Talk:Spin group

merge

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, Spin_c, 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)[reply]

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)[reply]

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)[reply]

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)

Spin

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

SO vs O

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)[reply]

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 (talk • contribs) at 09:24, 9 October 2007 (UTC) on User talk:Fropuff. Copied here for convenience.[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

In fact, for

n = 2

,

Spin(n, R)

≈

SO(2)

≈

S 1

. Hence,

π 1 (Spin(n, R))

≈

π 1 (SO(2))

≈

π 1 (S 1)

≈

Z

is not the trivial group

{e

} which contains only the unit element.Mgvongoeden (talk) 16:18, 18 May 2013 (UTC)[reply]

Center of Spin group

The reference to Varadarajan (p.208) states that for p and q both even the center of $\operatorname {Spin} (p,q)$ coincides with that of $\operatorname {Spin} (p q,\mathbf {C} )$ then makes a mistake when giving the explicit answer. Namely, it is $\mathrm {Z} _{4}$ for $p q=4k 2$ and $\mathrm {Z} _{2}\oplus \mathrm {Z} _{2}$ for $p q=4k$ . I'm correcting that now in the article. 69.141.211.196 (talk) 03:34, 5 November 2016 (UTC)[reply]

Incorrect use of “⊗”

Since the edits of 67.198.37.16 (talk · contribs · WHOIS) the article acquired such things as $e_{i}\otimes e_{j}=-e_{j}\otimes e_{i}$ , that is, the symbol “⊗” abusively denotes the Clifford product. Isn’t just the empty space—that is, $e_{i}e_{j}=-e_{j}e_{i}$ —a better notation? Incnis Mrsi (talk) 18:27, 1 May 2019 (UTC)[reply]

It is just fine to drop the tensor symbol, and use the empty-space, once the student understands that the empty-space is an implicit tensor product, and can keep that in their head, correctly, and substitute it into all the correct places, on demand. However, for an encyclopedia, where we are explaining things for the first time, it is convenient to tell the reader "hey there is a tensor product right here, please do not imagine that it is some weirdo scalar product of these weird numbers that seem to magically anti-commute". It's not magic. Clifford algebras are quotients of tensor algebras. I suspect that you won't like what I just wrote, but please consult your favorite reference on the topic, and you will find the author says, early on, "... and now, by abuse of notation, we drop the tensor symbol because it is painful to write it over and over." Which is what this article currently says. 67.198.37.16 (talk) 17:22, 3 May 2019 (UTC)[reply]

Holy shit… the Clifford product is not outer product because the Clifford algebra is not tensor algebra but its quotient over certain ideal. In the tensor algebra the equality «

e_{1}\otimes e_{2}=-e_{2}\otimes e_{1}

» is bluntly false, but difference of the sides belongs to said ideal (of course, assuming an orthogonal

{e j}

). That’s why

e_{1}e_{2}=-e_{2}e_{1}

is true. Incnis Mrsi (talk) 17:37, 3 May 2019 (UTC)[reply]

Calm down. When defining a tensor algebra, the

e_{i}

start out being just plain-old-ordinary college-freshman basis vectors:

e_{i}=(0,0,0,\cdots ,1,0,\cdots ,0)

and so

e_{i}\otimes e_{j}

starts out being just the ordinary outer product -- (dyadic product)-- its just an ordinary NxN matrix that is all zero except a 1 in the (i,j)'th entry. When constructing the Clifford algebra, The

e_{i}\otimes e_{j}

becomes just one representative element of a coset: the coset of all tensors

\cdots \otimes e_{i}\otimes e_{j}\otimes (v\otimes v-|v|^{2})\otimes (z\otimes z-|z|^{2})\otimes \cdots

(or call it an "orbit" or call it an "ideal" or call it an "element of the kernel of a short exact sequence" ... all the same idea). So, my goal of using the tensor symbol was to remind the reader that, yes, these are just ordinary college-freshman basis vectors that are being tensored together, and/but they're also representative elements of this large coset/ideal -- that they are these rather explicit algebraic objects, with concrete constructions, and not some airy-fairy magic numbers that magically anti-commute due to the arbitrary whims of some mathematician. Also, I am not just making this up -- this is commonly found in textbooks. It's a notational device. I'm not the one who invented it.

Note also, for the quantum inclined, one tends to explicitly write the tensor symbol in about half the texts, e.g. on Bell-pairs, so

e_{i}=|i\rangle

and lots of texts write

|i\rangle \otimes |j\rangle

for a while, before getting lazy and abusing that down to

|i\rangle |j\rangle

or eventually

|i,j\rangle

so that the Pauli exclusion principle is

|i,j\rangle =-|j,i\rangle

... But also one can move backwards, now, inserting

\psi

for wave-functions, etc. until one finally gets back to explicit, concrete, ordinary real numbers, instead of ... cough cough, magically anti-commuting symbols. I think it would be wonderful if something like the above could be added either here or to the clifford-algebra article, since I think the quotient, and the resulting anti-commutation really really trips up students a lot, its a major point of confusion. 67.198.37.16 (talk) 18:08, 3 May 2019 (UTC)[reply]

I suppose you could do something like this: "Let us define

{\tilde {e}}_{i}{\tilde {e}}_{j}=\cdots \otimes e_{i}\otimes e_{j}\otimes (v\otimes v-|v|^{2})\otimes (z\otimes z-|z|^{2})\otimes \cdots

and now we will drop the tilde's ..." but I think that will ruin the flow of the rest of the article. These really are cosets/ideals, and it is perfectly valid to take the tensor product of cosets/ideals -- given two ideals, you take the tensor-product pair-wise, to create a new ideal. 67.198.37.16 (talk) 18:28, 3 May 2019 (UTC)[reply]

Who could become happy with these complications from heck? If there is something wrong with blank space denoting Clifford product, then let’s choose another symbol except the one reserved for outer product. Incnis Mrsi (talk) 18:39, 3 May 2019 (UTC)[reply]

I suppose one could write something like this: "Let us define the set (the set-theoretical set)

{\tilde {e}}_{i}{\tilde {e}}_{j}={\Big \{}x\in TV|x=\cdots \otimes e_{i}\otimes e_{j}\otimes (v\otimes v-|v|^{2})\otimes (z\otimes z-|z|^{2})\otimes \cdots \forall v,z,\cdots \in V{\Big \}}

and now we will drop the tilde's ..." but I think that will ruin the flow of the rest of the article. These really are cosets/ideals, and it is perfectly valid to take the tensor product of cosets/ideals -- given two ideals, you take the tensor-product pair-wise, to create a new ideal. That is, you can take the tensor product of these sets by taking the tensor product of the set-elements, pair-wise, and then observing that the set ${\tilde {e}}_{i}{\tilde {e}}_{j}$ is exactly the same set as ${\tilde {e}}_{j}{\tilde {e}}_{i}$ except that all of the set-elements now have a minus sign in front of them, which justifies the notation ${\tilde {e}}_{i}{\tilde {e}}_{j}=-{\tilde {e}}_{j}{\tilde {e}}_{i}$ . That is the meaning of the minus sign; that is where the minus sign is coming from. By abuse of notation, one commonly drops the tilde, and the tensor symbol, but given that you got upset about this, I think it illustrates that when these notational devices are dropped too soon, and the abuse-of-notation starts too early, everyone gets confused. This is all very concrete; there is nothing wrong with taking the tensor-product of sets (of elements of TV); its a well-defined operation. It's more: it is the technical, low-level definition of what it means to take products of elements from any quotient algebra obtained from the tensor algebra, not just the Clifford algebra. It is 100% valid to use the tensor symbol on any/all elements of any/all algebras that are obtained as quotients of the tensor algebra, precisely because the ideals are sets of elements taken from TV. 67.198.37.16 (talk) 18:55, 3 May 2019 (UTC)[reply]

Oh, well, OK, I guess I kind-of see what you are saying .. one could pick some symbol e.g. $\;\wedge \!\!\!\!\!\bigcirc$ and define

{\tilde {e}}_{i}\wedge \!\!\!\!\!\!\bigcirc {\tilde {e}}_{j}={\Big \{}x\in TV|x=\cdots {\Big \}}

and then drop the tilde. I like the symbol $\;\wedge \!\!\!\!\!\bigcirc$ , because of the anti-commuting flavor. However, at this point, we are inventing things. If you actually look at actual textbooks that actually define Clifford algebras (or other TV quotient algebras), you actually see the tensor product symbol being used shamlessly, even though one should have written something like $\otimes _{Cliff}$ for $\;\wedge \!\!\!\!\!\bigcirc$ . So, yes, I get your point. 67.198.37.16 (talk) 19:18, 3 May 2019 (UTC)[reply]

Whatever. I'm concluding that WP needs an article titled quotient space (tensor algebra) or quotient algebra (tensor algebra) which explains all of the above, and uses the symmetric algebra, the exterior algebra and the clifford algebra as examples. This is kind-of foundational for things like the PBW theorem and for representation theory in general. If I look at the article quotient algebra and squint real hard, I can see that it is saying exactly what I'm saying above, except that it has already flown out into a very abstract definition, lacking in the concreteness and constructability and example-cases I'm giving above. And that's the sort-of meta-issue with WP ... the math articles start out abstract, and get more abstract, which is fine if you're a math grad-student, but tends to frustrate undergrad engineers (and professional software developers) who are honestly interested in the subject, but don't have the pre-requisites. (and, BTW, more than a few of them are regular editors and admins here at WP -- smart, capable, interested in physics/math, but missing the foundations needed to understand things, and floundering more than a bit.) 67.198.37.16 (talk) 20:17, 3 May 2019 (UTC)[reply]

I think this whole situation is analogous to the use of the normal multiplication symbol being used for modular arithmetic even though it changes meaning (operating on cosets of some ideal of Z rather than the integers themselves) after the quotient is taken. However I do not endorse the use of the tensor product symbol in an analogous way. For one, you could also argue that the tensor product is an operation on vector spaces and not members thereof. Also anticommuting nature of the exterior product only comes after taking the quotient, and this anticommuting behavior is not observed in the original tensor product, whereas in modular arithmetic the induced multiplication operation in the quotient group does not have such a striking difference from that of the original ring.--Jasper Deng (talk) 23:07, 3 May 2019 (UTC)[reply]

There is a superficial analogy, but normal multiplication symbols (“⋅” or blank) have a firm tradition of use for arbitrary rings. Even non-commutative (although the original multiplication is commutative). Who uses “⊗” for everything related to tensors? Note that the Clifford algebra is not graded, unlike the tensor algebra. Apart of by 67.198.37.16, I never saw such notation. Incnis Mrsi (talk) 05:58, 4 May 2019 (UTC)[reply]

If your readings are in ring theory, then yes, it is always 100% blank, or a dot, always. If your readings are in affine Lie algebras, then the tensor symbol is widespread, and a recurring touchstone, even though it is eventually dropped because its just annoyingly verbose. If your readings are in Riemannian geometry, then its 50-50 depending on author; older books, pre-1975-ish use endless strings of indexes; new, post 1980-ish books use tensor symbols widely. The fashion for otimes in physics came in with Misner, Thorne, Wheeler, and also Abraham & Marsden. I'm not sure when spin manifolds came into fashion - post 1990's I think - but modern books on Riemannian geometry all cover spin manifolds (and thus necessarily the spin group) (and use the otimes symbol extensively, until they start abusing notation). (and I don't think anyone cared about the spin group much, until they realized there's a postnikov tower, and we all went, oh, ahah! The spin group gets exciting because of the postnikov tower) The tensor symbol is also widely used in PBW theorem approaches to representation theory. If you've never seen it, then you don't read in these topics. (I do.) I read very little ring theory; I got part-way into the "geometry of schemes" by Eisenbud, put it down, vowed to come back to it someday, haven't yet. I'm throwing out this wall of words at you, with the intent of impressing on you that I kind-of know what I'm talking about.67.198.37.16 (talk) 06:48, 4 May 2019 (UTC)[reply]

What do we discuss? For tensor algebras one can use either “⊗” or, say, abstract index notation (as Roger Penrose does) – both are valid, and each has its advantages over the rival in specific contexts. Our problem is indeed how we are about to denote Clifford product. Is \wedge\!\!\!\!\!\bigcirc a serious proposal? My proposal, again, is to use

e_{1}e_{2}

or, if zero-width assumed multiplication symbol insults some esthetes, then a full-width space infix, namely

e_{1}\ e_{2}

(HTML:

e 1 e 2

). Incnis Mrsi (talk) 08:30, 4 May 2019 (UTC)[reply]

(edit conflict) My differential geometry text was emphatic about using wedges for the exterior product. Here we're talking about a different Clifford algebra. I would greatly prefer the usage of another symbol just because the article glosses over the details of taking quotients. For example, to an untrained reader, it is not at all obvious that the structure obtained here is a group at all. IF we keep using the tensor product symbol, we should have a one-letter subscript attached to it because the idea of a quotient is foreign to those not trained in abstract algebra.--Jasper Deng (talk) 08:32, 4 May 2019 (UTC)[reply]

To me, an overlap of “∧” and “◯” is even more acceptable than one-letter subscript attached to [⊗]. The problem is, indeed, that the IP coded it in such a way that MathJax fails negative-width spaces. It could be \rlap{\wedge}\bigcirc then—Failed to parse (unknown function "\rlap"): {\displaystyle \rlap{\wedge}\bigcirc} —but \rlap seemingly doesn’t work in MathJax either, at least in the installation I use. Incnis Mrsi (talk) 08:50, 4 May 2019 (UTC)[reply]

As a temporary solution, may we settle on “⋆”

\star

? It is sometimes used as infix for “deformed” non-commutative products. Incnis Mrsi (talk) 09:04, 4 May 2019 (UTC)[reply]

Another possibility would be the IP’s overlap proposal, but (due to accessibility reasons) it has to be implemented in HTML:

\land ◯

, and the article should be converted to {{math}} for this reason. Incnis Mrsi (talk) 09:13, 4 May 2019 (UTC)[reply]

As a solution, I guess removing (most of) the tensor symbols is OK. I'll do that now. As a long-term solution, creating a quotient algebra (tensor algebra) would be correct. As anon IP I can't create new pages. I suppose a long section could be added to quotient algebra but I don't particularly want to write that because I don't have a cheat sheet to work from. I'd rather avoid the

\star

- it is indeed liable to be confused with the deformed non-commutative product aka Moyal product. I have two requests, though: (1) should

e_{i}^{2}=-1

or should

e_{i}^{2}= 1

? The edit history of this page shows that editors keep flipping the sign around, and I have zero clever words to say why one sign is better than another. (It should be plus to be consistent with clifford algebra.) Is it taste, or is there something more? (2) maybe I will take a whack at extending quotient algebra blindly, please take a look at it in 24 hours and let me know if it looks unreasonable/unholy. 67.198.37.16 (talk) 16:34, 4 May 2019 (UTC)[reply]

When I studied Clifford algebra, there was

e 12 = -1

certainly (for Euclidean space) or, more generally,

v 2 = - q (v)

. This convention likely was established by W. K. Clifford, and presumably it can be traced to works by W. R. Hamilton on quaternions, which are identical to the Clifford algebra of Euclidean plane. Substituting “ 1” instead, the resulting algebra would fail to be a division algebra. On the other hand… Pin and Spin groups don’t care about this sign. Incnis Mrsi (talk) 17:24, 4 May 2019 (UTC)[reply]

Since I don't see evidence of a universal choice for q, we should not speak of the Clifford algebra but a Clifford algebra. The article on Clifford algebras makes no such distinction. We certainly don't want the split-complex numbers where there are zero divisors galore (for example 1 i and 1 - i).--Jasper Deng (talk) 04:52, 5 May 2019 (UTC)[reply]

Wait: universal… in which world? For a vector space of any dimension greater than 0, of course, there is no universal quadratic form. For the Euclidean plane it definitely exists, hence the Clifford algebra of Euclidean plane. Incnis Mrsi (talk) 08:52, 5 May 2019 (UTC)[reply]

The Clifford algebras obtained by changing the sign above are clearly non-isomorphic, and clearly the one discussed by this article is not the exterior algebra which is the Clifford algebra with q = 0. Thus there is no unique Clifford algebra over any (nonzero) vector space.--Jasper Deng (talk) 10:35, 5 May 2019 (UTC)[reply]

Again, Jasper Deng’s edit [1] makes the article worse, not better – neither Cℓ nor Spin are possible for a bare vector space without an additional structure. Hence the Clifford algebra, but not merely “for

V

”. As for the choice of

- q (v)

– compatibility with 19th-century theories. It’s this thing defined as the Clifford algebra for the given scalar square

q

. With another sign Cℓ is certainly not isomorphic, although Pin and Spin are the same. Incnis Mrsi (talk) 11:25, 5 May 2019 (UTC)[reply]

No, my edit does not in any way, shape, or form imply that the spin group or a Clifford algebra is meaningful for a bare vector space. Constructing "a certain Clifford algebra" requires a choice of q before one can even start. Therefore, we implicitly are choosing q in the process of "constructing a certain Clifford algebra for V". I can make more edits to clarify that if you'd like, but as-is my edit isn't incorrect. We only speak of the Clifford algebra once q is chosen.--Jasper Deng (talk) 17:18, 5 May 2019 (UTC)[reply]

Jasper’s edit was formally correct, but pointless. May we say

“

Construction of the Spin group often starts with the construction of the Clifford algebra over a Euclidean vector space

V

”

instead? There is no ambiguity about the quadratic form

q

– the same the Spin’s action must preserve, not certain obscure one. Surely we can then generalize for arbitrary non-degenerate

q

, as well as arbitrary field (except for those of characteristic 2). Incnis Mrsi (talk) 18:10, 5 May 2019 (UTC)[reply]

There is no reason we should expect the reader to know it is this very particular quadratic form out of many, especially if they haven’t read about the group yet and therefore cannot know that it is invariant under the action of this group.—Jasper Deng (talk) 18:43, 5 May 2019 (UTC)[reply]

To me, Jasper presents the whole thing backwards. We want Spin, the thing which 2:1 covers SO. One can’t define SO on a bare vector space; a fixed quadratic (or bilinear) form—up to multiplication by scalars—is necessary. We already know what a thing must Spin be, the whole Cℓ stuff is nothing more than an explicit construction of the group we hope should exist. That’s why it is more reasonable to start from an Euclidean space which has the necessary structure in hand. Incnis Mrsi (talk) 19:04, 5 May 2019 (UTC)[reply]

Even in Euclidean space there are many quadratic forms, the usual inner product being only one. I do think describing the motivation of the choice of quadratic form would be germane, but we should either way be explicit that we are choosing one, not acting as if one was already chosen.—Jasper Deng (talk) 19:10, 5 May 2019 (UTC)[reply]

Quotient algebra

And IMHO nothing special should be written on this kind of construction besides that it’s the ~~quotient algebra~~ of tensor algebra. Which ideals are suitable? Again, topical to tensor algebra. Incnis Mrsi (talk) 18:51, 4 May 2019 (UTC)[reply]

"Nothing special should be written on this kind of construction" -- I disagree on two counts: describing the construction in great detail lowers the barrier-to-entry - e.g. to undergrads in math, grad students in physics who are usually rushed at high-speed through such topics, and thus have a poor foundation. The second reason is that, written correctly, it provides a stepping stone to enveloping algebras and more generally to "deformed/quantized" algebra topics. I mean, the tensor algebra has two coalgebras in it; one of them is a Hopf algebra. By quotienting, the coalgebras can be extended to symmetric an alternating algebras; but what, exactly happens if I try to extend the coalgebras to the quotient that is the Clifford algebra? When exactly is quotienting compatible with coalgebras? I dunno off the top of my head; do you? Is it trivial, or is it hard? If it's trivial, then why doesn't the Clifford algebra article mention it? I'd have to crack open some book on string groups and read it very carefully to find out. So you can't just say "oh la de dah, quotients are trivial", if you can't actually answer what happened to the Hopf structure, and why it would work for some quotients and not others, or if you cannot articulate the details. Extending the tensor algebra article is possible, but its already lengthly. Adding a subsection to quotient algebra is clearly wrong. I think there's a real opportunity to create a mid-level article that is accessible to a broader audience than usual, and would open the door to multiple topics. 67.198.37.16 (talk) 19:51, 4 May 2019 (UTC)[reply]

@67.198.37.16: this talk page is about the article Spin group! I now have {tensor,quotient,Clifford} algebra and a lot of other links from here in my watchlist. Please, go elsewhere with proposals not directly related to editing “Spin group”. Incnis Mrsi (talk) 20:19, 4 May 2019 (UTC)[reply]

I got it how – a helpful link should be ~~quotient ring~~, whereas “quotient algebra” should be dropped. Incnis Mrsi (talk) 21:29, 4 May 2019 (UTC)[reply]

The redirect quotient associative algebra is online. Incnis Mrsi (talk) 09:14, 7 May 2019 (UTC)[reply]