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.


1 Answer 1


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.


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