Clifford algebra - Knowlpedia

Let $V$ be a $k$ -module over a commutative ring $k$ in which $2$ is invertible, and let $q:V\to k$ be a quadratic form. The Clifford algebra $\operatorname{Cl}(V,q)$ is the unital associative algebra

\operatorname{Cl}(V,q) =T(V) \big/\langle v\otimes v+q(v)1:v\in V\rangle .

Here $T(V)$ is the tensor algebra. Thus the image of every $v\in V$ satisfies $v^2=-q(v)1$ . If $q(v)=g(v,v)$ comes from an inner product, polarization gives

vw+wv=-2g(v,w)1.

This sign is the common Riemannian-geometry convention; many algebra texts instead impose $v^2=q(v)1$ .

Universal property

The quotient is characterized without choosing a basis. If $A$ is any unital associative $k$ -algebra and $f:V\to A$ is linear with

f(v)^2=-q(v)1_A,

then there is a unique unital algebra homomorphism

\widetilde f:\operatorname{Cl}(V,q)\to A

whose restriction to $V$ is $f$ . This property makes Clifford multiplication natural under isometries of quadratic spaces.

Grading and dimension

The tensor-degree parity descends to a $\mathbb Z/2$ -grading

\operatorname{Cl}(V,q) =\operatorname{Cl}^0(V,q)\oplus\operatorname{Cl}^1(V,q).

Vectors lie in the odd part, while products of an even number of vectors lie in the even part. If $V$ is free of finite rank $n$ , then the usual ordered monomials in a basis show that $\operatorname{Cl}(V,q)$ has rank $2^n$ as a $k$ -module, without requiring $q$ to be nondegenerate.

This parity grading makes $\operatorname{Cl}(V,q)$ a superalgebra. It is generally not supercommutative: an odd vector can have the nonzero square $-q(v)1$ , whereas odd elements in a supercommutative algebra have square zero when $2$ is invertible.

Filtration and exterior algebra

Tensor degree does not descend to a $\mathbb Z$ -grading because the relation identifies a degree-two tensor with a scalar. It does descend to an increasing filtration $F^0\subseteq F^1\subseteq\cdots$ . For a finite-dimensional quadratic vector space over a field of characteristic different from $2$ , the associated graded Clifford algebra is

\operatorname{gr}_F\operatorname{Cl}(V,q)\cong \Lambda V.

Here $\Lambda V$ is the exterior algebra. Thus the Clifford algebra is a filtered deformation of the exterior algebra, not generally an isomorphic algebra.

The quadratic relation also has a Lie-superalgebra interpretation: $\operatorname{Cl}(V,q)$ is a central quotient of a super universal enveloping algebra.

Real and complex forms

For a real quadratic space one obtains a real Clifford algebra, whose isomorphism type depends on the signature and displays mod- $8$ periodicity. Complexification gives

\operatorname{Cl}_{\mathbb C}(V,q) =\operatorname{Cl}_{\mathbb R}(V,q)\otimes_{\mathbb R}\mathbb C.

Over $\mathbb C$ , the signature no longer affects the isomorphism type, although parity of $\dim V$ still matters.

Sign warning

Two standard conventions coexist:

v^2=-q(v)1 \qquad\text{or}\qquad v^2=q(v)1.

They interchange some signature labels and change the displayed Clifford relation. A formula for spinors or Dirac operators is therefore meaningful only after its sign convention has been fixed.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapter I, Clifford algebras and modules.
Nicole Berline, Ezra Getzler, and Michèle Vergne, Heat Kernels and Dirac Operators, Springer, 1992. DOI record. Relevant: Chapter 3, Clifford algebras.