nLab
Cayley-Dickson construction

CayleyDickson construction

Cayley–Dickson construction

Idea

The Cayley-Dickson construction or Cayley-Dickson double (Dickson 1919, (6)) takes a real star-algebra AA to a new real star-algebra whose elements are pairs of elements of AA, in generalization of how the complex numbers arise as a doubling of the real numbers.

When iteratively applied to the real numbers, regarded as a star-algebra with trivial involution, the Cayley-Dickson construction yields, consecutively, the complex numbers, then the quaternions, then the octonions (thus all four real normed division algebras), then the sedenions, …

Definition

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 in components

Definition

(Cayley-Dickson construction in components)

The Cayley–Dickson double of the real star-algebra AA is the real star-algebra CD(A)CD(A)

  • whose underlying real vector space is the direct sum AAA \oplus A,

  • whose multiplication is given by

    (1)(a,b)(c,d)(acdb¯,a¯d+cb) (a,b) (c, d) \;\coloneqq\; (a c - d \overline{b}, \overline{a} d + c b)
  • whose star-involution is given by

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

(Dickson 1919, (6))

Definition by generators and relations

Definition

(Cayley-Dickson double by generators and relations)

The Cayley-Dickson double of a real star-algebra AA is the real star-algebra CD˜(A)\widetilde{CD}(A) obtained by adjoining one generator \ell to AA subject to the following relations:

(3) 2=1,AAA¯= \ell^2 \;=\; -1 \,, \phantom{AAA} \overline{\ell} \;=\; - \ell

and

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

for all a,bAa, b \in A.

(Baez 02, second half of 2.2)

Lemma

(induced relations)

The relation in Def. imply the following further relations:

(5)a=a¯,AAa=a¯ a \ell \;=\; \ell \overline{a} \,, \phantom{AA} \ell a \;=\; \overline{a} \ell


(a)b=(ba),AAa(b)=(ba) (\ell a) b \;=\; \ell (b a) \,, \phantom{AA} a (b \ell) \;=\; (b a) \ell


(6)(a)(b)=ba¯,AA(a)(b)=b¯a (\ell a) (\ell b) \;=\; - b \overline{a} \,, \phantom{AA} (a \ell) (b \ell) \;=\; - \overline{b} a

for all a,bAa, b \in A.

Proof

Using (4) we have:

a =a(1) =(a¯1) =a¯AAA,AAAa =(1)a =(1a¯) =a¯ \begin{aligned} a \ell & = a (\ell 1) \\ & = \ell (\overline{a} 1) \\ & = \ell \overline{a} \end{aligned} \phantom{AAA} , \phantom{AAA} \begin{aligned} \ell a & = (1 \ell) a \\ & = (1 \overline{a}) \ell \\ & = \overline{a} \ell \end{aligned}

Using this and (4) we have:

(7)(a)b =(a¯)b =(a¯b¯) =a¯b¯¯ =(ba)AAA,AAAa(b) =a(b¯) =(a¯b¯) =a¯b¯¯ =(ba) \begin{aligned} (\ell a) b & = (\overline{a} \ell ) b \\ & = (\overline{a} \overline{b}) \ell \\ & = \ell \overline{ \overline{a} \overline{b} } \\ & = \ell (b a) \end{aligned} \phantom{AAA} , \phantom{AAA} \begin{aligned} a (b \ell) & = a (\ell \overline{b}) \\ & = \ell( \overline{a} \overline{b} ) \\ & = \overline{ \overline{a} \overline{b} } \ell \\ & = (b a) \ell \end{aligned}

and

(8)(a)(b) =(a)(b¯) =ab¯¯ =ba¯AAA,AAA(a)(b) =(a¯)(b) =a¯b¯ =b¯a \begin{aligned} (\ell a) (\ell b) & = (\ell a) (\overline{b} \ell) \\ & = - \overline{ a \overline{b} } \\ & = - b \overline{a} \end{aligned} \phantom{AAA} , \phantom{AAA} \begin{aligned} (a \ell) (b \ell) & = (\ell \overline{a}) (b \ell) \\ & = - \overline{ \overline{a} b } \\ & = - \overline{b} a \end{aligned}

Equivalence of the definitions

Proposition

Definition and Definition are equivalent, in that we have an isomorphism of real star-algebras:

ϕ DK(A) DK˜(A) (a,b) a+b \array{ \phi & DK(A) &\overset{\simeq}{\longrightarrow}& \widetilde{DK}(A) \\ & (a,b) &\mapsto& a + \ell b }
Proof

It is clear from Def. that for every element xDK˜(A)x \in \widetilde{DK}(A) there is a unique pair of elements a,bAa,b \in A such that

x=a+b=ϕ(a,b). x = a + \ell b \;=\; \phi(a,b) \,.

This means that ϕ\phi is a linear isomorphism of the underlying real vector spaces. Hence it only remains to check that ϕ\phi is indeed an algebra homomorphism and that it respects the involution.

To see that ϕ\phi is an algebra homomorphism, we multiply out and then use the relations (7) and (8) from Lemma :

ϕ(a,b)ϕ(c,d) =(a+b)(c+d) =ac+(b)(d)+a(d)+(b)c =acdb¯+(a¯d+cb) =ϕ(acdb¯,a¯d+cb) \begin{aligned} \phi(a,b) \phi(c,d) & = (a + \ell b) (c + \ell d) \\ & = a c + (\ell b)(\ell d) + a (\ell d) + (\ell b) c \\ & = a c - d \overline{b} + \ell ( \overline{a} d + c b ) \\ & = \phi(a c - d \overline{b}, \overline{a} d + c b) \end{aligned}

Here in the last line we indeed find the component formula (1).

To see that ϕ\phi respects the involution we use (3) from Def. and (5) from Lemma :

ϕ(a,b)¯ =a+b¯ =a¯+b¯ =a¯b¯ =a¯b =ϕ(a¯,b). \begin{aligned} \overline{\phi(a,b)} & = \overline{ a + \ell b } \\ & = \overline{a} + \overline{\ell b} \\ & = \overline{a} - \overline{b} \ell \\ & = \overline{a} - \ell b \\ & = \phi( \overline{a}, -b ) \,. \end{aligned}

Here in the last line we indeed find the component formula (2).


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},

These are the real normed division algebras

The diagram on the right shows a basis of imaginary octonions obtained via Cayley-Dickson doubling from a standard basis {i,j,k}\{i,j,k\} of imaginary quaternions.




Next, the CD-double of the octonions is the sedenions 𝕊\mathbb{S}, etc. , 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

Named after Arthur Cayley and Leonard Dickson.

The original article:

  • Leonard Dickson, On Quaternions and Their Generalization and the History of the Eight Square Theorem,

    Annals of Mathematics, Second Series, Vol. 20, No. 3 (Mar., 1919), pp. 155-171 (jstor:1967865)

Review and introduction:

See also

More:

Last revised on July 21, 2020 at 13:01:32. See the history of this page for a list of all contributions to it.