Incnis Mrsi (talk | contribs) →L(a*) = L(a)* and similar: new section |
Incnis Mrsi (talk | contribs) →L(a*) = L(a)* and similar: Something apparently is bad with the article |
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== ''L''(''a''*) = ''L''(''a'')* and similar == |
== ''L''(''a''*) = ''L''(''a'')* and similar == |
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Something is missing to make sense with its right-hand side. {{math|''L''(''a'')}} is an ''operator'' on algebra, not an element. One must first say how {{math|''O''*}} is defined where {{mvar|O}} is a unary operator. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 09:36, 6 May 2013 (UTC) |
Something [[Special:PermanentLink/553599562#Definition|is missing]] to make sense with its right-hand side. {{math|''L''(''a'')}} is an ''operator'' on algebra, not an element. One must first say how {{math|''O''*}} is defined where {{mvar|O}} is a unary operator. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 09:36, 6 May 2013 (UTC) |
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: For complex numbers we can guess that {{math|1=''O''* = * [[function composition|∘]] ''O'' ∘ * = [[abstraction operator|λ''x''.]] (''O''(''x''*))*}} is an intended definition, because it is the only possible combination of {{mvar|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 [[quaternion]]s: |
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{| align=center cellpadding=6px |
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| style="width:16em" |{{math|1=''L''(''i''*)1 = −''i''}} |
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|{{math|1=(''L''(''i'')*)1 = (''i'' 1*)* = ''i''*}}||{{math|1== −''i''}} |
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|- |
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|{{math|1=''L''(''i''*)''i'' = 1}} |
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|{{math|1=(''L''(''i'')*)''i'' = (''i'' (−''i''))* = 1*}}||{{math|1== 1}} |
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|- bgcolor=#FF9999 |
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|{{math|1=''L''(''i''*)''j'' = −''k''}} |
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|{{math|1=(''L''(''i'')*)''j'' = (''i'' (−''j''))* = (−''k'')*}}||{{math|1== ''k''}} |
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|- bgcolor=#FF9999 |
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|{{math|1=''L''(''i''*)''k'' = ''j''}} |
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|{{math|1=(''L''(''i'')*)''k'' = (''i'' (−''k''))* = ''j''*}}||{{math|1== −''j''}} |
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|} |
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: The right column is not a left multiplication to whatever quaternion, but is the ''right'' multiplication to {{math|−''i''}}. One should not necessarily be a genius of the algebra to realize that '''correct''' equalities are: |
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:: {{math|1=''R''(''a''*) = ''L''(''a'')*}}, expanded as {{math|1=''b'' ''a''* = (''a'' ''b''*)*}} |
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:: {{math|1=''L''(''a''*) = ''R''(''a'')*}}, expanded as {{math|1=''a''* ''b'' = (''b''* ''a'')*}} |
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: Something apparently is bad with the article. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 11:48, 6 May 2013 (UTC) |
Revision as of 11:48, 6 May 2013
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 MR0347870, published notes suitable for undergraduates. It was originally in Russian and was translated into German MR0485981 and English MR996029. 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)
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 [1]. In a series of edits [2] 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 [3] 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)
- The reasons for that reversion appear elsewhere, the relevant material being:
- 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 [1]. In a series of edits [2] 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)
- 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:
- Faraut, J.; Koranyi, A. (1994), Analysis on symmetric cones, Oxford Mathematical Monographs, Oxford University Press, ISBN: 0198534779
- Springer, T. A.; Veldkamp, F. D. (2000), Octonions, Jordan Algebras and Exceptional Groups, Springer-Verlag, ISBN 3-540-66337-1
- Jacobson, N. (1968), Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, vol. 39, American Mathematical Society
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:
L(i*)1 = −i | (L(i)*)1 = (i 1*)* = i* | = −i |
L(i*)i = 1 | (L(i)*)i = (i (−i))* = 1* | = 1 |
L(i*)j = −k | (L(i)*)j = (i (−j))* = (−k)* | = k |
L(i*)k = j | (L(i)*)k = (i (−k))* = j* | = −j |
- The right column is not a left multiplication to whatever quaternion, but is the right multiplication to −i. One should not necessarily be a genius of the algebra to realize that correct equalities are:
- R(a*) = L(a)*, expanded as b a* = (a b*)*
- L(a*) = R(a)*, expanded as a* b = (b* a)*
- Something apparently is bad with the article. Incnis Mrsi (talk) 11:48, 6 May 2013 (UTC)