nLab Cayley-Dickson construction

CayleyDickson construction

Cayley–Dickson construction


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, …


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


(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


(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


(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)


(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.


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}


(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


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 }

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).


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).


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}.

Cayley-Dickson-Albert construction

The Cayley-Dickson construction described above was slightly generalized in Section 5 of Albert (1942), where the multiplication is further modified by a parameter γ\gamma in the ground field. This process constructs an algebra A γA^{\gamma} starting from a unital algebra AA of order nn over a field FF endowed with an involution J:AAJ:A\to A over FF such that for all xAx\in A:

  • x+J(x)Fx+ J(x) \in F,

  • xJ(x)=J(x)xFx J(x) = J(x) x \in F.

To construct A γA^{\gamma}, one chooses a parameter γF\gamma\in F, and on the vector space AAA\oplus A one defines a product

(g 1,g 2)(h 1,h 2)=(g 1h 1+γJ(h 2)g 2,h 2g 1+g 2J(h 1)) (g_1 ,g_2 )\cdot (h_1, h_2 ) = (g_1 h_1 + \gamma J(h_2) g_2 , h_2 g_1 +g_2 J(h_1 ))

and an involution KK on A γA^{\gamma} as

K((g 1,g 2))=(J(g 1),g 2) K((g_1, g_2 ))= (J(g_1) , -g_2)

In particular, for F=F=\mathbb{R} this process allows to get the split- variants of the real division algebra, cf.:

Higher algebra

Albuquerque & Majid (1999) describe the Cayley-Dickson process from a different point of view. The quaternions, and octonions are realized as 2 n\mathbb{Z}_2^n group algebras over the reals twisted by a group nn-cocycle ωZ n( 2 n, ×)\omega\in Z^n(\mathbb{Z}_2^n,\mathbb{R}^{\times}), for n=2,3n=2,3, respectively.

In more detail, the quaternions are described as the group algebra [ 2× 2]\mathbb{R}[\mathbb{Z}_2\times\mathbb{Z}_2] where the group multiplication is twisted by a cocycle ω(g,h)\omega(g,h). The 2-cocycle condition of such a function implies that the quaternions are associative. On the other hand, the octonions are realized as a group algebra [ 2× 2]\mathbb{R}[\mathbb{Z}_2\times\mathbb{Z}_2] where the function ω(g,h)\omega(g,h) twisting the multiplication is no longer a 2-cocycle. One of the effects of this is that the octonions are no longer associative or, rather, that they satisfy a nontrivial associativity condition, in the sense of a quasi-Hopf algebra. This nontrivial associativity condition is witnessed by a group 33-cocycle in Z 3( 2× 2× 2, ×)Z^3(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2, \mathbb{R}^{\times}). Notably, this 33-cocycle turns out to be a 33-coboundary, so that its cohomology class is trivial.

This point of view also clarifies the common claim that algebras in the Cayley-Dickson process lose “nice” properties after more iterations. Rather, a more precise claim is that these nice properties are weakened. In the case of the octonions, these are better regarded as associative up to a nontrivial cocycle.

This highlights that a more suitable point of view to study the algebras arising in the Cayley-Dickson process is higher algebra. As noted in e.g. (p.8 of Baez (2002)), the octonions are indeed associative, when viewed as a monoid object internal to the tensor category which is the fusion category of 2 3\mathbb{Z}_2^3-graded real vector spaces with nontrivial associator defined by the twisting 33-cocycle.

This seems to suggest a pattern regarding the rest of the 2 n2^n-gons, as of now unproven(?). For instance, the sedenions could be realized as a twist of the group algebra [ 2 4]\mathbb{R}[\mathbb{Z}_2^4] by a 4-cocycle Z 4( 2 4, ×)Z^4(\mathbb{Z}_2^4,\mathbb{R}^{\times}), and as such would have fewer nice properties as a 1-algebra. Yet as a higher algebra it would retain its nice properties: it would correspond to the fusion 2-category of 2 4\mathbb{Z}_2^4-graded 2-vector spaces twisted by a group 4-cocycle.


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


On the generalization of the Cayley-Dickson construction:

On the Cayley-Dickson construction in terms of quasi-Hopf algebras:

  • Helena Albuquerque, Shahn Majid. Quasialgebra Structure of the Octonions. Journal of Algebra Volume 220, Issue 1, 1 October 1999, Pages 188-224. (doi)

Last revised on February 8, 2024 at 18:02:21. See the history of this page for a list of all contributions to it.