Algebraic structures |
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In mathematics, a **composition algebra**A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies

for all x and y in A.

A composition algebra includes an involution called a **conjugation**: The quadratic form 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.^{ [1] } When *x* is *not* a null vector, the multiplicative inverse of *x* is When there is a non-zero null vector, *N* is an isotropic quadratic form, and "the algebra splits".

Every unital composition algebra over a field K can be obtained by repeated application of the Cayley–Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(*K*) = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8.^{ [2] }^{ [3] }^{ [4] }

- 1-dimensional composition algebras only exist when char(
*K*) ≠ 2. - Composition algebras of dimension 1 and 2 are commutative and associative.
- Composition algebras of dimension 2 are either quadratic field extensions of K or isomorphic to
*K*⊕*K*. - Composition algebras of dimension 4 are called quaternion algebras. 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*.^{ [5] }

When the field K is taken to be complex numbers **C** and the quadratic form *z*^{2}, then four composition algebras over **C** are **C** itself, the bicomplex numbers, the biquaternions (isomorphic to the 2×2 complex matrix ring M(2, **C**)), and the bioctonions **C** ⊗ **O**, which are also called complex octonions.

Matrix ring M(2, **C**) has long been an object of interest, first as biquaternions by Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra.

The squaring function *N*(*x*) = *x*^{2} on the real number field forms the primordial composition algebra. When the field K is taken to be real numbers **R**, then there are just six other real composition algebras.^{ [3] }^{:166} In two, four, and eight dimensions there are both a division algebra and a "split algebra":

- binarions: complex numbers with quadratic form
*x*^{2}+*y*^{2}and split-complex numbers with quadratic form*x*^{2}−*y*^{2}, - quaternions and split-quaternions,
- octonions and split-octonions.

Every composition algebra has an associated bilinear form B(*x,y*) constructed with the norm N and a polarization identity:

^{ [6] }

The composition of sums of squares was noted by several early authors. Diophantus 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 discussed the four-square identity in 1748, and it led W. R. Hamilton to construct his four-dimensional algebra of quaternions.^{ [5] }^{:62} 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 octonion 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...
^{ [5] }^{:61}

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 e, and for quaternions *q* and *Q* writes a Cayley number *q* + *Q*e. Denoting the quaternion conjugate by *q*′, the product of two Cayley numbers is^{ [7] }

The conjugate of a Cayley number is *q'* – *Q*e, and the quadratic form is *qq*′ + *QQ*′, obtained by multiplying the number by its conjugate. The doubling method has come to be called the Cayley–Dickson construction.

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.^{ [8] } 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.^{ [9] } Nathan Jacobson described the automorphisms of composition algebras in 1958.^{ [2] }

The classical composition algebras over **R** and **C** are unital algebras. Composition algebras *without* a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.^{ [10] }^{:463–81}

In abstract algebra, an **alternative algebra** is an algebra in which multiplication need not be associative, only alternative. That is, one must have

In mathematics, a **Clifford algebra** is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As *K*-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford.

In mathematics, the **octonions** are a normed division algebra over the real numbers, meaning it is a hypercomplex number system; Octonions are usually represented by the capital letter O, using boldface **O** or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.

In mathematics, **hypercomplex number** is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

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.

In mathematics, and more specifically in abstract algebra, a ***-algebra** is a mathematical structure consisting of two **involutive rings**R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution at all.

In mathematics, a **square** is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3^{2}, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations ` x^2` or

In abstract algebra, a **split complex number** has two real number components *x* and *y*, and is written *z* = *x* + *y j*, where *j*^{2} = 1. The *conjugate* of *z* is *z*^{∗} = *x* − *y j*. Since *j*^{2} = 1, the product of a number *z* with its conjugate is *zz*^{∗} = *x*^{2} − *y*^{2}, an isotropic quadratic form, *N*(*z*) = *x*^{2} − *y*^{2}.

In abstract algebra, the **biquaternions** are the numbers *w* + *x***i** + *y***j** + *z***k**, where *w*, *x*, *y*, and z are complex numbers, or variants thereof, and the elements of {**1**, **i**, **j**, **k**} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof:

In abstract algebra, the **split-quaternions** or **coquaternions** are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. But unlike the quaternions, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. As an algebra over the real numbers, they are isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the Synonyms section below.

In mathematics, a **quaternion algebra** over a field *F* is a central simple algebra *A* over *F* that has dimension 4 over *F*. Every quaternion algebra becomes a matrix algebra by *extending scalars*, i.e. for a suitable field extension *K* of *F*, is isomorphic to the 2×2 matrix algebra over *K*.

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: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0).

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

A **non-associative algebra** is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure *A* is a non-associative algebra over a field *K* if it is a vector space over *K* and is equipped with a *K*-bilinear binary multiplication operation *A* × *A* → *A* which may or may not be associative. Examples include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed that the multiplication is associative, using parentheses to indicate the order of multiplications is necessary. For example, the expressions (*ab*)(*cd*), *d* and *a*(*b* ) may all yield different answers.

In mathematics, an **octonion algebra** or **Cayley algebra** over a field *F* is an algebraic structure which is an 8-dimensional composition algebra over *F*. In other words, it is a unital non-associative algebra *A* over *F* with a non-degenerate quadratic form *N* such that

In mathematics, **Hurwitz's theorem** is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions. Such algebras, sometimes called **Hurwitz algebras**, are examples of composition algebras.

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.

In mathematics, the **Hurwitz problem**, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables.

In algebra, an **Okubo algebra** or **pseudo-octonion algebra** is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras are composition algebras, flexible algebras, Lie admissible algebras, and power associative, but are not associative, not alternative algebras, and do not have an identity element.

In mathematics, a **bioctonion**, or **complex octonion**, is a pair (*p,q*) where *p* and *q* are biquaternions.

Wikibooks has a book on the topic of: Associative Composition Algebra |

- ↑ Springer, T. A.; F. D. Veldkamp (2000).
*Octonions, Jordan Algebras and Exceptional Groups*. Springer-Verlag. p. 18. ISBN 3-540-66337-1. - 1 2 Jacobson, Nathan (1958). "Composition algebras and their automorphisms".
*Rendiconti del Circolo Matematico di Palermo*.**7**: 55–80. doi:10.1007/bf02854388. Zbl 0083.02702. - 1 2 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, ISBN 978-0-8218-4459-5 - ↑ Schafer, Richard D. (1995) [1966].
*An introduction to non-associative algebras*. Dover Publications. pp. 72–75. ISBN 0-486-68813-5. Zbl 0145.25601. - 1 2 3 Kevin McCrimmon (2004)
*A Taste of Jordan Algebras*, Universitext, Springer ISBN 0-387-95447-3 MR 2014924 - ↑ Arthur A. Sagle & Ralph E. Walde (1973)
*Introduction to Lie Groups and Lie Algebras*, pages 194−200, Academic Press - ↑ 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, ISSN 0003-486X, JSTOR 1967865 - ↑ Max Zorn (1931) "Alternativekörper und quadratische Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(3/4): 395–402
- ↑ Albert, Adrian (1942). "Quadratic forms permitting composition".
*Annals of Mathematics*.**43**: 161–177. doi:10.2307/1968887. Zbl 0060.04003. - ↑ 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, American Mathematical Society ISBN 0-8218-0904-0

- Faraut, Jacques; Korányi, Adam (1994).
*Analysis on symmetric cones*. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. pp. 81–86. ISBN 0-19-853477-9. MR 1446489. - Lam, Tsit-Yuen (2005).
*Introduction to Quadratic Forms over Fields*. Graduate Studies in Mathematics.**67**. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023. - Harvey, F. Reese (1990).
*Spinors and Calibrations*. Perspectives in Mathematics.**9**. San Diego: Academic Press. ISBN 0-12-329650-1. Zbl 0694.53002.

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