# Talk:Spin group

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Field: Algebra

I suggest merging this into the spinor article (or maybe vice versa). Any comments? R.e.b. 03:53, 30 Apr 2005 (UTC)

I was thinking rather that we should keep it separate (maybe rename it to Spin group) and move much of the material that you added to Clifford algebra (on the Clifford group, and the Spin and Pin groups) to here. That article is getting rather long as it is. -- Fropuff 04:44, 2005 Apr 30 (UTC)
There really is quite a lot of this stuff: Spin, Pin, Spinc, spin structures on manifolds, spin bundles and their construction(s), the various physics names for types of spinors, Dirac operator(s). I'm not 100% sure of the right way to go. I do think that spinor should probably be a meaty article. Perhaps like tensor it will have to try to please everyone, and link off to fuller treatments. So maybe the thing is to get the spinor article structured first. Charles Matthews 10:27, 30 Apr 2005 (UTC)

Agree with Charles and Fropuff. linas 03:27, 1 May 2005 (UTC)

Sounds as if there is no support for the idea of merging, so I wont bother. R.e.b. 04:17, 1 May 2005 (UTC)

Gack, I just looked at Clifford algebra, indeed, its a very long article. For us attention-span-limited internet junkies, moving some of that article here would be good. If this results in some minor duplication due to a need for a cohernet introduction/overview, that's OK. linas 00:40, 5 May 2005 (UTC)
Since there is little enthusiasm for merging, I shall remove the notice. And after all, it's pretty clear that we can, and may often want to, talk about Spin groups and friends without getting into spinors. KSmrq 08:08, 2005 August 10 (UTC)

Should that be Spin+(2,2) = SO(2,2), rather than just Spin(2,2)? Tom Huckstep 10:54, 28 August 2007 (UTC)

This page needs to be cleaned up, and in particular needs some references. I haven't dealt much with Spin groups myself, but I know for instance that Spin(2)=U(1)=O(2) is just plain wrong. U(1) is *not* the same as O(2), but *is* the same as SO(2). Also, this is not a double cover, which seems to violate the definition at the top -- however, the comment towards the bottom of the page (which *does* say Spin(2)=SO(2)) says that there are special exceptions, when the fundamental group does not contain Z_2. It seems, therefore, that the definition is probably incorrect, and the "accidental isomorphisms" section contains at least one error. Could an expert please clear this up, and add at least one good reference? 131.215.118.194 01:03, 11 September 2007 (UTC)

## Errors and other problems

Dear Fropuff,

I've checked the spin groups for indefinite signature, and they seem to be accurate. I've therefore reverted to the old page on spin groups. Please let me know if there are errors there, but I think the groups are correct (and I am a mathematician doing research into Lie groups; but still, mistakes can happen; let me know if this is the case). All the best, Cheesfondue —Preceding comment added by Cheesefondue (talkcontribs) at 09:24, 9 October 2007 (UTC) on User talk:Fropuff. Copied here for convenience.

First off, I think it's bad form to list examples of the indefinite spin groups before defining or even mentioning them. They should probably be handled in a separate section. Secondly, the recent edits have introduced numerous errors:
• Spin(2) ≠ O(2) as noted above
• Spin+(1,1) = GL(1,R) not R (which is connected) or O+(1,1) (which is nonabelian).
• Spin+(3,3) = SL(4,R) not SL(2,R)
• Spin+(2,2) is usually given as SL(2,R)×SL(2,R) and not SO+(2,2). I'm not 100% certain that the latter is wrong but it certainly seems like it to me.
Finally the comment in the last paragraph is nonsense. The fundamental group of SO+(2,1) contains no Z_2 factors and yet its double cover is SL(2,R) which is not isomorphic to SO+(2,1).
--Fropuff 16:55, 9 October 2007 (UTC)
Dear Fropuff,
Your comments are all substantially correct (and the ones I still disagree about are due to differences in notation). I apologise for the mistakes, and will put up a fully corrected version some day.
Cheesefondue 19:13, 10 October 2007 (UTC)

## Changes

Have reworked and corrected the spin group article, added separate sections for indefinite signature and topology. Let me know if there still are errors in the article.

Cheesefondue 11:47, 13 October 2007 (UTC)

The new version looks much better, thanks. I guess some choice needs be made about the connectivity of Spin(p,q). All but one of the references I have define Spin(p,q) to be a double cover of SO(p,q) rather than SO+(p,q), so I would lean towards this convention. I am, however, aware of the reasons for not wanting to do so. Probably both conventions should be stated and explained at some point. -- Fropuff 02:10, 14 October 2007 (UTC)

## constructive definition?

I'm a bit befuddled by the exact sequence definition. Is there a more constructive definition of Spin(n)?

On a related note, is Spin(n) the unit elements of the even subalgebra of Cl(n)?

Thanks, 70.116.91.46 (talk) 04:36, 12 March 2012 (UTC)

## Unclear

What is meant by $Spin(2)=U(1)$ and te others? Surely they are not isomorph as topological groups, since $Spin(2)$ as to be simply connected, while $U(1)$ is not...

Moreover the symbol $=$ shouldn't be used here $\simeq$ is more apropriate, IF the groups are homoeomorph — Preceding unsigned comment added by 212.23.103.73 (talk) 11:54, 17 May 2013 (UTC)

Indeed, n = 2 is the only case where Spin(n, R) is not simply connected. You apparently do not understand that the Spin group has not necessarily to be the universal cover of the orthogonal group. It is a double cover (topology), which may or may not be universal. Incnis Mrsi (talk) 12:45, 17 May 2013 (UTC)
In fact, for n = 2, Spin(n, R)SO(2)S1. Hence, π1(Spin(n, R))π1(SO(2))π1(S1)Z is not the trivial group {e} which contains only the unit element.Mgvongoeden (talk) 16:18, 18 May 2013 (UTC)