Clifford algebra as a super-enveloping quotient

Statement

Let $k$ be a field of characteristic $0$ , let $(V,q)$ be a quadratic vector space, and put $b_q(v,w)=q(v+w)-q(v)-q(w)$ . Define a Lie superalgebra

\mathfrak h_q=kz\oplus V

by declaring $z$ even and central, every element of $V$ odd, and

[v,w]=-b_q(v,w)z.

For the Clifford convention $v^2=-q(v)1$ , there is an isomorphism of superalgebras

\operatorname{Cl}(V,q) \cong U(\mathfrak h_q)/(z-1),

where $U(\mathfrak h_q)$ is the universal enveloping algebra.

Verification of the relation

Because $v,w$ are odd, the defining enveloping-algebra relation reads

vw+wv=[v,w]=-b_q(v,w)z.

After imposing $z=1$ , this becomes the polarized Clifford relation. Setting $w=v$ gives

2v^2=-b_q(v,v)=-2q(v),

and hence $v^2=-q(v)$ . The universal properties of the two quotient algebras then give the stated isomorphism.

Interpretation

The odd bracket in $\mathfrak h_q$ stores the quadratic form as a central even value. The quotient $z=1$ chooses a nonzero central character. Choosing $z=0$ instead gives

U(\mathfrak h_q)/(z)\cong\Lambda V,

so the exterior and Clifford algebras appear as two central fibers of the same super-enveloping construction.

Conventions

If the alternative Clifford convention $v^2=+q(v)$ is used, define $[v,w]=+b_q(v,w)z$ instead. The sign in the Lie-superalgebra bracket and the sign in the Clifford relation must be changed together.

The construction also admits versions over fields of characteristic different from $2$ , provided Lie superalgebras and their enveloping algebras are defined in that broader setting. The statement above uses characteristic $0$ so that its terminology agrees exactly with the linked knowls.

References

P. Deligne and J. W. Morgan, “Notes on supersymmetry (following Joseph Bernstein),” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999. Relevant: Sections 1–2.
I. M. Musson, Lie Superalgebras and Enveloping Algebras, American Mathematical Society, 2012. DOI record. Relevant: Chapters 1 and 6.