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. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. 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 product of an element and its conjugate (or sometimes the square root of this product) is called the norm.
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.
More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.
Hurwitz's theorem states that the reals, complex numbers, quaternions, and octonions are the only finite-dimensional normed division algebras over the real numbers, while the Frobenius theorem states that the first three are the only finite-dimensional associative division algebras over the real numbers.
Contents
Synopsis
The Cayley–Dickson construction is due to Leonard Dickson in 1919 showing how the octonions can be constructed as a two-dimensional algebra over quaternions. In fact, starting with a field F, the construction yields a sequence of F-algebras of dimension 2n. For n = 2 it is an associative algebra called a quaternion algebra, and for n = 3 it is an alternative algebra called an octonion algebra. These instances n = 1, 2 and 3 produce composition algebras as shown below.
The case n = 1 starts with elements (a, b) in F × F and defines the conjugate (a, b)* to be (a*, –b) where a* = a in case n = 1, and subsequently determined by the formula. The essence of the F-algebra lies in the definition of the product of two elements (a, b) and (c, d):
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Stages in construction of real algebras
Details of the construction of the classical real algebras are as follows:
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-wise and with multiplication defined by
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{\displaystyle (a,b)(c,d)=(ac-bd,ad+bc).\,}
A complex number whose second component is zero is associated with a real number: the complex number (a, 0) is associated with the real number a.
The complex conjugate (a, b)* of (a, b) is given by
(
Quaternions
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
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{\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}).\,}
Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.
Octonions
All the steps to create further algebras are the same from octonions onwards.
This time, form ordered pairs (p, q) of quaternions p and q, with multiplication and conjugation defined exactly as for the quaternions:
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{\displaystyle (p,q)(r,s)=(pr-s^{*}q,sp+qr^{*}).\,}
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 r*q rather than
Sedenions
The algebra immediately following the octonions is called the sedenions. It retains the algebraic property of power associativity, meaning that if s is a sedenion, snsm = sn + m, but loses the property of being an alternative algebra and hence cannot be a composition algebra. It is also at this point that the algebras formed by the Cayley-Dickson construction begin to have nontrivial zero divisors, in that this and every further algebra created by the construction will have pairs of nonzero values (for example,
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{\displaystyle (e_{3}+e_{10})}
and
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{\displaystyle (e_{6}-e_{15})}
) which when multiplied give
0
{\displaystyle 0}
.
Trigintaduonions
The algebra immediately following the sedenions is the trigintaduonions, which form a 32-dimensional algebra over the real numbers and can be represented by blackboard bold
T
{\displaystyle \mathbb {T} }
.
Further algebras
The Cayley–Dickson construction can be carried on ad infinitum, at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. 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. All the algebras generated in this way over a field are quadratic: that is, each element satisfies a quadratic equation with coefficients from the field.
In 1954, R. D. Schafer proved that the algebras generated by the Cayley–Dickson process over a field F satisfy the flexible identity. He also proved that any derivation algebra of a Cayley–Dickson algebra is isomorphic to the derivation algebra of Cayley numbers, a 14-dimensional Lie algebra over F.
Modified Cayley–Dickson construction
The Cayley–Dickson construction, starting from the real numbers
R
{\displaystyle \mathbb {R} }
, generates the composition algebras
C
{\displaystyle \mathbb {C} }
(the complex numbers),
H
{\displaystyle \mathbb {H} }
(the quaternions), and
O
{\displaystyle \mathbb {O} }
(the octonions). 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:
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General Cayley–Dickson construction
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
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