In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford (1845–1879).
The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct from symplectic Clifford algebras.
Introduction and basic properties
A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a quadratic form . The Clifford algebra is the "freest" unital associative algebra generated by subject to the condition
where the product on the left is that of the algebra, and the is its multiplicative identity. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below.
When is a finite-dimensional real vector space and is nondegenerate, may be identified by the label , indicating that has an orthogonal basis with elements with , with , and where indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. This basis may be found by orthogonal diagonalization.
The free algebra generated by may be written as the tensor algebra , that is, the direct sum of the tensor product of copies of over all . Therefore one obtains a Clifford algebra as the quotient of this tensor algebra by the two-sided ideal generated by elements of the form for all elements . The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. ). Its associativity follows from the associativity of the tensor product.
The Clifford algebra has a distinguished subspace , being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a -algebra isomorphic to the Clifford algebra.
If the characteristic of the ground field is not , then one can rewrite the fundamental identity above in the form
where
is the symmetric bilinear form associated with , via the polarization identity.
Quadratic forms and Clifford algebras in characteristic form an exceptional case. In particular, if it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying , nor that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not , and are false if this condition is removed.
As a quantization of the exterior algebra
Clifford algebras are closely related to exterior algebras. Indeed, if then the Clifford algebra is just the exterior algebra . For nonzero there exists a canonical linear isomorphism between and whenever the ground field does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the exterior product since it makes use of the extra information provided by .
The Clifford algebra is a filtered algebra, the associated graded algebra is the exterior algebra.
More precisely, Clifford algebras may be thought of as quantizations (cf. quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.
Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.
Universal property and construction
Let be a vector space over a field , and let be a quadratic form on . In most cases of interest the field is either the field of real numbers , or the field of complex numbers , or a finite field.
A Clifford algebra is a pair , where is a unital associative algebra over and is a linear map satisfying for all in , defined by the following universal property: given any unital associative algebra over and any linear map such that
