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. |