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Cayley-Dickson construction

Cayley–Dickson construction

Idea

Consider how the complex numbers are formed from the real numbers. If generalized carefully, this kind of operation may be performed again to yield the quaternions, then the octonions (hence the four real normed division algebra), then the sedenions, and so on.

This is a special case of a construction which takes a real star-algebra AA to a new star-algebra whose elements are pairs of elements of AA. This operation is the Cayley–Dickson construction.

Definitions

Let AA be an possibly nonassociative star-algebra over the field \mathbb{R} of real numbers: an algebra equipped with an involution (¯):xx¯\overline(-) \colon x \mapsto \overline{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.)

Definition

(Cayley-Dickson construction, first version)

The Cayley–Dickson double A 2A^2 of AA is the real algebra whose underlying \mathbb{R}-vector space is is the direct sum AAA \oplus A, and whose multiplication is given by

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

and the formula

(a,b)¯(a¯,b) \widebar{(a,b)} \coloneqq (\overline{a},-b)

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

The following description is different but equivalent:

Definition

(Cayley-Dickson double by generators and relations)

The Cayley-Dickson double A 2A^2 of AA is the real algebra obtained by adjoining one generator \ell to AA subject to the following relations:

2=1 \ell^2 = -1

and

a(b)=(a¯b),AAa(b)=(ab¯),AA(a)(b 1)=ab¯ a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} a(\ell b) = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell^{-1}) = \overline{a b}

for all a,bAa, b \in A.

Properties

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,\overline{b})). For this reason, we may write A[i]A[\mathrm{i}] in place of A 2A^2, at least when AA is unital.

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\overline{\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 = \overline{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, flexible, and unital, and have all inverse elements; the subalgebra with x¯=x\overline{x} = x is always just \mathbb{R}.

References

Last revised on April 23, 2018 at 08:39:19. See the history of this page for a list of all contributions to it.