Talk:Hurwitz's theorem (composition algebras)

Moved here from Hurwitz's theorem

Moved from "Hurwitz's theorem (composition algebra)" to agree with wikipedia (and standard) terminology.

more details about the proof?
Is it possible to elaborate a bit on this theorem? For instance, what are the key insights of the proof? TotientDragooned (talk) 19:49, 18 February 2010 (UTC)

I second the motion! —Preceding unsigned comment added by 69.171.128.107 (talk) 22:00, 14 March 2010 (UTC)

Kantor and Solodnikov
Referred to but no specifics given as to title, year, publisher etc. Brews ohare (talk) 15:36, 1 June 2010 (UTC)
 * It appears to be, published notes suitable for undergraduates. It was originally in Russian and was translated into German  and English . JackSchmidt (talk) 16:02, 1 June 2010 (UTC)

I have found this book, unfortunately out of print; the author's name was misspelled. Brews ohare (talk) 16:48, 1 June 2010 (UTC)

Cayley-Dickson doubling method
This method is referred to, but no details given and no source where it can be pursued. Brews ohare (talk) 15:38, 1 June 2010 (UTC)

Is there no hope for starting with clear, simple generalizations?
Here is a quote "the Hurwitz so-called 1,2,4,8 theorem ... proved that numbers associated with a numbers algebra, or 'normed division algebra' cannot occur in any other than in 1, 2, 4, or 8 dimensions." http://www.alenspage.net/ComplexNumbers.htm

From a perspective in the weeds, this may seem too simple, or uninteresting. But for those interested in a key insight to take away, it's the dimensional pattern that stands out. There's something not working if an encyclopedia page doesn't meet even the slightest journalistic function, which is to put the one most important thing first in language anyone understands. Then qualify it as needed. BRM Coyle] — Preceding [[Wikipedia:Signatures|unsigned comment added by 208.80.117.214 (talk) 20:08, 30 April 2014 (UTC)

Status of this article
This article is of no value other than the statement of the theorem. Sources are missing, mathematical mumbo-jumbo is not explained. Brews ohare (talk) 15:40, 1 June 2010 (UTC) I've added some sources. The article is still a mess. Brews ohare (talk) 16:49, 1 June 2010 (UTC)

Merger proposal
I suggest this article could be merged into Normed division algebra which is currently unreferenced and rather short. Deltahedron (talk) 18:51, 26 August 2012 (UTC)
 * Boldly done. Deltahedron (talk) 08:48, 27 August 2012 (UTC)

Algebraic aspects

 * Hurwitz stated his theorem for sums of squares over C and it holds for regular quadratic forms over any field of characteristic not 2 (see Lam 2005 pp.127-130). So it seems strictly more general than a statement about real non-associative algebras.  Deltahedron (talk) 16:59, 26 April 2013 (UTC)
 * So I rewrote the introduction in accordance with that comment to make it clear that the modern version of the theorem is considerably more general than Hurwitz's original statement and to put in in the algebraic context . In a series of edits  those changes have been almost completely reverted with the only explanation being the rather obscure "no need to ediorialize the lede".  In accordance with WP:BRD it would be good to have a clear explanation and discussion of the reasons here.  We have the situation that the current version again appears to assert that Hurwitz's formulation was in terms of real algebras whereas Lam and Rajwade (both now cited) state that it was for forms over the complexes.  This needs to be rectified.  Deltahedron (talk) 06:28, 29 April 2013 (UTC)
 * The reasons for that reversion appear elsewhere, the relevant material being:
 * Your error is to imagine that the lede of an extended article can be treated as if it were a stub. On the contrary, the lede summarises the article and the article is a summary of material from the sources selected. So please don't editorialise in a lede to write content at odds with the main body of the article, treating it as if it were a stub.
 * If an article is drawn from sources that use the real and complex numbers, suggesting otherwise in the lede is misleading and unhelpful to the reader. The normal method of teaching and presentation in textbooks (wikipedia articles are no different) is first to treat the standard cases where matters are simple (e.g. Lie elgebras over R and C, compact Lie groups, etc) and then mention greater generality in later subsections, comments or footnotes.
 * If you want to add comments about what might happen in general for arbitrary fields, do not do so prominently in the lede. That is WP:UNDUE. You can easily a section called "Further directions" summarising generalisations and surveying the literature. That is normal practice. No article on Lie algebras would treat anything other than R and C first and foremost.
 * So in summary please don't treat ledes as if they are stubs, with your own WP:OR editorialising. A lede just summarises an article and needless generality confuses the reader.
 * The "WP:OR editorialising" is not original at all, but comes from the reliable sources cited.
 * Schafer (1995) pp.72-73: The following celebrated theorem on quadratic forms permitting composition has been developed through the work of many authors, including Hurwitz, Dickson, Albert, Kaplansky and Jacobson. Schafer then states Theorem 3.25 (Hurwitz) for an algebra over a field of characteristic not 2.
 * Lam (2005) p.130: Theorem 5.10 (Hurwitz) again for a regular quadratic form over a general field (characteristic not 2 being assumed throughout the book).
 * Rajwade (2003) p.3: Actually Hurwitz proved this only over C but his proof generalises to any field K with char.K not 2.
 * I think that disposes of the notion that the statement of the theorem is drawn from sources that use the real and complex numbers. The sources I quote were taken  from the article Normed division algebra, the foundation of this version.  If these sources are content to state the theorem in algebraic generality, then so am I.  Deltahedron (talk) 17:09, 29 April 2013 (UTC)


 * We also have to issue that although "composition algebra" and "Hurwitz algebra" are bolded as synonyms, Hurwitz algebra redirects here but Composition algebra is an independent article, currently linked under See also, which defines the algebra over a general field. So there is some work to be done to disentangle the related but not identical concepts here.  Deltahedron (talk) 16:20, 28 April 2013 (UTC)
 * This issue is being discussed at Talk:Hurwitz algebra. Deltahedron (talk) 06:20, 29 April 2013 (UTC)

Composition algebras?
According to sources, "composition algebras" or "composition of quadratic forms" are terms in common usage. The title "normed divison algebra" can be found, but is not that common. Here are some sources that use the term composition algebra:



All of these are excellent sources. Another title for ths article could be "Hurwitz's theorem (composition of quadratic forms)". Normed division algebras is ambiguous, because of the Gelfand–Mazur theorem. The issue with the naming is a minor issue, as the content is what is important. If there are mutliple naming conventions, wikipedia can state that without a great song and dance. Creating the content is the main problem, not trivial naming matters. That would be like MOS debates where wikipedians get themselves tied up in knots without adding anything constructive to this encyclopedia. It is waste of time and can distract from the more important task of adding content. New content added to this article includes the proof of the classification theorem for Euclidean Hurwitz algebras, the proofs of the 1, 2, 4, 8 theorem of Eckmann and Chevalley and Fredenthal's diagonalization theorem as a tool for checking Jordan algebar axioms. Mathsci (talk) 19:18, 29 April 2013 (UTC)

L(a*) = L(a)* and similar
Something is missing to make sense with its right-hand side. $L(a)$ is an operator on algebra, not an element. One must first say how $O*$ is defined where $O$ is a unary operator. Incnis Mrsi (talk) 09:36, 6 May 2013 (UTC)
 * For complex numbers we can guess that $O* = * ∘ O ∘ * = λx. (O(x*))*$ is an intended definition, because it is the only possible combination of $O$ with involutions which keeps an operator complex-linear, which it must be so to satisfy the equality. Henceforth assuming this definition, we see that the equality fails for quaternions:


 * The right column is not a left multiplication to whatever quaternion, but is the right multiplication to $L(i*)1 = −i$. One should not necessarily be a genius of the algebra to realize that correct equalities are:
 * $(L(i)*)1 = (i 1*)* = i*$, expanded as $= −i$
 * $L(i*)i = 1$, expanded as $(L(i)*)i = (i (−i))* = 1*$
 * Something apparently is bad with the article. Incnis Mrsi (talk) 11:48, 6 May 2013 (UTC)
 * No you're just misreading, because you can check the pages in Faraut & Koranyi from which the proof is summarised. The material is from pages 81-91. Given an element of the algebra it defines an operators on the underlying finite-dimensional inner product space by L(a)b = ab. If your difficulties, my advice is not have a mathematical discussion, but for you to access the book and check directly with that.  That is how—even for mathematics articles—wikipedia is edited. I didn't invent these proofs nor could I. That would be original research. One place to access the book is http://en.bookfi.org . Mathsci (talk) 12:18, 6 May 2013 (UTC)

That's what I've just done anyway (adjoint was linked lower down). I explained the notation for an adjoint operator on a real i.p. space before stating the properties. (All the notation is standard in Jordan algebras from which this was detached. It's lucky I don't use the notation in Jacobson which involves indecipherable upper case Gothic letters.) The argument for quaternions above doesn't work for octonions since they are not associative. The argument in the text must be used: these properties are proved starting from the polarized version of the identity $= 1$:
 * The definition of the conjugate on a Euclidean Hurwitz algebra is given in the text as a* = 2(a,1) - a. (It's a reflection.) So for a quaternion (α + βi + γj + δk)* = α − βi  − γj − δk, the well-known formula. For the multiplication operators we use the inner product coming from the norm ||α + βi + γj + δk||2 = α2 + β2 + γ2 + δ2. It is (α1 + β1i + γ1j + δ1k,α2 + β2i + γ2j + δ2k) = α1α2 + β1 β2 + γ1γ2 + δ1δ2. In the notation of the text it is (a,b)=Re ab* = Re b*a. The adjoint of L(a) is defined with respect to this real inner product and turns out to be  L(a*) as indicated in the text. The statement and proof are Part (i) of Proposition V.1.2 half way down page 82 of Faraut & Koranyi. (They use overline for the * operation on the Euclidean Hurwitz algebra.) Perhaps what confused you is that there is an involution * on the algebra (as defined above) and an adjoint operation on real-linear operators on the underlying real inner product space.  I see no inner products in what you've written, so that seems to be where your error came from. So your second column seems to be where that has gone wrong. There is no inner product in sight, just some guesswork that isn't quite right and has nothing to do with what is written in the article or the source.  In the text there is an explicit page number (page 82). Please could you read the source instead of bothering me to act as a some kind of mathematics tutor? The argument is just transcribed from that text (with a slight change in notation because of the awkward way some mathematical notation is rendered on wikipedia). Here, for your peace of mind, is another way to see this: (L(a)b,c) = (ab,c) = Re c*ab = Re (a*c)*b = (b,a*c) =  (b, L(a*)c). Hence L(a)* = L(a*), by the definition of the adjoint on a finite-dimensional real inner product space. Mathsci (talk) 12:42, 6 May 2013 (UTC)
 * It is just confused me… oh cute. Surely there was nothing wrong with you who confusingly mapped both algebra involution and adjoint operator to the same symbol *? Faraut & Koranyi said literally L(u)* = L($\overline{u}$). Incnis Mrsi (talk) 13:09, 6 May 2013 (UTC)
 * As a bystander I just want to say I don't see anything wrong with math; true, maybe a minor notational change can help improving readability. For example, it doesn't hurt to link adjoint operator. (by the way, if you just follow the proof, it's pretty clear what was meant.) -- Taku (talk) 13:56, 6 May 2013 (UTC)


 * $$\displaystyle{2(a,b)(c,d)=(ac,bd) + (ad,bc).}$$

Setting $L(i*)j = −k$ or $(L(i)*)j = (i (−j))* = (−k)*$ yields $= k$ and $L(i*)k = j$.

To spell it out, set b = 1. Then 2(a,1) (c,d) = (ac,d) + (c,ad). So (L(a),cd) = (ac,d) = (c,(a – 2(a,1)1)d) = (c,L(a*)d). And so on. Mathsci (talk) 14:24, 6 May 2013 (UTC)

Euclidean Hurwitz algebra
I suggest that this section would be better placed at Composition algebra, since Euclidean Hurwitz algebra ⇒ Hurwitz algebra ⇒ composition algebra. Deltahedron (talk) 17:55, 3 February 2014 (UTC) Or possibly a standalone article, considering its length? Deltahedron (talk) 18:55, 3 February 2014 (UTC)
 * The confusion between composition algebras and “Euclidean” composition algebras, that remains from the times of user:Mathsci, should be healed. The former is an abstract structure over an arbitrary field. Remove from here all general facts about composition algebras that are not directly relevant to the proof. Note the problem in the lead: it suggests a confusing terminology with respect to composition algebras, instead of explaining that Euclidean composition algebras are composition algebras over reals with the supplementary positive definiteness condition.
 * What can you say about Euclidean composition algebras in general? Incnis Mrsi (talk) 22:00, 3 February 2014 (UTC)

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Not necessarily NON-associative?
Section "Definition" says: "a finite-dimensional not necessarily nonassociative algebra"... I do not say that this makes no sense, but probably "not necessarily associative" is meant (as in "Non-associative algebra":  'In other words, "non-associative" means "not necessarily associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings.'  Boris Tsirelson (talk) 09:43, 10 May 2017 (UTC)

Normed division algebra
Normed division algebra redirects to this article. I suggest the the title of this article be changed to it and that the theorem herein should be a topic in this article. — Preceding unsigned comment added by 108.41.98.105 (talk) 04:17, 17 May 2023 (UTC)

Discussion on Twitter
Some recent discussion which led me here and may have interesting material for others who understand it better: Tophtucker (talk) 15:26, 24 April 2024 (UTC)
 * https://x.com/slbrandin/status/1782845815423177081
 * https://x.com/rbehiel/status/1782938050991194213
 * https://x.com/getjonwithit/status/1783121616227995902
 * https://x.com/rbehiel/status/1783131428294893746