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Cayley-Dickson construction

In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley-Dickson algebras; since they extend the complex numbers, they are hypercomplex numbers.

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.

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Complex numbers as ordered pairs

The complex numbers can be written as ordered pairs <math>(a, b)</math> of real numbers <math>a</math> and <math>b</math>, with the addition operator being component-by-component 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)</math> is the real number <math>a</math>.

Another important operation on complex numbers is conjugation. The conjugate <math>(a, b)^*</math> of <math>(a, b)</math> is given by

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

The conjugate has the property that

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

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 <math>z</math> is

<math>|z| = (z^* z)^{1/2}.</math>

Furthermore, for any nonzero complex number <math>z</math>, conjugation gives a multiplicative inverse,

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

Inasmuch 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 <math>(a, b)</math> of complex numbers <math>a</math> and <math>b</math>, with multiplication defined by

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

The order of the factors seems odd now, but will be important in the next step. Define the conjugate <math>(a, b)^*</math> of <math>(a, b)</math> by

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

These operators are direct extensions of their complex analogs: if <math>a</math> and <math>b</math> 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:

<math>(a, b)^* (a, b)
  = (a^*, -b) (a, b)
  = (a^* a + b b^*, a b - a b)
  = (|a|^2 + |b|^2, 0 ).</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 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 isn't quite like the multiplication of real numbers, though. It isn't commutative, that is, if <math>q</math> and <math>q</math> are quaternions, it isn't generally true that <math>p q = q p</math>.

Yet another step: the octonions

From now on, all the steps will look the same.

This time, form ordered pairs <math>(p, q)</math> of quaternions <math>p</math> and <math>q</math>, 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 <math>b c</math> rather than <math>c b</math>, the formula for the 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 Graves[?] in 1844, and is called the octonions 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 isn't associative, that is, if <math>p</math>, <math>q</math>, and <math>r</math> are octonions, it isn't generally true that <math>(p q) r = p (q r)</math>.

And so forth

The algebra immediately following the octonions is called the sedenions. It retains an algebraic property called power associativity, meaning that if <math>s</math> is a sedenion, <math>s^n s^m = s^{n + m}</math>, but loses the property of being an alternative algebra.

The Cayley-Dickson construction can be carried on ad infinitum, at each step producing an algebra whose dimension is double that of algebra of the preceding step.

After the octonions, though, the algebras even contain zero divisors, that is, if <math>p</math> and <math>q</math> are elements of one of these algebras, then <math>p q = 0</math> no longer implies <math>p = 0</math> or <math>q = 0</math>.

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