Editing Cayley–Dickson construction

{{Short description|Method for producing composition algebras}}
In [[mathematics]], the '''Cayley–Dickson construction''', sometimes also known as the '''Cayley–Dickson process''' or the '''Cayley–Dickson procedure''' produces a sequence of [[algebra over a field|algebras]] over the [[field (mathematics)|field]] of [[real number]]s, each with twice the [[dimension of a vector space|dimension]] of the previous one. It is named after [[Arthur Cayley]] and [[Leonard Eugene Dickson]]. The algebras produced by this process are known as '''Cayley–Dickson algebras''', for example [[complex number]]s, [[quaternion]]s, and [[octonion]]s. These examples are useful [[composition algebra]]s frequently applied in [[mathematical physics]].

The Cayley–Dickson construction defines a new algebra as a [[Cartesian product]] of an algebra with itself, with [[multiplication]] defined in a specific way (different from the [[componentwise operation|componentwise]] multiplication) and an [[involution (mathematics)|involution]] known as ''conjugation''.  The product of an element and its [[complex conjugate|conjugate]] (or sometimes the square root of this product) is called the [[norm (mathematics)|norm]].

The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing [[ordered field|order]], then [[commutativity]] of multiplication, [[associativity]] of multiplication, and finally [[alternativity]].

More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.<ref name=Sch66>{{citation | first=Richard D. | last=Schafer | year=1995 | orig-year=1966 | zbl=0145.25601 | title=An introduction to non-associative algebras | publisher=[[Dover Publications]] | isbn=0-486-68813-5 | url-access=registration | url=https://archive.org/details/introductiontono0000scha }}</ref>{{rp|45}}

[[Hurwitz's theorem (composition algebras)|Hurwitz's theorem]] states that the reals, complex numbers, quaternions, and octonions are the only finite-dimensional [[normed algebra|normed]] [[division algebra]]s over the real numbers, while [[Frobenius theorem (real division algebras)|Frobenius theorem]] states that the first three are the only finite-dimensional associative division algebras over the real numbers.

==Synopsis==

{| class="wikitable floatleft"
|+ Cayley–Dickson algebras properties
|-
! rowspan=2|[[Algebra over a field|Algebra]]
! rowspan=2|[[Dimension (vector space)|Dimension]] !! rowspan=2|[[Ordered field|Ordered]] !! colspan=4|[[Multiplication]] properties !! rowspan=2|Nontriv.<br>[[zero divisor|zero<br>divisors]]
|-
! [[Commutative]] !! [[Associative]] !! [[Alternative algebra|Alternative]] !! [[Power associativity|Power-assoc.]]
|-
! [[Real number|Real&nbsp;numbers]]
| style="text-align:center" | 1 || {{Yes}} || {{Yes}} || {{Yes}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Complex number|Complex&nbsp;num.]]
| style="text-align:center" | 2 || {{No}} || {{Yes}} || {{Yes}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Quaternion]]s
| style="text-align:center" | 4 || {{No}} || {{No}} || {{Yes}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Octonion]]s
| style="text-align:center" | 8 || {{No}} || {{No}} || {{No}} || {{Yes}} || {{Yes}} || {{No}}
|-
! [[Sedenion]]s
| style="text-align:center" | 16 || {{No}} || {{No}} || {{No}} || {{No}} || {{Yes}} || {{Yes}}
|-
! [[Trigintaduonion]]s<br>and higher
| style="text-align:center" | ≥ 32 || {{No}} || {{No}} || {{No}} || {{No}} || {{Yes}} || {{Yes}}
|}{{clear}}

The '''Cayley–Dickson construction''' is due to [[Leonard Dickson]] in 1919 showing how the [[octonion]]s can be constructed as a two-dimensional algebra over [[quaternion]]s. In fact, starting with a field ''F'', the construction yields a sequence of ''F''-algebras of dimension 2<sup>''n''</sup>. For ''n'' = 2 it is an associative algebra called a [[quaternion algebra]], and for ''n'' = 3 it is an [[alternative algebra]] called an [[octonion algebra]].  These instances ''n'' = 1, 2 and 3 produce [[composition algebra]]s as shown below. 

The case ''n'' = 1  starts with elements (''a'', ''b'') in ''F'' × ''F'' and defines the conjugate (''a'', ''b'')* to be (''a''*, –''b'') where ''a''* = ''a'' in case ''n'' = 1, and subsequently determined by the formula. The essence of the ''F''-algebra lies in the definition of  the product of two  elements (''a'', ''b'') and (''c'', ''d''):
:<math>(a,b) \times (c,d) = (ac - d^*b, da + bc^*).</math>
'''Proposition 1:''' For <math>z = (a,b)</math> and <math>w = (c,d),</math> the conjugate of the product is <math>w^*z^* = (zw)^*.</math>
:proof: <math>(c^*,-d)(a^*,-b) = (c^*a^* + b^*(-d), -bc^*-da) = (zw)^*.</math>
'''Proposition 2:''' If the ''F''-algebra is associative and <math>N(z) = zz^*</math>,then <math>N(zw) = N(z)N(w).</math>
:proof: <math>N(zw) = (ac-d^*b, da+bc^*)(c^*a^*-b^*d, -da -bc^*) = (aa^* + bb^*)(cc^* + dd^*)</math> + terms that cancel by the associative property.

==Stages in construction of real algebras==

Details of the construction of the classical real algebras are as follows:
=== Complex numbers as ordered pairs ===
{{Main|Complex number}}
The [[complex numbers]] can be written as [[ordered pair]]s {{math|(''a'', ''b'')}} of [[real number]]s {{mvar|a}} and {{mvar|b}}, with the addition operator being component-wise and with multiplication defined by

: <math>(a, b) (c, d) = (a c - b d, a d + b c).\,</math>

A complex number whose second component is zero is associated with a real number: the complex number {{math|(''a'', 0)}} is associated with the real number&nbsp;{{mvar|a}}.

The [[complex conjugate]] {{math|(''a'', ''b'')*}} of {{math|(''a'', ''b'')}} is given by

: <math>(a, b)^* = (a^*, -b) = (a, -b)</math>

since {{mvar|a}} is a real number and is its own conjugate.

The conjugate has the property that

: <math>(a, b)^* (a, b)  = (a a + b b, a b - b a) = \left(a^2 + b^2, 0\right),\,</math>

which is a non-negative real number.  In this way, conjugation defines a ''[[norm (mathematics)|norm]]'', making the complex numbers a [[normed vector space]] over the real numbers:  the norm of a complex number&nbsp;{{mvar|z}} is

: <math>|z| = \left(z^* z\right)^\frac12.\,</math>

Furthermore, for any non-zero complex number&nbsp;{{mvar|z}}, conjugation gives a [[inverse element|multiplicative inverse]],

: <math>z^{-1} = \frac{z^*}{|z|^2}.</math>

As a complex number consists of two independent real numbers, they form a two-dimensional [[vector space]] over the real numbers.

Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.

=== Quaternions ===
{{Main|Quaternion}}
[[File:Cayley_Q8_multiplication_graph.svg|thumb|link={{filepath:Cayley_Q8_multiplication_graph.svg}}|Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of ''i'' (red), ''j'' (green) and ''k'' (blue). In [{{filepath:Cayley_Q8_quaternion_multiplication_graph.svg}} the SVG file,] hover over or click a path to highlight it.]]

The next step in the construction is to generalize the multiplication and conjugation operations.

Form ordered pairs {{math|(''a'', ''b'')}} of complex numbers {{mvar|a}} and {{mvar|b}}, with multiplication defined by

: <math>(a, b) (c, d)  = (a c - d^* b, d a + b c^*).\,</math>

Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.

The order of the factors seems odd now, but will be important in the next step.

Define the conjugate {{math|(''a'', ''b'')*}} of {{math|(''a'', ''b'')}} by

: <math>(a, b)^* = (a^*, -b).\,</math>

These operators are direct extensions of their complex analogs:  if {{mvar|a}} and {{mvar|b}} are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.

The product of a nonzero element with its conjugate is a non-negative real number:

: <math>\begin{align} (a, b)^* (a, b) &= (a^*, -b) (a, b) \\ &= (a^* a + b^* b, b a^* - b a^*) \\ &= \left(|a|^2 + |b|^2, 0 \right).\, \end{align}</math>

As before, the conjugate thus yields a norm and an inverse for any such ordered pair.  So in the sense we explained above, these pairs constitute an algebra something like the real numbers.  They are the [[quaternion]]s, named by [[William Rowan Hamilton|Hamilton]] in 1843.

As a quaternion consists of two independent complex numbers, they form a four-dimensional vector space over the real numbers.

The multiplication of quaternions is not quite like the multiplication of real numbers, though; it is not [[commutative]] – that is, if {{mvar|p}} and {{mvar|q}} are quaternions, it is not always true that {{math|''pq'' {{=}} ''qp''}}.

=== Octonions ===
{{Main|Octonion}}
All the steps to create further algebras are the same from octonions onwards.

This time, form ordered pairs {{math|(''p'', ''q'')}} of quaternions {{mvar|p}} and {{mvar|q}}, with multiplication and conjugation defined exactly as for the quaternions:
: <math>(p, q) (r, s) = (p r - s^* q, s p + q r^*).\,</math>

Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were {{math|''r''*''q''}} rather than
{{math|''qr''*}}, the formula for multiplication of an element by its conjugate would not yield a real number.

For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.

This algebra was discovered by [[John T. Graves]] in 1843, and is called the [[octonion]]s or the "[[Arthur Cayley|Cayley]] numbers".<ref>{{Cite journal | last1 = Baez | first1 = John C. | author-link = John Baez| title = The Octonions | journal = Bulletin of the American Mathematical Society |series=New Series | volume = 39 | issue = 2 | pages = 145–205 | year = 2002 | url = http://math.ucr.edu/home/baez/octonions/ | doi = 10.1090/S0273-0979-01-00934-X | arxiv = math/0105155| mr = 1886087 | s2cid = 586512}}</ref>

As an octonion consists of two independent quaternions, they form an eight-dimensional vector space over the real numbers.

The multiplication of octonions is even stranger than that of quaternions; besides being non-commutative, it is not [[associative]] – that is, if {{mvar|p}}, {{mvar|q}}, and {{mvar|r}} are octonions, it is not always true that {{math|(''pq'')''r'' {{=}} ''p''(''qr'')}}.

For the reason of this non-associativity, octonions have [[Octonion#Properties|no matrix representation]].

=== Sedenions ===
{{main|Sedenion}}
The algebra immediately following the octonions is called the [[sedenion]]s.<ref>{{Cite journal | last1=Imaeda | first1=K. | last2=Imaeda | first2=M. | title=Sedenions: algebra and analysis | doi=10.1016/S0096-3003(99)00140-X | mr=1786945 | year=2000 | journal=Applied Mathematics and Computation   | volume=115 | issue=2 | pages=77–88}}</ref> It retains the algebraic property of [[power associativity]], meaning that if {{mvar|s}} is a sedenion, {{math|''s<sup>n</sup>s<sup>m</sup>'' {{=}} ''s''<sup>''n''&thinsp;+&thinsp;''m''</sup>}}, but loses the property of being an [[alternative algebra]] and hence cannot be a [[composition algebra]].

=== Trigintaduonions ===
{{main|Trigintaduonion}}
The algebra immediately following the [[sedenion]]s is the [[trigintaduonion]]s,<ref>{{cite web | title=Trigintaduonion | website=University of Waterloo | url=https://ece.uwaterloo.ca/~dwharder/Java/doc/ca/uwaterloo/alumni/dwharder/Numbers/Trigintaduonion.html | access-date=2024-10-08}}</ref><ref>{{cite arXiv | last1=Cawagas | first1=Raoul E. | last2=Carrascal | first2=Alexander S. | last3=Bautista | first3=Lincoln A. | last4=Maria | first4=John P. Sta. | last5=Urrutia | first5=Jackie D. | last6=Nobles | first6=Bernadeth | title=The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion) | date=2009 | class=math.RA | eprint=0907.2047v3}}</ref><ref>{{cite journal | last1=Cariow | first1=A. | last2=Cariowa | first2=G. | title=An algorithm for multiplication of trigintaduonions | journal=Journal of Theoretical and Applied Computer Science | volume=8 | issue=1 | date=2014 | issn=2299-2634 | pages=50–75 | url=https://www.infona.pl/resource/bwmeta1.element.baztech-0c72d8f6-14c2-4959-8640-58b568cfaa97 | access-date=2024-10-10}}</ref> which form a 32-[[dimension of a vector space|dimensional]] [[algebra over a field|algebra]] over the [[real number]]s<ref>{{cite journal | last1=Saini | first1=Kavita | last2=Raj | first2=Kuldip | title=On generalization for Tribonacci Trigintaduonions | journal=Indian Journal of Pure and Applied Mathematics | publisher=Springer Science and Business Media LLC | volume=52 | issue=2 | year=2021 | issn=0019-5588 | doi=10.1007/s13226-021-00067-y | pages=420–428}}</ref> and can be represented by [[blackboard bold]] <math>\mathbb T</math>.<ref>{{cite arXiv | last1=Cawagas | first1=Raoul E. | last2=Carrascal | first2=Alexander S. | last3=Bautista | first3=Lincoln A. | last4=Maria | first4=John P. Sta. | last5=Urrutia | first5=Jackie D. | last6=Nobles | first6=Bernadeth | title=The Subalgebra Structure of the Cayley-Dickson Algebra of Dimension 32 (trigintaduonion) | date=2009-07-12 | class=math.RA | eprint=0907.2047 }}</ref>

=== Further algebras ===
The Cayley–Dickson construction can be carried on ''[[ad infinitum]]'', at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. These include the 64-dimensional sexagintaquatronions (or 64-nions), the 128-dimensional centumduodetrigintanions (or 128-nions), the 256-dimensional ducentiquinquagintasexions (or 256-nions), and ''ad infinitum''.<ref>{{cite journal | last=Cariow | first=Aleksandr | title=An unified approach for developing rationalized algorithms for hypercomplex number multiplication | journal=Przegląd Elektrotechniczny | publisher=Wydawnictwo SIGMA-NOT | volume=1 | issue=2 | date=2015 | issn=0033-2097 | doi=10.15199/48.2015.02.09 | pages=38–41}}</ref> All the algebras generated in this way over a field are ''quadratic'': that is, each element satisfies a quadratic equation with coefficients from the field.<ref name=Sch66/>{{rp|50}}

In 1954, [[Richard D. Schafer|R. D. Schafer]] proved that the algebras generated by the Cayley–Dickson process over a field {{mvar|F}} satisfy the [[flexible identity]]. He also proved that any [[derivation algebra]] of a Cayley–Dickson algebra is isomorphic to the derivation algebra of Cayley numbers, a 14-dimensional [[Lie algebra]] over {{mvar|F}}.<ref>Richard D. Schafer (1954) "On the algebras formed by the Cayley–Dickson process", [[American Journal of Mathematics]] 76: 435–46 {{doi|10.2307/2372583}}</ref>

== Modified Cayley–Dickson construction ==
{{further|Split-complex number|Split-quaternion|Split-octonion|Bicomplex number|Biquaternion|Bioctonion}}
The Cayley–Dickson construction, starting from the real numbers <math>\mathbb R</math>, generates the [[composition algebra]]s <math>\mathbb C</math> (the [[complex number]]s), <math>\mathbb H</math> (the [[quaternion]]s), and <math>\mathbb O</math> (the [[octonion]]s). There are also composition algebras whose norm is an [[isotropic quadratic form]], which are obtained through a slight modification, by replacing the minus sign in the definition of the product of ordered pairs with a plus sign, as follows:
<math display="block">(a, b) (c, d) = (a c + d^* b, d a + b c^*).</math>

When this modified construction is applied to <math>\mathbb R</math>, one obtains the [[split-complex number]]s, which are [[ring isomorphism|ring-isomorphic]] to the [[direct product]] <math>\mathbb R \times \mathbb R;</math> following that, one obtains the [[split-quaternion]]s, an [[associative algebra]] [[isomorphic]] to that of the 2&thinsp;×&thinsp;2 real [[matrix (mathematics)|matrices]]; and the [[split-octonion]]s, which are isomorphic to {{math|[[Split-octonion#Zorn's vector-matrix algebra|Zorn('''R''')]]}}. Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.<ref>Kevin McCrimmon (2004) ''A Taste of Jordan Algebras'', pp 64, Universitext, Springer {{ISBN|0-387-95447-3}} {{mr|id=2014924}}</ref>

== General Cayley–Dickson construction ==
{{harvtxt|Albert|1942|p= 171}} gave a slight generalization, defining the product and involution on {{math|''B'' {{=}} ''A'' ⊕ ''A''}} for {{mvar|A}} an [[*-algebra|algebra with involution]] (with {{math|(''xy'')* {{=}} ''y''*''x''*}}) to be 
: <math>\begin{align} (p, q) (r, s) &= (p r - \gamma s^* q, s p + q r^*)\, \\ (p, q)^* &= (p^*, -q)\, \end{align}</math>
for {{mvar|γ}} an additive map that commutes with {{math|*}} and left and right multiplication by any element. (Over the reals all choices of {{mvar|γ}} are equivalent to −1, 0 or 1.) In this construction, {{mvar|A}} is an algebra with involution, meaning:
* {{mvar|A}} is an [[abelian group]] under {{math|+}}
* {{mvar|A}} has a product that is left and right [[distributive property|distributive]] over {{math|+}}
* {{mvar|A}} has an involution {{math|*}}, with {{math|(''x''*)* {{=}} ''x''}}, {{math|(''x'' + ''y'')* {{=}} ''x''* + ''y''*}},  {{math|(''xy'')* {{=}} ''y''*''x''*}}.
The algebra {{math|''B'' {{=}} ''A'' ⊕ ''A''}} produced by the Cayley–Dickson construction is also an algebra with involution.

{{mvar|B}} inherits properties from {{mvar|A}} unchanged as follows. 
* If {{mvar|A}} has an identity {{math|1<sub>''A''</sub>}}, then {{mvar|B}} has an identity {{math|(1<sub>''A''</sub>, 0)}}.
* If {{mvar|A}} has the property that {{math|''x'' + ''x''*}}, {{math|''xx''*}} associate and commute with all elements, then so does {{mvar|B}}. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.
 
Other properties of {{mvar|A}} only induce weaker properties of {{mvar|B}}:
* If {{mvar|A}} is commutative and has trivial involution, then {{mvar|B}} is commutative.
* If {{mvar|A}} is commutative and associative then {{mvar|B}} is associative.
* If {{mvar|A}} is associative and {{math|''x'' + ''x''*}}, {{math|''xx''*}} associate and commute with everything, then {{mvar|B}} is an [[alternative algebra]].

== Notes ==
{{reflist}}

== References ==
* {{Citation | last1=Albert | first1=A. A. | author1-link=Abraham Adrian Albert | title=Quadratic forms permitting composition | jstor=1968887 | mr=0006140  | year=1942 | journal=[[Annals of Mathematics]] |series=Second Series | volume=43 | pages=161–177 | doi=10.2307/1968887 | issue=1}} (see p.&nbsp;171)
* {{Citation | last1=Baez | first1=John | author1-link=John Baez | title=The Octonions | url=http://math.ucr.edu/home/baez/octonions/octonions.html | year=2002 | journal=[[Bulletin of the American Mathematical Society]]  | volume=39 | pages=145–205 | doi=10.1090/S0273-0979-01-00934-X | issue=2| arxiv=math/0105155 | s2cid=586512 }}. ''(See "[http://math.ucr.edu/home/baez/octonions/node5.html Section 2.2, The Cayley–Dickson Construction]")''
* {{Citation | last1=Dickson | first1=L. E. | author1-link=Leonard Dickson | title=On Quaternions and Their Generalization and the History of the Eight Square Theorem | jstor=1967865 | publisher=Annals of Mathematics | series=Second Series | year=1919 | journal=[[Annals of Mathematics]] | volume=20 | issue=3 | pages=155–171 | doi=10.2307/1967865}}
* {{cite journal |last1=Biss |first1=Daniel K. |last2=Christensen |first2=J. Daniel |last3=Dugger |first3=Daniel| last4=Isaksen |first4=Daniel C.|date=2007 |title=Large annihilators in Cayley–Dickson algebras II |journal=Boletin de la Sociedad Matematica Mexicana |volume=3 |pages=269–292 |arxiv=math/0702075|bibcode=2007math......2075B }}
* {{Citation | last1=Hamilton | first1=William Rowan | author1-link=William Rowan Hamilton | title=On Quaternions | url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/Quatern2.html | year=1847 | journal=Proceedings of the Royal Irish Academy | issn=1393-7197 | volume=3 | pages=1–16}}
* {{Citation | last1=Kantor | first1=I. L. | last2=Solodownikow | first2=A. S. | title=Hyperkomplexe Zahlen | publisher=B.G. Teubner | location=Leipzig | year=1978}} (the following reference gives the English translation of this book)
* {{Citation | last1=Kantor | first1=I. L. | last2=Solodovnikov | first2=A. S. | title=Hypercomplex numbers | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-96980-0 | mr=996029 | year=1989 | url-access=registration | url=https://archive.org/details/hypercomplexnumb0000kant }}
* {{cite book |first=Guy |last=Roos |chapter=Exceptional symmetric domains §1: Cayley algebras |title=Symmetries in Complex Analysis |publisher=[[American Mathematical Society]] |year=2008 |isbn=978-0-8218-4459-5 |editor-first=Bruce |editor-last=Gilligan |editor2-first=Guy |editor2-last=Roos |volume=468 |series=Contemporary Mathematics}}

==Further reading==
* {{cite journal |first1=Jamil |last1=Daboul |first2=Robert |last2=Delbourgo |title=Matrix representations of octonions and generalizations |journal=[[Journal of Mathematical Physics]] |volume=40 |pages=4134–50 |year=1999 |issue=8 |doi=10.1063/1.532950 |arxiv=hep-th/9906065 |bibcode=1999JMP....40.4134D |s2cid=16932871 }}

{{Number systems}}
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[[Category:Composition algebras]]
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[[Category:Hypercomplex numbers]]