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Octonion

Index 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. [1]

66 relations: Adolf Hurwitz, Alternative algebra, Antisymmetric tensor, Arthur Cayley, Associative property, Automorphism, Birkhäuser, Bivector, Blackboard bold, Cayley–Dickson construction, Commutative property, Commutator, Compact group, Complex number, Composition algebra, Cross product, Cyclic permutation, Distributive property, E8 lattice, Fano plane, Finite field, G2 (mathematics), Gamma matrices, Geometric algebra, GF(2), Group (mathematics), Hodge star operator, Hurwitz's theorem (composition algebras), Inverse element, Involution (mathematics), Isomorphism, Isotopy of an algebra, John T. Graves, Kronecker delta, Lie group, Linear combination, Linear map, Mathematics, Mnemonic, Moufang loop, Multiplication table, Norm (mathematics), Octonion algebra, Okubo algebra, Order (ring theory), Philosophical Magazine, PSL(2,7), Quadratic residue code, Quantum logic, Quasigroup, ..., Quaternion, Real number, Sedenion, Seven-dimensional cross product, Simply connected space, Skew-symmetric matrix, SO(8), Special relativity, Split-octonion, String theory, Subalgebra, Triality, Up to, Vector (mathematics and physics), William Rowan Hamilton, Zero divisor. Expand index (16 more) »

Adolf Hurwitz

Adolf Hurwitz (26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.

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Alternative algebra

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

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Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.

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Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

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Associative property

In mathematics, the associative property is a property of some binary operations.

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Automorphism

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

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Birkhäuser

Birkhäuser is a former Swiss publisher founded in 1879 by Emil Birkhäuser.

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Bivector

In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors.

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Blackboard bold

Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.

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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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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.

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Compact group

In mathematics, a compact (topological) group is a topological group whose topology is compact.

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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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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.

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Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

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Cyclic permutation

In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of X. If S has k elements, the cycle is called a k-cycle.

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Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

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E8 lattice

In mathematics, the E8 lattice is a special lattice in R8.

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Fano plane

In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2.

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Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

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G2 (mathematics)

In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups.

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Gamma matrices

In mathematical physics, the gamma matrices, \, also known as the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cℓ1,3(R).

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Geometric algebra

The geometric algebra (GA) of a vector space is an algebra over a field, noted for its multiplication operation called the geometric product on a space of elements called multivectors, which is a superset of both the scalars F and the vector space V. Mathematically, a geometric algebra may be defined as the Clifford algebra of a vector space with a quadratic form.

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GF(2)

GF(2) (also F2, Z/2Z or Z2) is the '''G'''alois '''f'''ield of two elements.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Hodge star operator

In mathematics, the Hodge isomorphism or Hodge star operator is an important linear map introduced in general by W. V. D. Hodge.

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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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Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

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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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Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Isotopy of an algebra

In mathematics, an isotopy from a possibly non-associative algebra A to another is a triple of bijective linear maps such that if then.

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John T. Graves

John Thomas Graves (4 December 1806 – 29 March 1870) was an Irish jurist and mathematician.

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Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.

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Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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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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Mnemonic

A mnemonic (the first "m" is silent) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory.

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Moufang loop

In mathematics, a Moufang loop is a special kind of algebraic structure.

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Multiplication table

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

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Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

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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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Order (ring theory)

In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that.

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Philosophical Magazine

The Philosophical Magazine is one of the oldest scientific journals published in English.

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PSL(2,7)

In mathematics, the projective special linear group PSL(2, 7) (isomorphic to GL(3, 2)) is a finite simple group that has important applications in algebra, geometry, and number theory.

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Quadratic residue code

A quadratic residue code is a type of cyclic code.

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Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions that takes the principles of quantum theory into account.

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Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible.

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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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Sedenion

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions.

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Seven-dimensional cross product

In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space.

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Simply connected space

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.

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Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

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SO(8)

In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space.

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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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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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String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

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Subalgebra

In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.

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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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Up to

In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.

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Vector (mathematics and physics)

When used without any further description, vector usually refers either to.

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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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Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

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Cayley number, Cayley numbers, Integral octonion, Octernion, Octonian, Octonians, Octonion multiplication, Octonions, 𝕆.

References

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

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