- Cayley–Dickson construction
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
complex numberscan be written as ordered pairs of real numbers and , with the addition operator being component-by-component and with multiplication defined by
A complex number whose second component is zero is associated with a real number: the complex number is the real number .
Another important operation on complex numbers is conjugation. The conjugate of is given by
The conjugate has the property that
which is a non-negative real number. In this way, conjugation defines a "norm", making the complex numbers a
normed vector spaceover the real numbers: the norm of a complex number is
Furthermore, for any nonzero complex number , conjugation gives a multiplicative inverse,
In as much as complex numbers consist of two independent real numbers, they form a 2-dimensional
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 of complex numbers and , with multiplication defined by
: [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 of by
These operators are direct extensions of their complex analogs: if and 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:
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
The multiplication of quaternions is not quite like the multiplication of real numbers, though. It is not
commutative, that is, if and are quaternions, it is not generally true that .
Yet another step: the octonions
From now on, all the steps will look the same.
This time, form ordered pairs ofquaternions and , 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 rather than, 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. Gravesin 1843, and is called the octonionsor 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 , , and are octonions, it is generally not true that .
And so forth
The algebra immediately following the octonions is called the
sedenions. It retains an algebraic property called power associativity, meaning that if is a sedenion, , but loses the property of being an alternative algebraand 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.
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.
*. "(See " [http://math.ucr.edu/home/baez/octonions/node5.html Section 2.2, The Cayley-Dickson Construction] ")"
* Hyperjeff, " [http://history.hyperjeff.net/hypercomplex.html Sketching the History of Hypercomplex Numbers] " (1996-2006).
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