Associated graded Clifford algebra

Statement

Let $V$ be a finite-dimensional vector space over a field $k$ of characteristic different from $2$ , and let $q$ be a quadratic form on $V$ . Give the Clifford algebra $\operatorname{Cl}(V,q)$ the filtration $F^r$ induced by tensors of degree at most $r$ . Then there is a canonical isomorphism of graded algebras

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

where $\Lambda V$ is the exterior algebra.

Why the exterior relation appears

Under the convention $v^2=-q(v)1$ , polarization gives

vw+wv=-b_q(v,w)1, \qquad b_q(v,w)=q(v+w)-q(v)-q(w).

The right side has filtration degree $0$ , while the left side has degree $2$ . Passing to the leading symbols therefore gives

\sigma(v)\sigma(w)+\sigma(w)\sigma(v)=0.

These are exactly the exterior-algebra relations. The theorem says that they are all the leading relations: the induced surjection $\Lambda V\to\operatorname{gr}\operatorname{Cl}(V,q)$ is an isomorphism.

Consequences

If $n=\dim V$ , then

\dim_k\operatorname{Cl}(V,q)=2^n,

even when $q$ is degenerate. An ordered basis $e_1,\ldots,e_n$ yields a basis of Clifford monomials

e_{i_1}\cdots e_{i_r}, \qquad i_1<\cdots<i_r,

parallel to the usual basis of $\Lambda V$ .

What the theorem does not say

The isomorphism is an isomorphism with the associated graded algebra, not an isomorphism of $\operatorname{Cl}(V,q)$ with $\Lambda V$ . The Clifford product is a filtered deformation of the exterior product: the quadratic form is visible in lower filtration degree and disappears only after taking leading symbols.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapter I, Section 1.
Claude Chevalley, The Algebraic Theory of Spinors and Clifford Algebras, Springer, collected works edition, 1997. DOI record. Relevant: Chapters I–II.