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Cayley-Dickson construction

Cayley–Dickson construction

Idea

Consider how the complex numbers are formed from the real numbers. If you generalise this carefully, then you can perform this operation again to get the quaternions, and so on. This operation is the Cayley–Dickson construction.

Definitions

Let AA be an possibly nonassociative **-algebra over the field \mathbb{R} of real numbers: an algebra equipped with an involution xx¯x\mapsto \bar{x} which is an antiautomorphism. (Actually, \mathbb{R} could be replaced by any commutative ring in the definitions, although some properties may depend on this ring.)

Then one defines a new algebra, the Cayley–Dickson double A 2A^2 of AA which is the direct sum AAA \oplus A as an \mathbb{R}-vector space, and with the multiplication rule given by

(a,b)(u,v)(auv¯b,bu¯+va), (a,b)\cdot(u,v) \coloneqq (a u - \bar{v} b, b \bar{u} + v a),

and the formula

(a,b)¯:=(a¯,b) \widebar{(a,b)} := (\bar{a},-b)

defines an involutive antiautomorphism on A 2A^2, so the doubling procedure can be iterated.

The map a(a,0)a\mapsto (a,0) is a monomorphism AA 2A\to A^2. If AA is unital with unit 11 then A 2A^2 is unital with unit (1,0)(1,0). In the unital case, the element i(0,1)\mathrm{i} \coloneqq (0,1) has the property i 2=1(1,0)\mathrm{i}^2 = -1 \coloneqq (-1,0), and we may write (a,b)(a,b) as a+bia + b \mathrm{i} (while a+ib=(a,b¯)a + \mathrm{i} b = (a,\bar{b})). For this reason, we may write A[i]A[\mathrm{i}] in place of A 2A^2, at least when AA is unital.

Properties

Generally speaking, the double A 2A^2 of an algebra AA has a nice property iff AA is one level nicer. For simplicity, assume that AA is unital (so that \mathbb{R} is a subalgebra). Since i¯=i\bar{\mathrm{i}} = -\mathrm{i}, we see that the involution on A 2A^2 is trivial iff the involution on AA is trivial and AA further has 2=02 = 0. Since ia=a¯i\mathrm{i} a = \bar{a} \mathrm{i}, A 2A^2 is commutative iff AA is commutative and the involution in AA is trivial. Since a(bi)=(ba)ia (b \mathrm{i}) = (b a) \mathrm{i}, A 2A^2 is associative iff AA is associative and commutative. Finally, A 2A^2 is alternative iff AA is associative (and hence also alternative).

Examples

The standard example is the sequence of consecutive doubles starting with \mathbb{R} itself (with the identity map as involution); these are the Cayley–Dickson algebras: the real numbers \mathbb{R}, the complex numbers \mathbb{C}, the quaternions \mathbb{H}, the octonions or Cayley numbers 𝕆\mathbb{O}, the sedenions 𝕊\mathbb{S}, etc. These are the normed division algebras (\mathbb{R}, \mathbb{C}, \mathbb{H}, and 𝕆\mathbb{O}), followed by further algebras which are not division algebras. All of these algebras are power-associative and unital and have all inverse elements; the subalgebra with x¯=x\bar{x} = x is always just \mathbb{R}.

References

Revised on October 22, 2013 17:41:35 by Toby Bartels (98.19.41.40)