# Why are Clifford Algebras Superalgebras?

In Mathematical Gauge Theory With Applications to the Standard Model of Particle Physics Hamilton on page 332 proves that Clifford algebras are superalgebras by the following:

First he defines two vector subspaces of the tensor algebra as

$$Cl^0(V,Q) = T^0(V)/(T^0(V)\cap I(Q))$$

$$Cl^1(V,Q) = T^0(V)/(T^1(V)\cap I(Q))$$

Then he states that because $$I(Q)= (T^0(V)\cap I(Q))\oplus(T^1(V)\cap I(Q))$$ it follows that $$CL(V,Q) = CL^0(V,Q)\oplus CL^1(V,Q)$$

I do not see why this is the case. Could someone explain this to me? It seems like I'm missing something trivial here.

Clifford algebras are superalgebras — i.e., they are $$\mathbb{Z}_2$$-graded — because they can be expressed as a quotient of the tensor algebra by an ideal which is $$\mathbb{Z}_2$$-graded. There is a theorem which states a graded ideal results in a graded quotient algebra. These main points are summarised below.

Def. An algebra $$A$$ is $$R$$-graded for a monoid $$R$$ (e.g., $$\mathbb{Z}$$ or $$\mathbb{Z}_2$$) if there is a decomposition $$A = \bigoplus_{k \in R} A_k$$ which preserves multiplication, $$A_p A_q \subseteq A_{p + q} \qquad\text{i.e.,}\qquad a \in A_p \text{ and }b \in A_q \implies ab \in A_{p + q} .$$

For example, the tensor algebra $$V^\otimes = \bigoplus_{k=0}^\infty {V_{(1)} \otimes \cdots \otimes V_{(k)}}$$ is $$\mathbb{Z}$$-graded, since the tensor product of rank $$p$$ and $$q$$ tensors is rank $$p + q$$. In fact, $$V^\otimes$$ is $$\mathbb{Z}_n$$-graded for any $$n$$.

Def. An ideal $$I \subseteq A$$ of an $$R$$-graded algebra $$A$$ is itself $$R$$-graded (a.k.a. homogeneous) if $$I = \bigoplus_{k \in R} I_k \quad\text{where}\quad I_k = I \cap A_k .$$

For example, the ideal $$I_{alt}$$ of $$V^\otimes$$ generated by $$\{a\otimes b + b\otimes a\}$$ is $$\mathbb{Z}$$-graded, because $$I_{alt} = I_{alt} \cap (V\otimes V)$$.

Thm. If $$A$$ is an $$R$$-graded algebra, and $$I$$ is an $$R$$-graded ideal, then the quotient algebra $$A/I$$ is also $$R$$-graded.

For example, this implies $$V^\otimes \big/ I_{alt} \cong \bigwedge V$$ is $$\mathbb{Z}$$-graded.

On the other hand, a Clifford algebra is a quotient $$Cl(V, Q) = V^\otimes \big/ \langle a \otimes a - Q(a) \rangle$$ by the ideal generated by $$a\otimes a \sim Q(a)$$. This ideal is not $$\mathbb{Z}$$-graded. However, it is $$\mathbb{Z}_2$$-graded (it is generated by the sum of a scalar and rank-$$2$$ tensor). Since $$V^\otimes$$ is also $$\mathbb{Z}_2$$-graded, the theorem above shows that $$Cl(V, Q)$$ is itself $$\mathbb{Z}_2$$-graded (a.k.a., a superalgebra).

Proofs can be found e.g., in my unpublished thesis, where I copied these from.