)[reply]

I'm fairly sure that any Lie group that has a smooth embedding (not necessarily with closed image) into a linear group also has an embedding with closed image and that people mostly don't care about that---unless you can prove the contrary with precise citations there is no reason for this confusing passage to remain in the article.jraimbau (talk) 13:52, 31 August 2025 (UTC)[reply]

No, that’s not my understanding; yes, you can take the closure but that simply gives a different Lie group. It’s somehow well-known that the term “linear Lie group” is ambiguous so, mathematically, we need to be specific about the definition. I suppose one option is “linear Lie group in the sense of an immersed submanifold”, though a bit long. Taku (talk) 10:15, 14 September 2025 (UTC)[reply]

"yes, you can take the closure but that simply gives a different Lie group" : obviously it's not the same embedding.

"It’s somehow well-known that the term “linear Lie group” is ambiguous" : i don't trust this statement at all, give me a proper source for this or stay silent.

All that you said on this topic smells of WP:OR, so unless you back it up quickly i'm going to purge the article. jraimbau (talk) 17:08, 14 September 2025 (UTC)[reply]

Not really an original research. For the need to distinguish between immersed subgroups and closed subgroups, see for example Definition 7.3.3. of [1]. Some people especially physicists are sloppy with math terms, which doesn't mean we can be sloppy too. I get you don't like the current terminology but then what terminology do you propose to use? Taku (talk) 18:35, 14 September 2025 (UTC)[reply]

We're not disputing the need to distinguish between immersed and closed Lie subgroups, only the need to distinguish them for linear Lie groups. As far as I can tell you're not using this distinction in your proof - it would work just as well with closed linear Lie groups. It would be nice to see a source for this topological definition in general, not just the matter of immersed linear Lie groups. Sheddow (talk) 00:00, 15 September 2025 (UTC)[reply]

Ah, thank you. Now, I think I understand the issue. According to [2], a connected immersed subgroup of a general linear group is indeed isomorphic to a closed one, but not generally so. If I remember right, there are some subtle topology matters so that you have to use not-necessarily-closed subgroups of a general Lie group in the discussion. At least, the reference used there didn’t use closed ones so if we were to use only them, that would be an original research. (The definition looks contrived and there is a reason.) Of course, we can switch to another ref that gives a simpler definition if there is any especially English one.Taku (talk) 05:22, 15 September 2025 (UTC)[reply]

Thanks for the reference doing the exercise proving what i was claiming. There are infinite countable linear groups which are not discretely linear, that's true. OTOH Lie theory typically deals with groups with finitely many connected components for which this is not an issue.

I still think the paragraph is unnecessarily complicated and very much not standard: the correct "topological definition" of a Lie group is just that it is a topological group locally homeomorphic to a euclidean space, by the solution to Hilbert's fifth problem. This is much stronger than what is described in the current state of the article and there are plenty of references for that (e.g. Tao's book).

I propose that : 1. the section on "topological definition of Lie groups" be removed ; 2. a mention of "topological description of Lie groups" be added after the mention of Hilbert's problem in the previous paragraph (possibly with a citation to Tao where he uses this exact terminology) ; 3. possibly, a very short mention of the non-standard definition of Kobayashi--Oshima (for example in a footnote). jraimbau (talk) 06:44, 15 September 2025 (UTC)[reply]

I agree the subtle point doesn't arise in usual applications, but since this article deals with Lie groups in general, we kind of have to not discard pathological ones.

As for the proposal, I think that misses the point: by the standard definition, a Lie group already locally looks like a Euclidean space. The point of the topological definition was that it is possible to approach Lie theory completely independent of differential topology, which is not obvious and is worth mentioned in my opinion (how that's done can be debated). Also, I don't think there is a reference that *defines* a Lie group via the fifth problem. On the other hand, there is indeed a reference that does define a Lie group as a topological group that is locally linear. Having said that, I do concede the topological definition is contrived and so if sufficiently many editors think it's just not worth included, I don't have a problem with the removal. -- Taku (talk) 10:29, 15 September 2025 (UTC)[reply]

There is no differential topology involved in defining Lie groups as topological groups on topological manifolds. If you are defining Lie groups as topological groups that are locally isomorphic to a linear Lie group, as in your definition, then you are pretty much requiring them to be topological manifolds anyways (if a topological group is locally homeomorphic to Euclidean space around the identity, then it's a topological manifold).

It's not enough to vaguely remember there is some subtle topology issue requiring us to use non-closed linear Lie groups. You actually have to give a source (PS: the problem isn't just that the given source is in Japanese, but I couldn't even find it online at all).

I'm in favor of removing or rewriting this section as suggested by Jean. It seems too obscure and poorly sourced. If the goal is to show that Lie groups can be defined without reference to smooth manifolds, then we should just reference Hilbert's fifth problem instead. Sheddow (talk) 14:06, 15 September 2025 (UTC)[reply]

Maybe I should have added not involving manifolds as well; for laypeople, the difference between smooth manifolds and topological manifolds isn't big. Of course, the topological definition is *equivalent* to the standard definition but I thought and still think it is somehow remarkable that no geometry like topological manifolds or smooth manifolds need to be involved in the theory of Lie groups. As for "vaguely remember there is some subtle..." by which I was thinking of the situations discussed in the pages I linked above. In particular, according to them, it is *not* enough to consider closed subgroups of a general linear group in the topological definition. But again having made my point, I can see the subtlety involved can be confusing and so understand the desire to remove it. Oh and I should add that the reference, while not in English and not online, seems reliable and so I don't think there is a reference issue. Taku (talk) 06:09, 16 September 2025 (UTC)[reply]

"it is *not* enough to consider closed subgroups of a general linear group" : i don't think that's true at all, i.e. if a topological group is locally isomorphic to an immersed Lie subgroup of GLn then it is locally isomorphic to a closed subgroup of GLn (e.g. a closed embedding of the identity component of the immersed Lie group). jraimbau (talk) 07:59, 16 September 2025 (UTC)[reply]

Yes, you're right! Sorry for missing something obvious. I remembered Lie's third theorem required immersed subgroups but according to [3] as well as the early discussion, immersed subgroups are actually not needed. So, I now admit I was wrong; we don't need immersed subgroups to give a topological definition. (So, the definition in reference, while not wrong, could be simplified after all, contrary to what I thought.) Taku (talk) 05:43, 17 September 2025 (UTC)[reply]

How is locally isomorphic to a linear Lie group not geometry? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:49, 16 September 2025 (UTC)[reply]

For me, linear Lie groups are something more like objects in the representation theory or the invariant theory. Obviously topology isn't and doesn't contain geometry... Taku (talk) 05:46, 17 September 2025 (UTC)[reply]

Hi. So following the discussion so far, I have begun a draft of the new version of the section at Draft:Lie group (which tries to implement what Sheddow pointed out). As I understand, I don't think the topological definition follows from solving Hilbert's fifth problem, which is about whether "smooth" can be weakened or replaced. So, it is still my position that giving a topological definition in some form has a point. Needless to say, anyone is entitled and welcome to edit the draft. -- Taku (talk) 07:13, 19 September 2025 (UTC)[reply]

I think that apart from yourself no one has so far voiced support for keeping this information in the article so unless you can present a coherent objection i'm going to remove it from the article. jraimbau (talk) 15:08, 21 September 2025 (UTC)[reply]

??? I thought my objection was clear: the topological definition in the section and now in the draft doesn’t follow from the matter of Hilbert’s fifth problem (since the definition has to do with linear Lie groups). Also, as far as we can tell, there is no reference that defines a Lie group using a solution to the fifth problem. So, the definition is not redundant. But if anyone thinks any topological definition is just not worth mentioned, then of course I can understand. Taku (talk) 08:29, 22 September 2025 (UTC)[reply]

While your objection is clear it is far from coherent: as noted above the characterisation of Lie groups through the solution to Hilbert 5 trivially implies that the definition by "local isomorphism to a linear Lie group" works. And it is much more standard and useful as a characterisation of Lie groups than the one you claim is given in the book by Kobayashi--Oshima (which is not indexed in Zentrablatt nor in mathscinet) which does not seem to be used anywhere else. Thus it appears there is no reason in leaving this confusing paragraph in the article.jraimbau (talk) 11:23, 22 September 2025 (UTC)[reply]

Well, it probably depends on the formulation of the fifth problem and the meaning of “trivially”. For me, the fifth problem asks whether “smooth” in the definition of a Lie group can be weakened or not. While important, it itself doesn’t directly address whether a Lie group is the same thing as a locally linear group. We still need things like Ado’s theorem and Lie’s third theorem to show a Lie group is locally linear and the closed subgroup theorem for the converse. For me, these results are not trivial. That is, locally linear and locally Euclidean are not *trivially* equivalent. Finally about the reference, it is actually an expanded version of a text in a textbook series and while not English so unfamiliar to English speakers, there shouldn’t be any reliability doubt on the book (we even have a Wikipedia article on one of the authors Toshiyuki Kobayashi and according to it, there doesn't seem any question on the expertise.) Taku (talk) 12:47, 22 September 2025 (UTC)[reply]

I'm sure the book is fairly reliable, but it does not mean every piece of information it contains is important. The one you insist on certainly does not seem so to me and as such i don't think it should be included in the article. jraimbau (talk) 16:25, 22 September 2025 (UTC)[reply]

Yes, so if the editorial consensus is that this non-standard definition is not worth mentioned, I don't mean to insist it be kept. (I'm just saying it's not wrong and not redundant.) Taku (talk) 08:08, 23 September 2025 (UTC)[reply]

I haven't really followed everything here, but not every Lie group can be embedded in a linear Lie group (e.g., the metaplectic group). On the other hand, every Borel measurable group homomorphism of a Lie group into GL(n,R) is a Lie group homomorphism. Tito Omburo (talk) 20:43, 22 September 2025 (UTC)[reply]

The initial question was whether a Lie group with an immersion into a linear group admits a closed embedding into a linear group. The answer is yes for groups with finitely many components, and no in full generality (e.g. for certain for certain discrete groups such as the rationals, or the linear Baumslag--Solitar groups). jraimbau (talk) 05:04, 23 September 2025 (UTC)[reply]

As jraimbau said, there was some math confusion due to me not knowing a fact about a linear Lie group. I think that was cleared up and that meant we can actually simplify the topological definition currently given in the article to the one given in the draft Draft:Lie group. The current contention is whether even that simplified definition should be mentioned at all. I think while not too important, it may be interesting, while the other editors disagree (but I don't want to insist it be in the article.) Taku (talk) 08:13, 23 September 2025 (UTC)[reply]

So, the discussion has died down but I have just updated the section (topological definition), following the discussion above. The update implements the suggested simplification and so at least it should address the concern the definition was unnecessarily complicated. Maybe there is still a will to remove the section altogether; I don't know and I will leave the ultimate decision on that to other editors. -- Taku (talk) 12:26, 6 November 2025 (UTC)[reply]