Cayley–Dickson construction

In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, 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. They are useful composition algebras frequently applied in mathematical physics.

The Cayley–Dickson construction defines a new algebra similar to the direct sum of an algebra with itself, with multiplication defined in a specific way (different from the multiplication provided by the genuine direct sum) 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, and next associativity of multiplication.

More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.^[1]

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 over the real numbers.

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.

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

$(a, b) (c, d) = (a c - d^* b, d a + b c^*).\,$

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 real number:

$(a, b)^* (a, b) = (a^*, -b) (a, b) = (a^* a + b^* b, b a^* - b a^*) = (|a|^2 + |b|^2, 0 ).\,$

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 over the real numbers.

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 always true that $p q = q p$, but it is true that $p q = (q p)'$, where $(a, b)' = (a, -b)$.

Octonions

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

This time, form ordered pairs $(p, q)$ of quaternions $p$ and $q$, with multiplication and conjugation defined exactly as for the quaternions:

$(p, q) (r, s) = (p r - s^* q, s p + q r^*).\,$

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 $qr^*$, the formula for multiplication of an element by its conjugate would not 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 over the real numbers.

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 not always true that

$(p q) r = p (q r).\$

For the reason of this non-associativity, octonions have no matrix representation.

Further algebras

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 the algebra of the preceding step. 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.^[2]

Modified Cayley–Dickson construction

The Cayley–Dickson construction, starting from the reals, generates all of the remaining Euclidean composition algebras. The remaining anisotropic composition algebras can be 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:

$(a, b) (c, d) = (a c + d^* b, d a + b c^*).\,$

When this modified construction is applied to the reals, one obtains the split-complex numbers, which are ring-isomorphic to the direct sum R ⊕ R (also written ²R); following that, one obtains the split-quaternions, isomorphic to M₂(R), and the split-octonions Zorn(R). Applying the original Cayley-Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.^[3]

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

$(p, q) (r, s) = (p r - \gamma s^* q, s p + q r^*)\,$

$(p, q)^* = (p^*, -q)\$

for γ an additive map that commutes with * and left and right multiplication by any element. (Over the reals all choices of γ are equivalent to −1, 0 or 1.) In this construction, A is an algebra with involution, meaning:

• A is an abelian group under +
• A has a product that is left and right distributive over +
• A has an involution *, with x** = x, (x + y)* = x* + y*, (xy)*  = y*x*.

The algebra B=A⊕A produced by the Cayley–Dickson construction is also an algebra with involution.

B inherits properties from A unchanged as follows.

• If A has an identity 1_A, then B has an identity (1_A, 0).
• If A has the property that x + x^*, xx^* associate and commute with all elements, then so does B. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.

Other properties of A only induce weaker properties of B:

• If A is commutative and has trivial involution, then B is commutative.
• If A is commutative and associative then B is associative.
• If A is associative and x + x^*, xx^* associate and commute with everything, then B is an alternative algebra.

Notes

1. Schafer (1995) p.45
2. Schafer (1995) p.50
3. Kevin McCrimmon (2004) A Taste of Jordan Algebras, pp 64, Universitext, Springer ISBN 0-387-95447-3 MR 2014924

References

• Albert, A. A. (1942), "Quadratic forms permitting composition", Annals of Mathematics. Second Series 43 (1): 161–177, doi:10.2307/1968887, ISSN 0003-486X, JSTOR 1968887, MR 0006140 (see p. 171)
• Baez, John (2002), "The Octonions", Bulletin of the American Mathematical Society 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X, ISSN 0002-9904. (See "Section 2.2, The Cayley-Dickson Construction")
• Dickson, L. E. (1919), "On Quaternions and Their Generalization and the History of the Eight Square Theorem", Annals of Mathematics, Second Series (Annals of Mathematics) 20 (3): 155–171, doi:10.2307/1967865, ISSN 0003-486X, JSTOR 1967865
• Kantor, I. L.; Solodownikow, A. S. (1978), Hyperkomplexe Zahlen, Leipzig: B.G. Teubner
• Hamilton, William Rowan (1847), "On Quaternions", Proceedings of the Royal Irish Academy 3: 1–16, ISSN 1393-7197
• Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society, ISBN 978-0-8218-4459-5 .
• Schafer, Richard D. (1995) [1966], An introduction to non-associative algebras, Dover Publications, ISBN 0-486-68813-5, Zbl 0145.25601

External links

• Hyperjeff, Sketching the History of Hypercomplex Numbers (1996–2006).