Cayley–Dickson construction

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.

Complex numbers as ordered pairs

The complex numbers can be written as ordered pairs (a, b) of real numbers 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 thanbc^*, 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 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 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 called power associativity, meaning that if s is a sedenion, s^n s^m = s^{n + m}, but loses the property of being an alternative 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 numbers "C"ℓ1,0(R); CD(+1, −1) gives split-quaternions, which turn out to be the coquaternions, "C"ℓ1,1(R) ≈ "C"ℓ"2","0"(R); CD(+1, −1, −1) gives split-octonions 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



*. "(See " [ Section 2.2, The Cayley-Dickson Construction] ")"

External links

* Hyperjeff, " [ Sketching the History of Hypercomplex Numbers] " (1996-2006).

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Construcción de Cayley-Dickson — En matemáticas, la construcción de Cayley Dickson produce una secuencia de álgebras sobre el cuerpo de los números reales, cada una con dimensión doble que la anterior. Las álgebras producidas por este proceso son conocidas como álgebras de… …   Wikipedia Español

  • Arthur Cayley — Pour les articles homonymes, voir Cayley.  Ne doit pas être confondu avec Arthur Caley. Arthur Cayley …   Wikipédia en Français

  • Arthur Cayley — Infobox Scientist name = Arthur Cayley |242px image width = 242px caption = Portrait in London by Barraud Jerrard birth date = birth date|1821|8|16|mf=y birth place = Richmond, Surrey, UK residence = England nationality = British death date =… …   Wikipedia

  • Leonard Eugene Dickson — Pour les articles homonymes, voir Dickson. Leonard Eugene Dickson (22 janvier 1874 à Independence (Iowa) 17 janvier 1954 à Harlingen (Texas)) est un mathématicien américain, spécialiste en théorie des nombres et en algèbre.… …   Wikipédia en Français

  • Octonion — In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold . There are only four such algebras, the other three being the real numbers R, the …   Wikipedia

  • Hypercomplex number — The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic.Hypercomplex numbers have had a long lineage of devotees including Hermann Hankel, Georg Frobenius, Eduard… …   Wikipedia

  • Sedenion — In abstract algebra, sedenions form a 16 dimensional algebra over the reals. The set of sedenions is denoted as mathbb{S}. Two types are currently known: # Sedenions obtained by applying the Cayley Dickson construction # Conic sedenions ( 16… …   Wikipedia

  • Octonion déployé — En mathématiques, les octonions déployés ou octonions fendus sont une extension non associative des quaternions (ou des coquaternions). Ils diffèrent des octonions par la signature de la forme quadratique : les octonions déployés ont une… …   Wikipédia en Français

  • Complex number — A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the square root of –1. A complex… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.