Talk:Closed-subgroup theorem

A fact from Closed-subgroup theorem appeared on Wikipedia's Main Page in the Did you know column on 2 August 2014 (check views). The text of the entry was as follows:

Did you know... that John von Neumann's theorem that every closed real matrix group is a Lie group inspired Élie Cartan to prove a generalization, the closed subgroup theorem?

A record of the entry may be seen at Wikipedia:Recent additions/2014/August.

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Something wrong in the informal description

Condition (1) of the informal description's first paragraph states

"

log(e H) = H, H \in U

(1)"

How can this be right? H is a subgroup, so it has the wrong type to be a member of an open set of group elements. Also, I think the style of writing "proposition(v), condition(v)" as a shorthand for "proposition(v) for all v such that condition(v)" is a very bad idea. It should be spelled out in words, per WP:TECHNICAL: we should not be writing this in a style that puts obstacles in the way of reading for people who do not already understand this material well. —David Eppstein (talk) 06:58, 18 July 2014 (UTC)[reply]

I really agree about the style. Some of the ladies in my grandmother's bridge club were discussing just this point. "EEng", said Mrs. Abramowitz, "we got as far as embedded submanifolds always having the subspace topology, but after that we were really stumped. This should really be written more for people who don't already understand this kind of material well. Sheila, weren't you just saying that it would help if von Neumann's proof for the special case of groups of linear transformations was explained first? Five no-trump." EEng (talk) 11:06, 18 July 2014 (UTC)[reply]

I was following Rossmann's style. He hates wasting alphabetical resources almost as much as he hates parentheses and consistently uses

H \in h

(element of Lie algebra) and

h \in H

(element of the corresponding Lie group), sometimes within one single equation. I agree with the objections against "proposition(v), condition(v)" and Rossmann's style. But I don't think the proof should go before the informal description of what the problem is with general subgroups. YohanN7 (talk) 15:28, 18 July 2014 (UTC)[reply]

Please tell me you knew I was kidding. EEng (talk) 15:59, 18 July 2014 (UTC)[reply]

Your grandmothers friends are to be taken seriously. They probably knew both von Neumann and Cartan in their youth, or, at least, went to the same nightclubs as them. YohanN7 (talk) 17:54, 18 July 2014 (UTC)[reply]

Oh, the stories these gals tell! Apparently one of them was present at Abel's duel. And that Poincaré was a real partier. The time they accidentally invited Gauss and Lobachevsky to the same dinner -- whoa! The fur really flew then! See [1]. EEng (talk) 18:18, 18 July 2014 (UTC)[reply]

Notation for the exponential map

Ok, that seems to be cleared up. Here's another notational issue. Sometimes in this article we're using $e x$ and sometimes we're using $exp(x)$ for what I think are the same thing, the exponential map. I think it would be less confusing to use only one of these, and to add a comment linking to exponential map the first time we do. Since this is not the same thing as the real-number powering operation with base 2.71828..., my preference would be to use the $exp(x)$ notation, but (especially as my understanding of this material is still somewhat incomplete) I don't want to just do it unilaterally without getting some comments first. Anyone else have an opinion? —David Eppstein (talk) 05:10, 21 July 2014 (UTC)[reply]

Linking to the section exponential map#Lie theory is a good idea. That section uses

exp(x)

and it would be good to be consistent with that section. In practice I've seen both

e x

and

exp(x)

used to denote the exponential map. --Mark viking (talk) 06:21, 21 July 2014 (UTC)[reply]

I prefer

e X

because

exp(X)

is ugly. Only when talking about the function (without its argument) one should use

exp

imo. YohanN7 (talk) 14:06, 21 July 2014 (UTC)[reply]

To me e^x looks like a two-parameter operator rather than a one-argument function. But in the case of the exponential map what is the first argument (e) supposed to mean? Also typographically it leads to lots of stuff in small harder-to-read superscript fonts. —David Eppstein (talk) 16:14, 21 July 2014 (UTC)[reply]

I see little point in changing from one accepted notation to another. If you need arguments in favor of one or the other, then apart from visual appeal, the

e X

is much more practical in actual calculations (think about it and you'll agree). Moreover, all functions I can think of

(sin, cos, log,

etc) that are generalized to have a wider domain (the

exp(X)

=

e X

comes from

m n

for natural numbers) keep their names and notation. The

exp

is simply an alternative, probably originally invented because of poor typesetting capabilities. Sure, if you insist, and if you are really unlucky, you can misinterpret

e X

(once in you lifetime) as expressing a function taking two arguments.

Besides, the article is now internally inconsistent. YohanN7 (talk) 14:01, 22 July 2014 (UTC)[reply]

It was already internally consistent, only the boundaries of the inconsistency have shifted. Basically, sin, cos, log etc already look like functions to me, as does exp. x^y does not. Per WP:TECHNICAL we should be avoiding things that confuse novice readers of this subject. Experts will not be confused by either notation. This is one of the things that was confusing me. That's why I brought it up. —David Eppstein (talk) 17:45, 22 July 2014 (UTC)[reply]

No, the article wasn't internally inconsistent before. Part of the

e X

notation is that when you talk about the function, then it is

exp

. There is no function

e

(or

exp(X)

for that matter). I have a feeling (but can't back it up) that it is not the novices that will be confused. I believe the

e X

-notation is favored by the younger generation. YohanN7 (talk) 18:10, 22 July 2014 (UTC)[reply]

To paraphrase, "Part of the notation is you use a completely different notation in certain circumstances, to mean the same thing"? That's just confusing. —David Eppstein (talk) 18:31, 22 July 2014 (UTC)[reply]

You need only to be able to distinguish between a function and the value of a function evaluated at a point, something non-mathematicians usually don't care much about. The situation parallels exactly the situation for the exponential mapping for real numbers. The functions name is "the exponential" or "exp" for short. Values at points are commonly denoted

e x

than

exp(x)

, especially in newer literature. YohanN7 (talk) 02:21, 23 July 2014 (UTC)[reply]

The notation is natural, well-established and used in the literature, and it is undoubtedly more practical in calculations. But I'm not going to insist. Arguing about notational issues is a waste of time. I objected because it is not standard to change notation in an article, just as little as it is standard to change from British spelling to US spelling. Then, if you do change, please do it everywhere, not only in places. YohanN7 (talk) 19:02, 22 July 2014 (UTC)[reply]

Well, it is not generally a good idea to change notation without a discussion. That's why we're having this discussion here. At the time when I started making this change to the actual article it seemed like the discussion had found fairly strong support for it but since then you've been arguing vociferously in favor of (1) being different from the notation in our exponential map article, and (2) maintaining two parallel notations for the name of the function and the values of the function. Nobody but me has been arguing in the other direction, and while I do feel extremely confident in my assessment that your choice of notation is confusing to novices, I don't feel at all confident in my understanding of what Lie theorists think should the notation should be. And I've been going very slowly in making changes to the article because I only want to change the parts I'm confident I understand, which (in part because of the confusing notation) is so far only a little of it. So again, before we go forwards or backwards from the current inconsistent state, can we perhaps get some more people than Yohan and me to chime in with opinions? It's hard to reach a consensus with only two opposite views. —David Eppstein (talk) 03:14, 23 July 2014 (UTC)[reply]

Here's a better idea: We add a note to the effect that

exp(X)

and

e X

can be used interchangeably — it is common for

exp(X)

to be used inline and

e X

to be used in displayed equations — and we can leave things as they are.

I can give you an argument in favor of

exp(X)

as well. Some use

e x

for matrix groups and

exp(X)

for general groups. This is rather strange because

exp

for general groups is not defined in terms of a power series. The power series definition is otherwise used as an excuse for naming the function (for matrices) the same way as for complex or real numbers. While the power series definition is gone, the algebraic properties are the same, so

e (X)

can be used confidently here too.

I don't believe you a split second when you say you have a hard time understanding the notation. A five-year old would understand it. Surely, you understand it for the exponential function for real numbers. Nothing changes here. You just want to "get right". This is rather silly when notation and conventions are involved.

Finally, the vociferous argumentation on my part isn't about one notation or the other. It is against changing the style of an article (half-ways) without consensus. I told you that. But since you persistently say that the notation confuses you and the rest of the world, I have to defend it. YohanN7 (talk) 12:59, 23 July 2014 (UTC)[reply]

Move discussion in progress

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Move discussion in progress

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Second coutable

I think the assumption -- that a closed subgroup here is required to be second countable -- need to be mentioned somewhere somehow (to avoid a trivial situation every subgroup is a Lie subgroup by equipping the subgroup with discrete topology). Note to myself: if my memory serves, you also need "second countable" in the proof of the lemma or the theorem; but maybe I misremembered. P.S. the article itself is very nice. -- Taku (talk) 21:09, 3 January 2019 (UTC)[reply]

I don't understand why you undid my changes: the hypothesis that you have at most countably many components is superfluous (it will be automatically satisfied if the subgroup is closed, meaning if you want that this is part of the conclusion). Also, I find it easier to say that the smooth structure is the one given by the embeddeding, which implies that also the group topology is the subspace topology. Third, "referred to Cartan's theorem" is not correct English: the correct wording is "referred to as Cartan's theorem". Nothing personal, I just want to understand! Kappaenne (talk) 21:19, 17 March 2019 (UTC)[reply]

Sorry and thank you; you’re correct: we are assuming the topology is a subspace topology so “second countable” is automatic (I must have confused some stuff with zero-dimensional Lie group). I will restore your changes. —- Taku (talk) 09:02, 18 March 2019 (UTC)[reply]

Thank you for your kind and quick reply! Kappaenne (talk) 10:49, 18 March 2019 (UTC)[reply]