- Cayley–Dickson construction
In

mathematics , the**Cayley–Dickson construction**produces a sequence of algebras over the field ofreal number s, each with twice the dimension of the previous one. The algebras produced by this process are known as**Cayley–Dickson algebras**; since they extend thecomplex number s, they arehypercomplex number s.These algebras all have a notion of norm and conjugate, with the general idea being that the product of an element and its conjugate should equal the square of its norm.

The surprise is that for the first several steps, besides having a higher dimensionality, the next algebra loses a specific algebraic property.

**Complex numbers as ordered pairs**The

complex numbers can be written asordered pair s $(a,\; b)$ ofreal number s $a$ and $b$, with the addition operator being component-by-component and with multiplication defined by: $(a,\; b)\; (c,\; d)\; =\; (a\; c\; -\; b\; d,\; a\; d\; +\; b\; c).,$

A complex number whose second component is zero is associated with a real number: the complex number $(a,\; 0)$ is the real number $a$.

Another important operation on complex numbers is conjugation. The conjugate $(a,\; b)^*,$ of $(a,\; b)$ is given by

: $(a,\; b)^*\; =\; (a,\; -b).,$

The conjugate has the property that

: $(a,\; b)^*\; (a,\; b)\; =\; (a\; a\; +\; b\; b,\; a\; b\; -\; b\; a)\; =\; (a^2\; +\; b^2,\; 0),,$

which is a non-negative real number. In this way, conjugation defines a "norm", making the complex numbers a

normed vector space over the real numbers: the norm of a complex number $z$ is: $|z|\; =\; (z^*\; z)^\{1/2\}.,$

Furthermore, for any nonzero complex number $z$, conjugation gives a multiplicative inverse,

: $z^\{-1\}\; =\; \{z^*\; /\; |z|^2\}.,$

In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional

vector space .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.

**Another step: the quaternions**The next step in the construction is to generalize the multiplication and conjugation operations. What to do is easy, if not quite obvious.

Form ordered pairs $(a,\; b)$ of complex numbers $a$ and $b$, with multiplication defined by

: $(a,\; b)\; (c,\; d)\; =\; (a\; c\; -\; d^*\; b,\; d\; a\; +\; b\; c^*).,$ [

*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 $(a,\; b)^*,$ of $(a,\; b)$ by

: $(a,\; b)^*\; =\; (a^*,\; -b).,$

These operators are direct extensions of their complex analogs: if $a$ and $b$ 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 an element with its conjugate is a non-negative number:

: $(a,\; b)^*\; (a,\; b)\; =\; (a^*,\; -b)\; (a,\; b)\; =\; (a^*\; a\; +\; b\; b^*,\; a\; b\; -\; a\; b)\; =\; (|a|^2\; +\; |b|^2,\; 0\; ).,$

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

quaternions , named by Hamilton in 1843.Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional

vector space .The multiplication of quaternions is not quite like the multiplication of real numbers, though. It is not

commutative , that is, if $p$ and $q$ are quaternions, it is not generally true that $p\; q\; =\; q\; p$.**Yet another step: the octonions**From now on, all the steps will look the same.

This time, form ordered pairs $(p,\; q)$ ofquaternions $p$ and $q$, with multiplication and conjugation defined exactly as for the quaternions.

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 $c^*b$ rather than$bc^*$, the formula for multiplication of an element by its conjugate wouldn't 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 theoctonions or the "Cayley numbers".Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space.

The multiplication of octonions is even stranger than that of quaternions. Besides being non-commutative, it is not

associative , that is, if $p$, $q$, and $r$ are octonions, it is generally not true that $(p\; q)\; r\; =\; p\; (q\; r)$.**And so forth**The algebra immediately following the octonions is called the

sedenions . It retains an algebraic property calledpower associativity , meaning that if $s$ is a sedenion, $s^n\; s^m\; =\; s^\{n\; +\; m\}$, but loses the property of being analternative algebra and hence cannot be a composition algebra.The Cayley-Dickson construction can be carried on "ad infinitum", at each step producing a power-associative algebra whose dimension is double that of algebra of the preceding step.

**Schafer's extension**The process was generalized by R. D. Schafer ("Amer. J. Math."

**76**, (1954), 435–446) to allow new elements that square to +1 instead of −1. Lounesto, "Clifford Algebras and Spinors", p. 285, uses CD(−1, −1, ...) to describe the standard process, and replaces the −1's by +1 where appropriate.Starting with reals, CD(+1) gives the so-called

split-complex number s "C"ℓ_{1,0}(**R**); CD(+1, −1) gives split-quaternions, which turn out to be thecoquaternion s, "C"ℓ_{1,1}(**R**) ≈ "C"ℓ_{"2","0"}(**R**); CD(+1, −1, −1) givessplit-octonion s and so on.Other combinations, such as CD(−1, +1, +1, ...), give other series of algebras describing 2, 4, 8, ... complex planes.

**ee also***

split-complex number **Notes****References***

*

*. "(See " [*http://math.ucr.edu/home/baez/octonions/node5.html Section 2.2, The Cayley-Dickson Construction*] ")"**External links*** Hyperjeff, " [

*http://history.hyperjeff.net/hypercomplex.html Sketching the History of Hypercomplex Numbers*] " (1996-2006).

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