Composition algebra and Hurwitz's theorem (composition algebras)

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Difference between Composition algebra and Hurwitz's theorem (composition algebras)

Composition algebra vs. Hurwitz's theorem (composition algebras)

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. 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.

Abraham Adrian Albert

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

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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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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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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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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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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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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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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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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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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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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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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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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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