Cayley–Dickson construction
In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one.
It is named after Arthur Cayley and Leonard Eugene Dickson.
These examples are useful composition algebras 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 multiplication) and an involution known as conjugation.
The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multiplication, and finally alternativity.
[1]: 45 Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed) division algebras (over the real numbers).
In fact, starting with a field F, the construction yields a sequence of F-algebras of dimension 2n.
The essence of the F-algebra lies in the definition of the product of two elements (a, b) and (c, d): Proposition 1: For
The next step in the construction is to generalize the multiplication and conjugation operations.
Form ordered pairs (a, b) of complex numbers a and b, with multiplication defined by Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.
So in the sense we explained above, these pairs constitute an algebra something like the real numbers.
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".
[2] As an octonion consists of two independent quaternions, they form an eight-dimensional vector space over the real numbers.
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.
[1]: 50 In 1954, R. D. Schafer proved that the algebras generated by the Cayley–Dickson process over a field F satisfy the flexible identity.
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:
, one obtains the split-complex numbers, which are ring-isomorphic to the direct product
Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.
[11] Albert (1942, p. 171) gave a slight generalization, defining the product and involution on B = A ⊕ A for A an algebra with involution (with (xy)* = y*x*) to be for γ an additive map that commutes with * and left and right multiplication by any element.
