# Isomorphisms

The Clifford algebra can be viewed as a $${\mathbb{Z}_{2}}$$-graded algebra, in that it can be decomposed into a direct sum of two vector subspaces generated by $${k}$$-vectors with $${k}$$ either even or odd. The even subspace, denoted $${C_{0}(r,s)}$$, is also a subalgebra, since the Clifford multiplication of two even $${k}$$-vectors remains even.

By choosing $${\hat{e}_{0}^{2}=-1\in C(r,s)}$$ and considering the algebra generated by the orthonormal basis $${\hat{e}_{0}\hat{e}_{i}\:(i\neq0)}$$, it is not hard to show that

$$\displaystyle C_{0}(r,s)\cong C(r,s-1)$$.

Then the relationship $${C_{0}(r,s)\cong C_{0}(s,r)}$$ leads to the isomorphism

$$\displaystyle C(r,s-1)\cong C(s,r-1)$$.

One can also show that:

• $${C(r,s)\otimes C(2,0)\cong C(r,s)\otimes\mathbb{R}(2)\cong C(s+2,r)\cong C(r+1,s+1)}$$
• $${C(r,s)\otimes C(0,2)\cong C(r,s)\otimes\mathbb{H}\cong C(s,r+2)}$$
• $${C(r,s)\otimes C(0,4)\cong C(r,s)\otimes\mathbb{H}(2)\cong C\left(r,s+4\right)}$$
• $${C\left(r-4,s+4\right)\cong C\left(r,s\right)}$$
• The periodicity theorem (related to and sometimes referred to as Bott periodicity): $${C\left(r+8,s\right)\cong C\left(r,s+8\right)\cong C\left(r,s\right)\otimes\mathbb{R}\left(16\right)}$$

The first isomorphism $${C(r+1,s+1)\cong C(r,s)\otimes\mathbb{R}(2)}$$ means that we need only consider classifying Clifford algebras based on the values of $${r-s}$$, and the periodicity theorem means that we can focus on values of $${r-s}$$ mod 8.

In physics, the most important signatures are Euclidean and Lorentzian; specific isomorphisms for some of these Clifford algebras are listed in the following table. Note that since the first column covers all values of $${r-s}$$ mod 8, it can be used to easily determine any other Clifford algebra.

\begin{aligned}n\end{aligned}\begin{aligned}C\left(n,0\right) & \cong C\left(1,n-1\right)\\
& \cong C_{0}\left(n,1\right)\\
& \cong C_{0}\left(1,n\right)
\end{aligned}
\begin{aligned}C\left(0,n\right) & \cong C_{0}\left(n+1,0\right)\\
& \cong C_{0}\left(0,n+1\right)
\end{aligned}
\begin{aligned}C\left(n-1,1\right)\end{aligned}
1$${\mathbb{R}\oplus\mathbb{R}}$$$${\mathbb{C}}$$$${\mathbb{C}}$$
2$${\mathbb{R}\left(2\right)}$$$${\mathbb{H}}$$$${\mathbb{R}\left(2\right)}$$
3$${\mathbb{C}\left(2\right)}$$$${\mathbb{H}\oplus\mathbb{H}}$$$${\mathbb{R}\left(2\right)\oplus\mathbb{R}\left(2\right)}$$
4$${\mathbb{H}\left(2\right)}$$$${\mathbb{H}\left(2\right)}$$$${\mathbb{R}\left(4\right)}$$
5$${\mathbb{H}\left(2\right)\oplus\mathbb{H}\left(2\right)}$$$${\mathbb{C}\left(4\right)}$$$${\mathbb{C}\left(4\right)}$$
6$${\mathbb{H}\left(4\right)}$$$${\mathbb{R}\left(8\right)}$$$${\mathbb{H}\left(4\right)}$$
7$${\mathbb{C}\left(8\right)}$$$${\mathbb{R}\left(8\right)\oplus\mathbb{R}\left(8\right)}$$$${\mathbb{H}\left(4\right)\oplus\mathbb{H}\left(4\right)}$$
8$${\mathbb{R}\left(16\right)}$$$${\mathbb{R}\left(16\right)}$$$${\mathbb{H}\left(8\right)}$$

Note: Clifford multiplication corresponds to matrix multiplication in the isomorphic matrix algebra. Recall that our notation denotes e.g. the algebra of $${2\times2}$$ matrices of quaternions as $${\mathbb{H}(2)}$$.

We can also form the complexified version of the Clifford algebra $${C(r,s)}$$, for which the signature is irrelevant and we simply write $${C\mathbb{^{C}}(n)}$$, where $${r+s=n}$$. The complex Clifford algebras can be completely described by the following isomorphisms:

• $${C\mathbb{^{C}}(2n)\cong\mathbb{C}(2^{n})}$$
• $${C\mathbb{^{C}}(2n+1)\cong\mathbb{C}(2^{n})\oplus\mathbb{C}(2^{n})}$$

Note that this yields an isomorphism $${C\mathbb{^{C}}(2n)\cong C(n,n+1)\cong C(n+2,n-1)}$$; in contrast, $${C\mathbb{^{C}}(2n+1)}$$ is not isomorphic to any real Clifford algebra. Also note that $${C_{0}\mathbb{^{C}}(n)\cong C\mathbb{^{C}}(n-1)}$$.

 Δ Note that when complexifying, we have e.g. $${\left(iv\right)\left(iw\right)=\left\langle iv,iw\right\rangle =-vw=-\left\langle v,w\right\rangle}$$, so that $${\left\langle \right\rangle}$$ is not a complex inner product as we have defined it; in particular, $${\left\langle v,v\right\rangle}$$ can have an imaginary part.
 Δ Although the above are all valid algebra isomorphisms, the original formulation of a Clifford algebra includes an extra structure: the generating vector space $${\mathbb{R}^{n}}$$ that is explicitly embedded in $${C(r,s}$$). This extra structure is lost in these isomorphisms, since the choice of such an embedding is not in general unique.