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 \({C(r,s)}\), which is equivalent to considering the Clifford algebra generated by an inner product space \({\mathbb{C}^{n}}\). Since the signature is irrelevant in this case, 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 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.

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