Talk:SL2(R)

Untitled
I noticed that there wasn't an article on this subject, so I went ahead and made one, mostly because there ought to be some article discussing the hyperbolic/parabolic/elliptic trichotomy. It could use a better summary of the representation theory, more details on the algebraic structure, and a discussion on the relationship to moduli space. Jim 07:29, 25 July 2007 (UTC)

The page says SL2(R) consists of all area-preserving 2x2 matrices. But if det(A)=-1 then A is also area-preserving. So this needs an additional caveat. I'm not quite sure how to phrase it without messing up the discourse with something too tangential. "...that preserve clockwiseness" seemed awkwards. Barak 10:28, 17 August 2007 (UTC)


 * I changed the wording. It now says that elements of SL2 preserve "oriented area", with links to orientation (mathematics) and area (geometry). There really ought to be an article on oriented area or signed area.  Jim 03:37, 18 August 2007 (UTC)

Blackboard bold discussion
There is consensus that standard bold (e.g., R, C, Z) are better supported on most browsers, and it makes sense to use these rather than the unicode blackboard bold fonts (ℝ, ℂ, ℤ). This consensus is codified in our manual of style: WP:MSM. Recently an editor changed from the MoS recommendation to the unicode blackboard bold fonts. I reverted this change, per the MSM page, and then I was reverted by the same editor, who claimed that WP:IAR policy trumped the manual of style guideline. I disagree with this application of IAR. The consensus preference of regular bold over blackboard bold exists for the reason already articulated: it is universally supported in all browsers (even text-only browsers and users with screen-readers). Therefore, I contest that this is actually an improvement of Wikipedia, which IAR clearly requires in order to be invoked. Not to mention, IAR is not the best policy to try to win an argument with anyway, for reasons that should be obvious. For these reasons, our manual of style, together with WP:CON policy, should be adhered to. Sławomir Biały (talk) 13:15, 16 October 2010 (UTC)
 * Wrong. In Wikipedia the should be bold when first mentioned, but if you use bold to show "Common sets of numbers", it would be confusing when mentioned in the PAGENAME. There's no bold and bold again style in Wikipedia currently. –– 虞海 ( Yú Hǎi ) 14:53, 16 October 2010 (UTC)
 * For example, you may write "ABCDPQℝST is a term refers to ..." but not "ABCDPQRST is a term refers to ...". You may also write "ABCDPQ𝐑ST is a term refers to ...", but its support is even worse than the blackboard bold. –– 虞海 ( Yú Hǎi ) 15:21, 16 October 2010 (UTC)
 * So indeed, my edit was actually an improvement of Wikipedia and I was following IAR. –– 虞海 ( Yú Hǎi ) 15:24, 16 October 2010 (UTC)

Somebody has moved this talk to Wikipedia talk:WikiProject Mathematics without any notify. –– 虞海 ( Yú Hǎi ) 16:04, 16 October 2010 (UTC)
 * It seems clear to me that the guideline for not using blackboard bold takes precedence over any trivial title bolding issues in the interest of cross-article uniformity. If you think otherwise you are probably in a minority. In any case now that you know it's contentious you must get a consensus before making this change again.
 * By the way, there seems to be a problem with your signature. It currently seems to be spamming the talk page source code with many lines of irrelevant stuff. That's quite irritating, so I suggest you change it. Hans Adler 16:01, 16 October 2010 (UTC)
 * For you, the title bolding issue is "trivial", but remember, the Wikipedia guideline "we want to make the creation of a print version of Wikipedia as easy as possible" told us it is important. As for the signing, I used ––zh (Yú Hǎi) as my signing source and I do also dislike he long code, but I think of "print version of Wikipedia". –– 虞海 ( Yú Hǎi ) 16:10, 16 October 2010 (UTC)
 * I personally have no problems with blackboard bold but it is inappropriate to change them for no reason other than your preference. According to the manual of style
 * "The Arbitration Committee has ruled that editors should not change an article from one guideline-defined style to another without a substantial reason unrelated to mere choice of style, and that revert-warring over optional styles is unacceptable. Where there is disagreement over which style to use in an article, defer to the style used by the first major contributor."
 * As there seems to be no substantial reason for this change other it should be reverted.
 * And I have also noticed the problems with 虞海's signature and have raised it on his talk page, as it makes replying here very difficult.-- JohnBlackburne wordsdeeds 16:22, 16 October 2010 (UTC)
 * There IS substantial reason: to solve the title bolding issue. And the signature issue is partially solved. ––虞海 ( Yú Hǎi ) 16:24, 16 October 2010 (UTC)


 * I'm with 虞海 here: without the blackboard bold it's much harder to see what's going on. I think we should separate the issue of whether blackboard bold should be used from that of the particular technical solution to use: U+1D411, U+211D, R', $$\mathbb{R}$$, etc.
 * CRGreathouse (t | c) 16:53, 16 October 2010 (UTC)
 * I personally don't think it is a good idea since neither $$\mathbb{\mathbf{R}}$$ nor $$\mathbf{\mathbb{R}}$$ solve the problem. ––虞海 ( Yú Hǎi ) (talk) 17:03, 16 October 2010 (UTC)
 * In fact, this ("ABCDPQ𝐑ST is a term refers to ...") do solve the problem, since R is bold and bold again. ––虞海 ( Yú Hǎi ) (talk) 17:04, 16 October 2010 (UTC)
 * Which solution or solutions are you saying are good and which bad? CRGreathouse (t | c) 17:27, 16 October 2010 (UTC)
 * I've had a look through and I fail to see the problem the blackboard bold was supposed to solve. As an aside I don't think people should bother with the Unicode template nowadays, my belief was it was a trick to get past a problem in older Internet Explorer browsers and was not much used. If they rely on that users will have lots of problems elsewhere where people don't use it. Dmcq (talk) 17:29, 16 October 2010 (UTC)
 * (Talking with CRGreathouse)To use Unicode to implement bold or doublestruck, such that it can be used parallelly with the bold title. ––虞海 ( Yú Hǎi ) (talk) 05:27, 17 October 2010 (UTC)
 * (Talking with Dmcq)It is quite clear that the Unicode blackboard bold will solve the problem, since ℝ can be used together with the title bold, say ℝ. ––虞海 ( Yú Hǎi ) (talk) 05:41, 17 October 2010 (UTC)
 * I could not see any use for the pair together, perhaps you could indicate the place in the article where there was a problem? And what's that about the title?, you weren't thinking of having things like ℝ in the article name were you or were you thinking of DISPLAYTITLE? I think DISPLAYTITLE has some restrictions too which would probably stop ℝ being used. Dmcq (talk) 13:55, 17 October 2010 (UTC)
 * The difference between the 2 double-struct-R is bold (used on quoting title), while the former is not. And about the title, we can modify the MediaWiki:Titleblacklist. And you comment "?, you weren't" and "were you or were you" - I'm sorry I don't understand the “?, you” and “were you or were you” syntax in English. ––虞海 ( Yú Hǎi ) (talk) 14:10, 17 October 2010 (UTC)

Again do not change the format from one to another without good reason. The consensus here together with the maths guidelines on the format to use and the general guideline on article stability all are against such a change without good reason. IAR on its own is not a good argument: it is rather a justification for ignoring rules if you have some other good reason to do so. So far you've not given such a reason or any clear argument, here or in your edit summaries.-- JohnBlackburne wordsdeeds 13:08, 17 October 2010 (UTC)
 * I don't know what your "consensus" is, but in the current "_talk page_", there's no consensus. And since nobody have any opinion of my comment there, they seem to have tacitly agreed. And the "clear MOS guidelines" you quoted does not have the force of policy. ––虞海 ( Yú Hǎi ) (talk) 13:17, 17 October 2010 (UTC)
 * I'm not sure what comment you are referring to. The MoS is pretty clear about this, and represents a wider consensus than any individual article talk page.  For reasons I have already stated very clearly (browser support and screen readers), the use of regular bold is preferred on Wikipedia over blackboard bold.  This is codified in our manual of style, which represents the consensus of hundreds of regular editors.  In addition, there certainly was never any consensus (implicit or otherwise) for a change in this article specifically.   Sławomir Biały  (talk) 13:28, 17 October 2010 (UTC)
 * MOS is a guideline, and does not impose anything on individual articles since IAR policy exist. As for your reason, it violated the nature of Wikipedia - an encyclopedia (so will eventually have a printed edition as the formal version). So the quality of printed version is far more important than that of old-browser-and-system-rendered-version. And since your MOS-math elimited the print version, it's trivial. And my comments (the most recent one "It is quite clear that" and older one "substantial reason: to solve the title bolding issue") is substantial. ––<Span xml:lang="zh-Hant-CN" lang="zh-Hant">虞海</Span> ( Yú Hǎi ) (talk) 13:40, 17 October 2010 (UTC)
 * There is no implication anywhere that Wikipedia will ever be printed. In fact other encyclopaedias are abandoning their print versions. As to printing many books do use bold instead of blackboard bold. Putting comments on a talk page for a guideline doesn't mean the guideline has changed. Dmcq (talk) 14:15, 17 October 2010 (UTC)
 * I'm not imposing the double struck here, (at least for me) plain bold is also acceptable, but must be distinguished with the title-quoting-bold. The style must be ensured. ––<Span xml:lang="zh-Hant-CN" lang="zh-Hant">虞海</Span> ( Yú Hǎi ) (talk) 14:54, 17 October 2010 (UTC)
 * (e/c) See WP:NOTPAPER. Wikipedia is not a print encyclopedia, and it is just silly to base editorial decisions on the speculation that some day it shall be one.  The title bolding issue is already discussed below, with consensus decidedly against it.  The "quite clear that" remark doesn't seem meaningful to me.  Anyway, I certainly don't see any reasons presented here that would justify bandying about IAR as you do so cavalierly.  IAR does not mean "I am right".   Sławomir Biały  (talk) 14:20, 17 October 2010 (UTC)
 * You must be GAMing the WP:NOTPAPER. WP:NOTPAPER says "no size limits" and remove other digital encyclopedia features with many "but". One of them of the style - must looks as good as a print encyclopedia. ––<Span xml:lang="zh-Hant-CN" lang="zh-Hant">虞海</Span> ( Yú Hǎi ) (talk) 14:52, 17 October 2010 (UTC)
 * Please provide a reference to where you saw a requirement that the style be as good as a print encyclopaedia. Dmcq (talk) 15:31, 17 October 2010 (UTC)

Keep reading. The MoS was written with this axiom firmly in mind. Sławomir Biały (talk) 15:18, 17 October 2010 (UTC)

Article name
Why is this at SL₂(R) and not SL(2, R) or similar? The former breaks two guidelines in the naming conventions, namely The current title is a subscript, in breach of the former guideline, then uses a very non-standard way to do it so breaches the second. The use of special Unicode subscripts is also against the Math MOS, for reasons given here. It's also not used anywhere in the article, and surely the article title should match the lede and rest of the article?-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 17:41, 16 October 2010 (UTC)
 * Use italics but "Other types of formatting (bold type, superscript, etc.) [...] should not be used in Wikipedia article titles."
 * Do not use non-language characters: Non-language characters such as "♥", as sometimes found in advertisements or logos, should never be used in titles.


 * I tend to agree: The article should be at SL(2, R) (except we would have an edit war about the spacing) or SL2(R) displayed SL2(R).  — Arthur Rubin  (talk) 19:19, 16 October 2010 (UTC)


 * Agreed. The second option seems best. CRGreathouse (t | c) 19:37, 16 October 2010 (UTC)


 * I moved it. Ozob (talk) 21:27, 16 October 2010 (UTC)


 * That's better, though it still did not match the text so I've added it. It still seems odd that the name as used in the title doesn't match the bulk of the article, but the various issues wrt the naming conventions are fixed.-- JohnBlackburne words<sub style="margin-left:-2.0ex;">deeds 22:21, 16 October 2010 (UTC)
 * And the reason it doesn't match the text is that there might be an edit war about a space in the name? I think it might do me some good to get myself some bubble wrap so I can pop a few bubbles on occasions like this :) 86.9.212.44 (talk) 22:48, 16 October 2010 (UTC)
 * If you told me that "₂" is a "non-language character", then how can "SL" and "R" be "language character"? Then the title would be better moved to Special Linear group (Real) degree 2. ––<Span xml:lang="zh" lang="zh">虞海</Span> ( Yú Hǎi ) (talk) 05:32, 17 October 2010 (UTC)
 * A "non-language character" is a character that is not normally used in the language. The parentheses are questionable, but "S", "L", and "R" are not "non-language characters".  I reverted your move, and am deleting the "Special Linear group (Real) ..." files.  — Arthur Rubin  (talk) 07:34, 17 October 2010 (UTC)
 * That's merely an imposing on the definition. If such imposing may be accepted, "₂" will also be a language character, for its no difference from the number   ₂    ! ––<Span xml:lang="zh" lang="zh">虞海</Span> ( Yú Hǎi ) (talk) 07:45, 17 October 2010 (UTC)
 * This is an English language encyclopaedia and one of the more desirable things is that English readers be able to easily write the title of an article in the search box. That's why there is that naming convention. Occasionally there are problems without a simple solution where one needs a character not on an English keyboard but this article is obviously not one of them. Dmcq (talk) 13:24, 17 October 2010 (UTC)
 * It is a desirable things for writers but not for readers. Writers should always think of readers. So the look-and-feel is more important. For the title, we can simply made redirection to "lead" the readers to the correct article. ––<Span xml:lang="zh-Hant-CN" lang="zh-Hant">虞海</Span> ( Yú Hǎi ) (talk) 13:43, 17 October 2010 (UTC)
 * Please follow the naming conventions or else discuss this there and get them changed first. This article does not need a change from them so should only be changed if the conventions say that is desirable. Dmcq (talk) 14:11, 17 October 2010 (UTC)

Assessment comment
Substituted at 02:34, 5 May 2016 (UTC)

Minor edit
Should the introductory sentence say "group of 2 x 2 real invertible matrices" ?

104.228.101.152 (talk) 22:30, 19 February 2017 (UTC)


 * No, that would be the general linear group GL(2,R). This article is about the compact group that is special since determinant is one. Rgdboer (talk) 02:27, 20 February 2017 (UTC)
 * You are right, of course, that GL(2,R) is the group of all invertible 2x2 real matrices, and SL(2,R) is its subgroup consisting of those matrices in GL(2,R) whose determinant equals 1.
 * But SL(2,R) is not compact. For instance, the sequence {{n, 0}, {0, 1/n}} ∊ SL(2,R), n = 1, 2, 3, ... contains no convergent subsequence. 2601:200:C000:1A0:F41B:AE21:91:D096 (talk) 01:22, 22 December 2022 (UTC)

Is this right?
The section Möbius transformations begins as follows:

"Elements of PSL(2,&thinsp;R) act on the complex plane by Möbius transformations:
 * $$z \mapsto \frac{az+b}{cz+d}\;\;\;\;\mbox{ (where }a,b,c,d\in\mathbf{R}\mbox{)}.$$

"This is precisely the set of Möbius transformations that preserve the upper half-plane."

But if a = -1, b = 0 = c, and d = 1 we get the Möbius transformation $$z \mapsto -z$$, which does not preserve the upper half-plane.

It's necessary to impose the condition that ad-bc = 1 to avoid this kind of thing.

Of course, we're talking about SL(2,R) and PSL(2,R), so maybe ad-bc = 1 is always assumed.

But it doesn't hurt to mention it again. 2601:200:C000:1A0:C9BF:3229:6680:A9F6 (talk) 22:52, 21 December 2022 (UTC)