Clifford algebra - Relation to the exterior 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 this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.) The Pin group PinV(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group SpinV(K) is the subgroup of elements of Dickson invariant 0 in PinV(K). When the characteristic is not 2, these are the elements of determinant ...

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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 - Spin and Pin groups

Suppose that p+q=2n is even. Then the Clifford algebra Cℓp,q(C) is a matrix algebra, and so has a complex representation of dimension 2n. By restricting to the group Pinp,q(R) we get a complex representation of the Pin group of the same dimension, called the spinor representation. If we restrict this to the spin group Spinp,q(R) then it splits as the sum of two half spin representations (or Weyl representatio ...

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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 - Spinors

Clifford algebra - Differential geometry. One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. ...

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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 - Applications

In this section we assume that V is finite dimensional and the bilinear form of Q is non-singular. The Clifford group Γ is defined to be the set of invertible elements x of the Clifford algebra such that xvα(x)−1 ∈ V for all v in V. This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group cont ...

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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 - The Clifford group Γ

In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. If V has even dimension then Cℓ(V,Q) is a central simple algebra over K. If V has even dimension then ...

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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 - Structure of Clifford algebras

If the dimension of V is n and {e1,…,en} is a basis of V, then the set is a basis for Cℓ(V,Q). The empty product (k = 0) is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. A ...

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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 - Basis and dimension

The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form: where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted Rp,qSee also:

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 - Examples: Real and complex Clifford algebras

Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. The Clifford algebra Cℓ(V,Q) is the "freest" algebra generated by V subject to the condition1 If the characteristic of the ground field K is not 2, then one can rewri ...

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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 - Introduction and basic properties

Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. In most cases of interest the field K is either R or C (which have characteristic 0) or a finite field. A Clifford algebra Cℓ(V,Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V,Q) defined by the following universal property: Given any associative algebra A over K a ...

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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 - Universal property and construction