mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a composition algebra over a field is a not necessarily associative

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

over together with a nondegenerate

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...

that satisfies :

N(xy) = N(x)N(y)

for all and in . A composition algebra includes an

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

called a conjugation:

x \mapsto x^*.

The quadratic form

N(x) = x x^*

is called the norm of the algebra. A composition algebra (''A'', ∗, ''N'') is either a

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 fie ...

or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a

null vector In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real bilinear forms, definite quadratic forms an ...

. When ''x'' is ''not'' a null vector, the

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''/' ...

of ''x'' is When there is a non-zero null vector, ''N'' 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 ...

, and "the algebra splits".

Structure theorem

Every unital composition algebra over a field can be obtained by repeated application of the

Cayley–Dickson construction 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 b ...

starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). The possible dimensions of a composition algebra are , , , and .Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in ''Symmetries in Complex Analysis'' by Bruce Gilligan & Guy Roos, volume 468 of ''Contemporary Mathematics'',

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meeting ...

, *1-dimensional composition algebras only exist when . *Composition algebras of dimension 1 and 2 are commutative and associative. *Composition algebras of dimension 2 are either quadratic field extensions of or isomorphic to . *Composition algebras of dimension 4 are called

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 ma ...

s. They are associative but not commutative. *Composition algebras of dimension 8 are called

octonion algebra In mathematics, an octonion algebra or Cayley algebra over a field ''F'' is a composition algebra over ''F'' that has dimension 8 over ''F''. In other words, it is a unital non-associative algebra ''A'' over ''F'' with a non-degenerate quadratic ...

s. They are neither associative nor commutative. For consistent terminology, algebras of dimension 1 have been called ''unarion'', and those of dimension 2 ''binarion''.

Instances and usage

When the field is taken to be

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 fo ...

s and the quadratic form , then four composition algebras over are , the

bicomplex number In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...

s, the biquaternions (isomorphic to the complex

matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...

), and the

bioctonion In mathematics, a bioctonion, or complex octonion, is a pair (''p,q'') where ''p'' and ''q'' are biquaternions. The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*: :(p,q)(r,s) = (pr - s^* q,\ ...

s , which are also called complex octonions. The matrix ring has long been an object of interest, first as biquaternions by

Hamilton Hamilton may refer to: People * Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname ** The Duke of Hamilton, the premier peer of Scotland ** Lord Hamilto ...

(1853), later in the isomorphic matrix form, and especially as Pauli algebra. The squaring function on the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

field forms the primordial composition algebra. When the field is taken to be real numbers , then there are just six other real composition algebras. In two, four, and eight dimensions there are both a

and a "split algebra": : binarions: complex numbers with quadratic form and

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 with quadratic form , :

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 quater ...

s and

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 th ...

s, :

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 and

split-octonion In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: t ...

s. Every composition algebra has an associated

bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...

B(''x,y'') constructed with the norm N and a

polarization identity In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product the ...

: :

B(x,y) \ = \ (x + y) - N(x) - N(y) 2 .

History

The composition of sums of squares was noted by several early authors.

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

was aware of the identity involving the sum of two squares, now called the

Brahmagupta–Fibonacci identity In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity say ...

, which is also articulated as a property of Euclidean norms of complex numbers when multiplied.

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

discussed the four-square identity in 1748, and it led W. R. Hamilton to construct his four-dimensional algebra of

s.Kevin McCrimmon (2004) ''A Taste of Jordan Algebras'', Universitext, Springer In 1848

tessarine In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...

s were described giving first light to bicomplex numbers. About 1818 Danish scholar Ferdinand Degen displayed the

Degen's eight-square identity In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: \begin & \left(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2+a_7^2+a_8^2\right)\le ...

, which was later connected with norms of elements of the

algebra: :Historically, the first non-associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras... In 1919

Leonard 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 reme ...

advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new

imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...

, and for quaternions and writes a Cayley number . Denoting the quaternion conjugate by , the product of two Cayley numbers is :

(q + Qe)(r + Re) = (qr - R'Q) + (Rq + Q r')e .

The conjugate of a Cayley number is , and the quadratic form is , obtained by multiplying the number by its conjugate. The doubling method has come to be called the

. In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras). In 1931

Max Zorn Max August Zorn (; June 6, 1906 – March 9, 1993) was a German mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a method used in set theory that is applicable to a wide range of m ...

introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate

Adrian Albert Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the A ...

also used the gamma in 1942 when he showed that Dickson doubling could be applied to any field with the squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms.

Nathan Jacobson Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician. Biography Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...

described the

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...

s of composition algebras in 1958. The classical composition algebras over and are

unital algebra 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 addition ...

s. Composition algebras ''without'' a

multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

were found by H.P. Petersson ( Petersson algebras) and Susumu Okubo ( Okubo algebras) and others.Max-Albert Knus,

Alexander Merkurjev Aleksandr Sergeyevich Merkurjev (russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev ...

, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in ''The Book of Involutions'', pp. 451–511, Colloquium Publications v 44,

Structure theorem

Instances and usage

History

See also

References

Further reading