Clifford algebra

In mathematics, a Clifford algebra^[a] is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As $K$ -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.^[1]^[2] 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.^[b]

Introduction and basic properties

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space $V$ over a field $K$ , where $V$ is equipped with a quadratic form $Q : V \to K$ . The Clifford algebra $Cl(V, Q)$ is the "freest" unital associative algebra generated by $V$ subject to the condition^[c]

v^{2}=Q(v)1\ {\text{ for all }}v\in V,

where the product on the left is that of the algebra, and the

1

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 $V$ is a finite-dimensional real vector space and $Q$ is nondegenerate, $Cl(V, Q)$ may be identified by the label $Cl p, q (R)$ , indicating that $V$ has an orthogonal basis with $p$ elements with $e i 2 = +1$ , $q$ with $e i 2 = -1$ , and where $R$ 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 $V$ may be written as the tensor algebra $⨁ n \geq0 V \otimes \dots \otimes V$ , that is, the direct sum of the tensor product of $n$ copies of $V$ over all $n$ . Therefore one obtains a Clifford algebra as the quotient of this tensor algebra by the two-sided ideal generated by elements of the form $v \otimes v - Q (v)1$ for all elements $v \in V$ . The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. $uv$ ). Its associativity follows from the associativity of the tensor product.

The Clifford algebra has a distinguished subspace $V$ , being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a $K$ -algebra that is isomorphic to the Clifford algebra.

If $2$ is invertible in the ground field $K$ , then one can rewrite the fundamental identity above in the form

uv+vu=2\langle u,v\rangle 1\ {\text{ for all }}u,v\in V,

where

\langle u,v\rangle ={\frac {1}{2}}\left(Q(u+v)-Q(u)-Q(v)\right)

is the symmetric bilinear form associated with

Q

, via the polarization identity.

Quadratic forms and Clifford algebras in characteristic $2$ form an exceptional case in this respect. In particular, if $char(K) = 2$ it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfies $Q (v) = ⟨ v, v ⟩$ ,^[3] Many of the statements in this article include the condition that the characteristic is not $2$ , 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 $Q = 0$ then the Clifford algebra $Cl(V, Q)$ is just the exterior algebra $⋀ V$ . Whenever $2$ is invertible in the ground field $K$ , there exists a canonical linear isomorphism between $⋀ V$ and $Cl(V, Q)$ . 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 $Q$ .

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 $V$ be a vector space over a field $K$ , and let $Q : V \to K$ be a quadratic form on $V$ . In most cases of interest the field $K$ is either the field of real numbers $R$ , or the field of complex numbers $C$ , or a finite field.

A Clifford algebra $Cl(V, Q)$ is a pair $(A, i)$ ,^[d]^[4] where $A$ is a unital associative algebra over $K$ and $i$ is a linear map $i : V \to Cl(V, Q)$ that satisfies $i (v) 2 = Q (v)1$ for all $v$ in $V$ , defined by the following universal property: given any unital associative algebra $A$ over $K$ and any linear map $j : V \to A$ such that

j(v)^{2}=Q(v)1_{A}{\text{ for all }}v\in V

[a]

[1]

[2]

[b]

[c]

[3]

[d]

[4]