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In mathematics, the Cayley–Dickson construction, named after
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
and
Leonard Eugene Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also rem ...
, produces a sequence of
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...
over the field of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, each with twice the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras, for example
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s,
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s, and
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s. These examples are useful
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involuti ...
s frequently applied in
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
. The Cayley–Dickson construction defines a new algebra as a
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
of an algebra with itself, with
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
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 Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
, then commutativity of multiplication,
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
of multiplication, and next alternativity. More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension. Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (
normed The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
)
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a f ...
s (over the real numbers).


Synopsis

The Cayley–Dickson construction is due to Leonard Dickson in 1919 showing how the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s can be constructed as a two-dimensional algebra over
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s. In fact, starting with a field ''F'', the construction yields a sequence of ''F''-algebras of dimension 2''n''. For ''n'' = 2 it is an associative algebra called a
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ...
, and for ''n'' = 3 it is an alternative algebra called an octonion algebra. These instances ''n'' = 1, 2 and 3 produce
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involuti ...
s 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''): :(a,b) \times (c,d) = (ac - d^*b, da + bc^*). Proposition 1: For z = (a,b) and w = (c,d), the conjugate of the product is w^*z^* = (zw)^*. :proof: (c^*,-d)(a^*,-b) = (c^*a^* + b^*(-d), -bc^*-da) = (zw)^*. Proposition 2: If the ''F''-algebra is associative and N(z) = zz^*,then N(zw) = N(z)N(w). :proof: 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:


Complex numbers as ordered pairs

The
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
can be written as
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
s of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s and , with the addition operator being component-wise and with multiplication defined by : (a, b) (c, d) = (a c - b d, a d + b c).\, A complex number whose second component is zero is associated with a real number: the complex number is associated with the real number . The
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of is given by : (a, b)^* = (a^*, -b) = (a, -b) since is a real number and is its own conjugate. The conjugate has the property that : (a, b)^* (a, b) = (a a + b b, a b - b a) = \left(a^2 + b^2, 0\right),\, which is a non-negative real number. In this way, conjugation defines a '' norm'', making the complex numbers a
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "leng ...
over the real numbers: the norm of a complex number  is : , z, = \left(z^* z\right)^\frac12.\, Furthermore, for any non-zero complex number , conjugation gives a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...
, : z^ = \frac. As a complex number consists of two independent real numbers, they form a two-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over the real numbers. Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.


Quaternions

The next step in the construction is to generalize the multiplication and conjugation operations. Form ordered pairs of complex numbers and , with multiplication defined by : (a, b) (c, d) = (a c - d^* b, d a + b c^*).\, 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 of by : (a, b)^* = (a^*, -b).\, These operators are direct extensions of their complex analogs: if and 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: : \begin (a, b)^* (a, b) &= (a^*, -b) (a, b) \\ &= (a^* a + b^* b, b a^* - b a^*) \\ &= \left(, a, ^2 + , b, ^2, 0 \right).\, \end 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
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s, 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
– that is, if and are quaternions, it is not always true that .


Octonions

All the steps to create further algebras are the same from octonions on. This time, form ordered pairs of quaternions and , with multiplication and conjugation defined exactly as for the quaternions: : (p, q) (r, s) = (p r - s^* q, s p + q r^*).\, Note, however, that because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were rather than , the formula for multiplication of an element by its conjugate would not yield a real number. For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element. This algebra was discovered by
John T. Graves John Thomas Graves (4 December 1806 – 29 March 1870) was an Irish jurist and mathematician. He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions in October 1843 and then discover ...
in 1843, and is called the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s or the " Cayley numbers". As an octonion consists of two independent quaternions, they form an eight-dimensional vector space over the real numbers. The multiplication of octonions is even stranger than that of quaternions; besides being non-commutative, it is not
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
– that is, if , , and are octonions, it is not always true that . For the reason of this non-associativity, octonions have no matrix representation.


Further algebras

The algebra immediately following the octonions is called the sedenions. It retains an algebraic property called power associativity, meaning that if is a sedenion, , but loses the property of being an alternative algebra and hence cannot be a
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involuti ...
. The Cayley–Dickson construction can be carried on ''
ad infinitum ''Ad infinitum'' is a Latin phrase meaning "to infinity" or "forevermore". Description In context, it usually means "continue forever, without limit" and this can be used to describe a non-terminating process, a non-terminating ''repeating'' pro ...
'', at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. All the algebras generated in this way over a field are ''quadratic'': that is, each element satisfies a quadratic equation with coefficients from the field. In 1954 R. D. Schafer examined the algebras generated by the Cayley–Dickson process over a field and showed they satisfy the flexible identity. He also proved that any derivation algebra of a Cayley–Dickson algebra is isomorphic to the derivation algebra of Cayley numbers, a 14-dimensional
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
over .


Modified Cayley–Dickson construction

The Cayley–Dickson construction, starting from the real numbers \mathbb R, generates the
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involuti ...
s \mathbb C (the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s), \mathbb H (the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s), and \mathbb O (the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s). There are also composition algebras whose norm is an
isotropic quadratic form In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...
, 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: (a, b) (c, d) = (a c + d^* b, d a + b c^*). When this modified construction is applied to \mathbb R, one obtains the
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s, which are ring-isomorphic to the
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
\mathbb R \times \mathbb R; following that, one obtains the
split-quaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers. After introduction in ...
s, an associative algebra isomorphic to that of the 2 × 2 real matrices; and the split-octonions, which are isomorphic to . Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.Kevin McCrimmon (2004) ''A Taste of Jordan Algebras'', pp 64, Universitext, Springer


General Cayley–Dickson construction

gave a slight generalization, defining the product and involution on for an algebra with involution (with ) to be : \begin (p, q) (r, s) &= (p r - \gamma s^* q, s p + q r^*)\, \\ (p, q)^* &= (p^*, -q)\, \end for an additive map that commutes with and left and right multiplication by any element. (Over the reals all choices of are equivalent to −1, 0 or 1.) In this construction, is an algebra with involution, meaning: * is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
under * has a product that is left and right distributive over * has an involution , with , , . The algebra produced by the Cayley–Dickson construction is also an algebra with involution. inherits properties from unchanged as follows. * If has an identity , then has an identity . * If has the property that , associate and commute with all elements, then so does . This property implies that any element generates a commutative associative *-algebra, so in particular the algebra is power associative. Other properties of only induce weaker properties of : * If is commutative and has trivial involution, then is commutative. * If is commutative and associative then is associative. * If is associative and , associate and commute with everything, then is an alternative algebra.


Notes


References

* (see p. 171) * . ''(See
Section 2.2, The Cayley–Dickson Construction
)'' * * * (the following reference gives the English translation of this book) * * *


Further reading

* {{DEFAULTSORT:Cayley-Dickson construction Composition algebras Historical treatment of quaternions