Clifford algebra

Clifford algebra - Relation to the exterior algebra. Given a vector space V one can construct the exterior algebra Λ(V), whose definition is independent of any quadratic form on V. It turns out that if F does not have characteristic 2 then there is a natural isomorphism between Λ(V) and Cℓ(V,Q) considered as vector spaces. This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra Cℓ(V,Q) ...

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Clifford algebra, Clifford algebra - Introduction and basic properties, Clifford algebra - Universal property and construction, Clifford algebra - Basis and dimension, Clifford algebra - Examples: Real and complex Clifford algebras, Clifford algebra - Properties, Clifford algebra - Relation to the exterior algebra, Clifford algebra - Grading, Clifford algebra - Antiautomorphisms, Clifford algebra - The Clifford scalar product, Clifford algebra - Structure of Clifford algebras, Clifford algebra - The Clifford group Γ, Clifford algebra - Spin and Pin groups, Clifford algebra - Spinors, Clifford algebra - Applications, Clifford algebra - Differential geometry, Clifford algebra - Physics, Clifford algebra - Footnotes

Read more here: » Clifford algebra: Encyclopedia II - Clifford algebra - Properties

In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. It is denoted by Λ(V) or Λ•(V) and its multiplication, known as the wedge product or the exterior product, is written as ∧. The wedge product is associative and bilinear; its essential property is that it is alternating on V: for all vectors which entails for all vectors , ...

Including:

• Exterior algebra - Basis and dimension
• Exterior algebra - Example: the exterior algebra of Euclidean 3-space
• Exterior algebra - Universal property and construction
• Exterior algebra - Anti-symmetric operators and exterior powers
• Exterior algebra - The interior product
• Exterior algebra - Index notation
• Exterior algebra - Differential forms
• Exterior algebra - Generalization
• Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia - Exterior algebra

In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. (Some authors use the term "algebra" synonymously with "associative algebra", but Wikipedia does not. Note also the other uses of the word listed in the algebra article.) Algebra over a field - Definitions. To be precise, let K ...

Including:

• Algebra over a field - Definitions
• Algebra over a field - Properties
• Algebra over a field - Kinds of algebras and examples
• Algebra over a field - Index-free notation
• Algebra over a field - K-algebra morphism

Read more here: » Algebra over a field: Encyclopedia - Algebra over a field

William Kingdon Clifford (May 4, 1845 - March 3, 1879) was an English mathematician and philosopher. Along with Hermann Grassmann he discovered what is now often called geometric algebra, which is a special case of the Clifford algebras named in his honor. He was also the first to suggest that gravitation might be due to an underlying geometry, and in his philosophical work coined the phrase "mind-stuff". William Kingdon Clifford - Biography. He was born at Exeter, and educated at a private school there. He ...

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• William Kingdon Clifford - Biography
• William Kingdon Clifford - Geometry
• William Kingdon Clifford - Philosopher
• William Kingdon Clifford - Books
• William Kingdon Clifford - Reference

Read more here: » William Kingdon Clifford: Encyclopedia - William Kingdon Clifford

A split-complex number is one of the form z = x + j y = e (jw) where x and y are real numbers and the quantity j satisfies j2 = +1. The collection of all such z is called the split-complex plane. Addition and multiplication of split-complex numbers are defined by (x + j y) + (u + j v) = (x + u) + j(y + v) (x + j y)(u + j v) = (xu + ...

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Split-complex number, Split-complex number - Definition, Split-complex number - Conjugate norm and inner product, Split-complex number - The diagonal basis, Split-complex number - Geometry, Split-complex number - Algebraic properties, Split-complex number - Matrix representations, Split-complex number - History, Split-complex number - Synonyms

Read more here: » Split-complex number: Encyclopedia II - Split-complex number - Definition

For vectors in R3, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {i, j, k}, the wedge product of a pair of vectors and is where {i ∧ j, i ∧ k, j ∧ k} is the basis for the three-space Λ2(R3). This imitates the usual definition of the cross product of vectors in three dimensions. Bringing in a third vector< ...

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Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product or insertion operator, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Example: the exterior algebra of Euclidean 3-space

For vectors in R3, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {i, j, k}, the wedge product of a pair of vectors and is where {i ∧ j, i ∧ k, j ∧ k} is the basis for the three-space Λ2(R3). This imitates the usual definition of the cross product of vectors in three dimensions. Bringing in a third vector< ...

See also:

Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Example: the exterior algebra of Euclidean 3-space

A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative. These include the most familiar kinds of algebras. Associative algebras: the algebra of all n-by-n matrices over the field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication. Group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication the commutative ...

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Algebra over a field, Algebra over a field - Definitions, Algebra over a field - Properties, Algebra over a field - Kinds of algebras and examples, Algebra over a field - Index-free notation, Algebra over a field - K-algebra morphism

Read more here: » Algebra over a field: Encyclopedia II - Algebra over a field - Kinds of algebras and examples

In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(n), roughly speaking) similar to vectors, but which change sign under a rotation of 2π radians. Spinor - Overview. A spinor of a certain type is an element of a specific projective representation of the rotation group SO(n,R), or more generally of the group SO(p,q,R), where p + q = n for spinors in a spa ...

Including:

• Spinor - Overview
• Spinor - Mathematical details
• Spinor - History
• Spinor - Examples in low dimensions

Read more here: » Spinor: Encyclopedia - Spinor

In higher mathematics, "algebraic structure" is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. The word "structure" can refer to a specific mathematical object or an even more abstract concept. For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups. This article uses both senses of the phrase. Algebraic structure - In the ...

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• Algebraic structure - In the sense of universal algebra
• Algebraic structure - Allowing axioms other than identities
• Algebraic structure - Allowing additional structure
• Algebraic structure - Categories

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If the dimension of V is n and {e1,...,en} is a basis of V, then the set is a basis for the k-th exterior power Λk(V). The reason is the following: given any wedge product of the form then every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the wedge product, this can be expanded to a linea ...

See also:

Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product or insertion operator, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Basis and dimension

Let V be a vector space over the field K (which in most applications will be the field of real numbers). The fact that Λ(V) is the "most general" unital associative K-algebra containing V with an alternating multiplication on V can be expressed formally by the following universal property: Given any unital associative K-algebra A and any K-linear map j : V → A such that j(v)j(v) = 0 for every v in V, th ...

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Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product or insertion operator, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Universal property and construction

Given two vector spaces V and X, an anti-symmetric operator from Vk to X is a multilinear map f: Vk → X such that whenever v1,...,vk are linearly dependent vectors in V, then f(v1,...,vk) = 0. The most famous example is the determinant, an anti-symmetric operator from (Kn)n to K. The map

The use of split-complex numbers dates back to 1848 when James Cockle revealed his Tessarines. William Kingdon Clifford used split-complex numbers to represent sums of spins in 1882. Clifford called the elements "motors". In the twentieth-century the split-complex numbers became a common platform to describe the Lorentz Boosts of special relativity, in a spacetime plane because a velocity change between frames of reference is n ...

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Split-complex number, Split-complex number - Definition, Split-complex number - Conjugate norm and inner product, Split-complex number - The diagonal basis, Split-complex number - Geometry, Split-complex number - Algebraic properties, Split-complex number - Matrix representations, Split-complex number - History, Split-complex number - Synonyms

Read more here: » Split-complex number: Encyclopedia II - Split-complex number - History

Grassmann algebras have some important applications in physics where they are used to model various concepts related to fermions and supersymmetry. For a physical description see Grassmann number. See also: superspace, superalgebra, supergroup (physics). ...

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Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product or insertion operator, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Physical applications

In the index notation, used primarily by physicists, ...

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Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product or insertion operator, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Index notation

If V* denotes the dual space to the vector space V, then for each , it is possible to define an antiderivation on the algebra , Suppose that . Then w is a multilinear mapping of V* to R, so it is defined by its values on the k-fold Cartesian product . If are k-1 elements of V*, then we define where in each term of the summation, "αi" occupies the i-th position among the argume ...

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Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product or insertion operator, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - The interior product or insertion operator

To be precise, let K be a field, and let A be a vector space over K. Suppose we are given a binary operation A×A→A, with the result of this operation applied to the vectors x and y in A written as xy. Suppose further that the operation is bilinear, i.e.: (x + y)z = xz + yz; x(y + z) = xy + xz; (ax)y = a(xy); and x(See also:

Algebra over a field, Algebra over a field - Definitions, Algebra over a field - Properties, Algebra over a field - Kinds of algebras and examples, Algebra over a field - Index-free notation, Algebra over a field - K-algebra morphism

Read more here: » Algebra over a field: Encyclopedia II - Algebra over a field - Definitions

In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring R[x] by the ideal generated by the polynomial x2 − 1, R[x]/(x2 − 1). The image of x in the quotient is imaginary unit j. With this description, it is clear that the split-complex numbers form a commutative ring with characteristic 0. Moreover if we define scalar multiplication in the obvious manner, the split-complex numbers ac ...

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Split-complex number, Split-complex number - Definition, Split-complex number - Conjugate norm and inner product, Split-complex number - The diagonal basis, Split-complex number - Geometry, Split-complex number - Algebraic properties, Split-complex number - Matrix representations, Split-complex number - History, Split-complex number - Synonyms

Read more here: » Split-complex number: Encyclopedia II - Split-complex number - Algebraic properties

If the dimension of V is n and {e1,...,en} is a basis of V, then the set is a basis for the k-th exterior power Λk(V). The reason is the following: given any wedge product of the form then every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the wedge product, this can be expanded to a linea ...

See also:

Exterior algebra, Exterior algebra - Basis and dimension, Exterior algebra - Example: the exterior algebra of Euclidean 3-space, Exterior algebra - Universal property and construction, Exterior algebra - Anti-symmetric operators and exterior powers, Exterior algebra - The interior product, Exterior algebra - Index notation, Exterior algebra - Differential forms, Exterior algebra - Generalization, Exterior algebra - Physical applications

Read more here: » Exterior algebra: Encyclopedia II - Exterior algebra - Basis and dimension