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Composition algebra

Index 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 for all and in. [1]

60 relations: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Abraham Adrian Albert, Academic Press, Alexander Merkurjev, Algebra over a field, American Mathematical Society, Annals of Mathematics, Automorphism, Bicomplex number, Bioctonion, Biquaternion, Brahmagupta–Fibonacci identity, Cayley–Dickson construction, Characteristic (algebra), Circolo Matematico di Palermo, Complex number, Definite quadratic form, Degen's eight-square identity, Degenerate bilinear form, Diophantus, Division algebra, Dover Publications, Euler's four-square identity, Field (mathematics), Freudenthal magic square, Graduate Studies in Mathematics, Hurwitz problem, Hurwitz's theorem (composition algebras), Imaginary unit, Involution (mathematics), Isotropic quadratic form, Leonard Eugene Dickson, Leonhard Euler, Markus Rost, Mathematics, Matrix ring, Max August Zorn, Multiplicative inverse, Nathan Jacobson, Non-associative algebra, Null vector, Octonion, Octonion algebra, Okubo algebra, Pauli matrices, Petersson algebra, Pfister form, Quadratic field, Quadratic form, Quaternion, ..., Quaternion algebra, Real number, Split-complex number, Split-octonion, Split-quaternion, Springer Science+Business Media, Square (algebra), Triality, William Rowan Hamilton, 1. Expand index (10 more) »

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (English: Papers from the Mathematical Seminar of the University of Hamburg) is a peer-reviewed mathematics journal published by Springer Science+Business Media.

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Abraham Adrian Albert

Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician.

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Academic Press

Academic Press is an academic book publisher.

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Alexander Merkurjev

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

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Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

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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, advocacy and other programs.

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Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

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Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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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)^*.

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Bioctonion

In mathematics, a bioctonion, or complex octonion, is a pair of biquaternions (p,q), p,q ∈. The product of two bioctonions is defined using biquaternion multiplication and the biconjugate p → p*: The bioctonion z.

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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.

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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.

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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.

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Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

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Circolo Matematico di Palermo

The Circolo Matematico di Palermo (Mathematical Circle of Palermo) is an Italian mathematical society, founded in Palermo by Sicilian geometer Giovanni B. Guccia in 1884.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.

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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.

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Degenerate bilinear form

In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V) given by is not an isomorphism.

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Diophantus

Diophantus of Alexandria (Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 201 and 215; died around 84 years old, probably sometime between AD 285 and 299) was an Alexandrian Hellenistic mathematician, who was the author of a series of books called Arithmetica, many of which are now lost.

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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.

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Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

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Euler's four-square identity

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Freudenthal magic square

In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups).

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Graduate Studies in Mathematics

Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS).

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Hurwitz problem

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.

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Hurwitz's theorem (composition algebras)

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.

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Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

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Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.

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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.

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Leonard Eugene Dickson

Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician.

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Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

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Markus Rost

Markus Rost is a German mathematician who works at the intersection of topology and algebra.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Matrix ring

In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication.

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Max August Zorn

Max August Zorn (June 6, 1906 – March 9, 1993) was a German mathematician.

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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.

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Nathan Jacobson

Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.

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Non-associative algebra

A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative.

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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.

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Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

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Octonion algebra

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 (called the norm form) such that for all x and y in A. The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers.

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Okubo algebra

In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by.

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Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary.

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Petersson algebra

In mathematics, a Petersson algebra is a composition algebra over a field constructed from an order-3 automorphism of a Hurwitz algebra.

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Pfister form

In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965.

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Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers.

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Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

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Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

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Quaternion algebra

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 the matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, A \otimes_F K is isomorphic to the 2×2 matrix algebra over K. The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Split-complex number

In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.

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Split-octonion

In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers.

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Split-quaternion

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.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Square (algebra)

In mathematics, a square is the result of multiplying a number by itself.

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Triality

In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces.

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William Rowan Hamilton

Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.

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1

1 (one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph.

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Redirects here:

Multiplicative quadratic form, Para-Hurwitz algebra.

References

[1] https://en.wikipedia.org/wiki/Composition_algebra

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