mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

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 t ...

that satisfies :

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

for all and in . A composition algebra includes an

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

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 or a split algebra, depending on the existence of a non-zero ''v'' in ''A'' such that ''N''(''v'') = 0, called a null vector. 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 Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...

of ''x'' is When there is a non-zero null vector, ''N'' is an isotropic quadratic form, 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, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...

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, meetings, ...

, *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 (mathematics), 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 (vector space), dimension 4 ove ...

s. They are associative but not commutative. *Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative. For consistent terminology, algebras of dimension 1 have been called ''unarion'', and those of dimension 2 ''binarion''. Every composition algebra is an alternative algebra. Using the doubled form ( _ : _ ): ''A'' × ''A'' → ''K'' defined by

(a:b) = N(a+b) - N(a) - N(b)=ab^*+ba^*,

then the trace of ''a'' is given by and the conjugate by where is multiplicative identity of . A series of exercises proves that a composition algebra is always an alternative algebra.

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 for ...

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

biquaternion In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions cor ...

s (isomorphic to the complex matrix ring ), and the bioctonions , which are also called complex octonions. The matrix ring has long been an object of interest, first as

s by

Hamilton Hamilton may refer to: * Alexander Hamilton (1755/1757–1804), first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States * ''Hamilton'' (musical), a 2015 Broadway musical by Lin-Manuel Miranda ** ''Hamilton'' (al ...

(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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

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 division algebra 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) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...

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. The algebra of quater ...

s and split-quaternions, :

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

s and split-octonions. 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: :

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 () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...

was aware of the identity involving the sum of two squares, now called the Brahmagupta–Fibonacci identity, 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 polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

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 tessarines were described giving first light to bicomplex numbers. About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, 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 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 mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

, 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 introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate split-octonions. Adrian Albert 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 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 automorphism ...

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 is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

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