What is Cayley-Dickson construction

[Greg Bernhardt]

Definition/Summary

The Cayley-Dickson construction is a way of getting algebras from other algebras. The starting algebra must have addition, multiplication, and conjugation defined for it, and the new algebra has pairs of elements of the starting algebra with appropriately defined operations on those pairs.

Starting with the real numbers, one finds the complex numbers, the quaternions, the octonions, the sedenions, and higher 2^n-ions. From the real numbers to the sedenions, one loses properties, though one does not lose properties after the sedenions.

Equations

The starting algebra A must have these operations:

Scalars: real numbers
Addition (+): A is an abelian group over +
Multiplication (.): distributive over +
Conjugation (*): satisfies (a*)* = a, (a + b)* = (a*) + (b*), (a.b)* = (b*).(a*)

The resulting algebra B has elements (a,b), with a,b in A, that satisfy these versions of addition, multiplication, and conjugation:
$(a_0,a_1) + (b_0,b_1) = (a_0+b_0,a_1+b_1)$
$(a_0,a_1)\cdot(b_0,b_1) = (a_0 b_0 - b_1 a_1^*, a_0^* b_1 + b_0 a_1)$ (Baez)
$(a_0,a_1)\cdot(b_0,b_1) = (a_0 b_0 - b_1^* a_1, b_1 a_0 + a_1 b_0^*)$ (Wikipedia)
$(a_0,a_1)^* = (a_0^*,-a_1)$

One can define a norm
$||a|| = a a^*$
and if it is nonzero for element a, a multiplicative inverse
$a^{-1} = \frac{a^*}{||a||}$

Also real and imaginary parts
$\text{Re } a = \frac12(a + a^*)$
$\text{Im } a = \frac12(a - a^*)$

Extended explanation

To illustrate the Cayley-Dickson construction in action, start with the real numbers and repeat that construction on it. For real numbers, conjugation is the identity operation, x^* = x. One finds:

• R: real numbers (1)
• C: complex numbers (2)
• H: quaternions (4)
• O: octonions (8)
• S: sedenions (16)
• Higher 2ⁿ-ions

Properties:

• Additive identity: all components zero
• Multiplicative identity: real part 1, imaginary part 0
• Self-conjugate: R
• Multiplication commutative: R, C
• Multiplication associative: R, C, H
• Multiplication alternative: R, C, H, O
• Normed multiplication: R, C, H, O
• Zero divisors: S, higher
• Multiplication power-associative: all

Notes:

• Matrix representations are only possible when multiplication is associative. Thus, the octonions and higher -ions do not have them.
• The alternative property is two-element associativity: a.(b.b) = (a.b).b, b.(a.b) = (b.a).b, b.(b.a) = (b.b).a
• Normed multiplication: ||a.b|| = ||a||.||b||
• Zero divisors are nonzero numbers which yield zero when multiplied by certain other nonzero numbers
• Power-associativity is forming a power of an element being independent of the order of multiplication

Power-associativity can be demonstrated from the construction of a power of an element. Let x have real part x_r and imaginary part x_i. Then for integer n >= 0,
$x^n = P(n,x_r,||x_i||) + x_i Q(n,x_r,||x_i||)$

where P and Q satisfy the addition laws
$P(n_1+n_2,a,b) = P(n_1,a,b) P(n_2,a,b) - b Q(n_1,a,b) Q(n_2,a,b)$
$Q(n_1+n_2,a,b) = P(n_1,a,b) Q(n_2,a,b) + Q(n_1,a,b) P(n_2,a,b)$

The norm satisfies
$||x^n|| = ||x||^n$
and the inverse power is
$x^{-n} = (P(n,x_r,||x_i||) - x_i Q(n,x_r,||x_i||))/||x||^n$

The first few values:
P(0,a,b) = 1, Q(0,a,b) = 0
P(1,a,b) = a, Q(1,a,b) = 1
P(2,a,b) = a² - b, Q(2,a,b) = 2a
P(3,a,b) = a³ - 3ab, Q(3,a,b) = 3a² - b

Automorphism groups:

• R: identity group
• C: Z(2) - conjugation
• H: SO(3) - rotation of the imaginary part
• O and higher: G2, the smallest exceptional Lie group

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

There are countless other ways to get new algebras from given ones:
Clifford algebras, Lie algebras, Jordan algebras, Tensor algebras, Grassmann algebras, universal envelopping algebras, etc.

It is told that it took Hamilton ten years to recognize, that there is no field extension of degree three over the reals and to find the quaternions - during a walk in the park on a small bridge.