In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.

The Cayley–Dickson construction defines a new algebra as a Cartesian product of an algebra with itself, with multiplication defined in a specific way (different from the componentwise multiplication) and an involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of this product) is called the norm.

The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multiplication, and finally alternativity.

More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.^[1]: 45

Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed) division algebras (over the real numbers).

Synopsis

The Cayley–Dickson construction is due to Leonard Dickson in 1919 showing how the octonions can be constructed as a two-dimensional algebra over quaternions. In fact, starting with a field F, the construction yields a sequence of F-algebras of dimension 2ⁿ. For n = 2 it is an associative algebra called a quaternion algebra, and for n = 3 it is an alternative algebra called an octonion algebra. These instances n = 1, 2 and 3 produce composition algebras as shown below.

The case n = 1 starts with elements (a, b) in F × F and defines the conjugate (a, b)* to be (a*, –b) where a* = a in case n = 1, and subsequently determined by the formula. The essence of the F-algebra lies in the definition of the product of two elements (a, b) and (c, d):

${\displaystyle (a,b)\times (c,d)=(ac-d^{*}b,da+bc^{*}).}$

Proposition 1: For ${\displaystyle z=(a,b)}$ and ${\displaystyle w=(c,d),}$ the conjugate of the product is ${\displaystyle w^{*}z^{*}=(zw)^{*}.}$

proof: ${\displaystyle (c^{*},-d)(a^{*},-b)=(c^{*}a^{*}+b^{*}(-d),-bc^{*}-da)=(zw)^{*}.}$

Proposition 2: If the F-algebra is associative and ${\displaystyle N(z)=zz^{*}}$,then ${\displaystyle N(zw)=N(z)N(w).}$

proof: ${\displaystyle N(zw)=(ac-d^{*}b,da+bc^{*})(c^{*}a^{*}-b^{*}d,-da-bc^{*})=(aa^{*}+bb^{*})(cc^{*}+dd^{*})}$ + terms that cancel by the associative property.

Stages in construction of real algebras

Details of the construction of the classical real algebras are as follows:

Quaternions

Main article: Quaternion

Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of i (red), j (green) and k (blue). In the SVG file, hover over or click a path to highlight it.

The next step in the construction is to generalize the multiplication and conjugation operations.

Form ordered pairs (a, b) of complex numbers a and b, with multiplication defined by

${\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}).\,}$

Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.

The order of the factors seems odd now, but will be important in the next step.

Define the conjugate (a, b)* of (a, b) by

${\displaystyle (a,b)^{*}=(a^{*},-b).\,}$

These operators are direct extensions of their complex analogs: if a and b are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.

The product of a nonzero element with its conjugate is a non-negative real number:

${\displaystyle {\begin{aligned}(a,b)^{*}(a,b)&=(a^{*},-b)(a,b)\\&=(a^{*}a+b^{*}b,ba^{*}-ba^{*})\\&=\left(|a|^{2}+|b|^{2},0\right).\,\end{aligned}}}$

As before, the conjugate thus yields a norm and an inverse for any such ordered pair. So in the sense we explained above, these pairs constitute an algebra something like the real numbers. They are the quaternions, named by Hamilton in 1843.

As a quaternion consists of two independent complex numbers, they form a four-dimensional vector space over the real numbers.

The multiplication of quaternions is not quite like the multiplication of real numbers, though; it is not commutative – that is, if p and q are quaternions, it is not always true that pq = qp.

Modified Cayley–Dickson construction

The Cayley–Dickson construction, starting from the real numbers ${\displaystyle \mathbb {R} }$, generates the composition algebras ${\displaystyle \mathbb {C} }$ (the complex numbers), ${\displaystyle \mathbb {H} }$ (the quaternions), and ${\displaystyle \mathbb {O} }$ (the octonions). There are also composition algebras whose norm is an isotropic quadratic form, which are obtained through a slight modification, by replacing the minus sign in the definition of the product of ordered pairs with a plus sign, as follows:

$${\displaystyle (a,b)(c,d)=(ac+d^{*}b,da+bc^{*}).}$$

When this modified construction is applied to ${\displaystyle \mathbb {R} }$, one obtains the split-complex numbers, which are ring-isomorphic to the direct product ${\displaystyle \mathbb {R} \times \mathbb {R}$ ;} following that, one obtains the split-quaternions, an associative algebra isomorphic to that of the 2 × 2 real matrices; and the split-octonions, which are isomorphic to Zorn(R). Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.^[3]

General Cayley–Dickson construction

Albert (1942, p. 171) gave a slight generalization, defining the product and involution on B = A ⊕ A for A an algebra with involution (with (xy)* = y*x*) to be

${\displaystyle {\begin{aligned}(p,q)(r,s)&=(pr-\gamma s^{*}q,sp+qr^{*})\,\\(p,q)^{*}&=(p^{*},-q)\,\end{aligned}}}$

for γ an additive map that commutes with * and left and right multiplication by any element. (Over the reals all choices of γ are equivalent to −1, 0 or 1.) In this construction, A is an algebra with involution, meaning:

• A is an abelian group under +
• A has a product that is left and right distributive over +
• A has an involution *, with (x*)* = x, (x + y)* = x* + y*, (xy)* = y*x*.

The algebra B = A ⊕ A produced by the Cayley–Dickson construction is also an algebra with involution.

B inherits properties from A unchanged as follows.

• If A has an identity 1_A, then B has an identity (1_A, 0).
• If A has the property that x + x*, xx* associate and commute with all elements, then so does B. This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative.

Other properties of A only induce weaker properties of B:

• If A is commutative and has trivial involution, then B is commutative.
• If A is commutative and associative then B is associative.
• If A is associative and x + x*, xx* associate and commute with everything, then B is an alternative algebra.

Notes

References

• Albert, A. A. (1942), "Quadratic forms permitting composition", Annals of Mathematics, Second Series, 43 (1): 161–177, doi:10.2307/1968887, JSTOR 1968887, MR 0006140 (see p. 171)
• Baez, John (2002), "The Octonions", Bulletin of the American Mathematical Society, 39 (2): 145–205, arXiv:math/0105155, doi:10.1090/S0273-0979-01-00934-X, S2CID 586512. (See "Section 2.2, The Cayley–Dickson Construction")
• Dickson, L. E. (1919), "On Quaternions and Their Generalization and the History of the Eight Square Theorem", Annals of Mathematics, Second Series, Annals of Mathematics, 20 (3): 155–171, doi:10.2307/1967865, JSTOR 1967865
• Biss, Daniel K.; Christensen, J. Daniel; Dugger, Daniel; Isaksen, Daniel C. (2007). "Large annihilators in Cayley–Dickson algebras II". Boletin de la Sociedad Matematica Mexicana. 3: 269–292. arXiv:math/0702075. Bibcode:2007math......2075B.
• Kantor, I. L.; Solodownikow, A. S. (1978), Hyperkomplexe Zahlen, Leipzig: B.G. Teubner (the following reference gives the English translation of this book)
• Kantor, I. L.; Solodovnikov, A. S. (1989), Hypercomplex numbers, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96980-0, MR 0996029
• Hamilton, William Rowan (1847), "On Quaternions", Proceedings of the Royal Irish Academy, 3: 1–16, ISSN 1393-7197
• Roos, Guy (2008). "Exceptional symmetric domains §1: Cayley algebras". In Gilligan, Bruce; Roos, Guy (eds.). Symmetries in Complex Analysis. Contemporary Mathematics. Vol. 468. American Mathematical Society. ISBN 978-0-8218-4459-5.

Further reading

• Daboul, Jamil; Delbourgo, Robert (1999). "Matrix representations of octonions and generalizations". Journal of Mathematical Physics. 40 (8): 4134–50. arXiv:hep-th/9906065. Bibcode:1999JMP....40.4134D. doi:10.1063/1.532950. S2CID 16932871.