Classification of complex Clifford modules

Statement

Let $V$ be an $n$ -dimensional complex vector space with a nondegenerate quadratic form. Up to isomorphism, its complex Clifford algebra is

\operatorname{Cl}(V)\cong \begin{cases} \operatorname{Mat}_{2^m}(\mathbb C),&n=2m,\\[2mm] \operatorname{Mat}_{2^m}(\mathbb C)\oplus \operatorname{Mat}_{2^m}(\mathbb C),&n=2m+1. \end{cases}

Consequently, in even dimension there is one irreducible ungraded complex Clifford module, of dimension $2^m$ . In odd dimension there are two inequivalent irreducible ungraded modules, each of dimension $2^m$ .

Restriction to the spin group

In dimension $2m$ , the restriction of the irreducible Clifford module to the even Clifford algebra, and hence to the spin group, decomposes as

\Delta_{2m}=\Delta_{2m}^{+}\oplus\Delta_{2m}^{-}.

The two summands are the irreducible half-spin modules, and Clifford multiplication by a vector interchanges them.

In dimension $2m+1$ , the two irreducible ungraded Clifford modules restrict to equivalent irreducible spin representations. That odd-dimensional spin representation has no intrinsic chiral decomposition.

Scope

Over $\mathbb C$ , the classification depends on dimension parity but not on the signature of a real form. The classification of real Clifford modules is different: it depends on the signature and is $8$ -periodic.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. DOI record. Relevant: Chapter I, §§4–5.
Pierre Deligne, “Notes on spinors,” in Quantum Fields and Strings: A Course for Mathematicians, Volume 1, American Mathematical Society, 1999, pp. 99–135.