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\newcommand{\id}{\mathop{\rm id}\nolimits} \)
BOTT PERIODICITY FOR CLIFFORD ALGEBRAS
ISHAN LEVY
For a real or complex finite dimensional vector space vector space with a nondegenerate
quadratic form, let’s try to identify the algebra we get as the Clifford algebra. Nondegenerate real
quadratic forms are classified by rank and signature, and complex ones by rank, so we need to
identify \(\Cl (p,q)\), the real Clifford algebra where the positive definite part is \(p\)-dimensional and the negative
definite part \(q\)-dimensional, and \(\Cl (n)\), the complex Clifford algebra coming from a complex \(n\) dimensional
space.
The complex case is easy to deal with. It is easy to identify \(\Cl (0) = \CC , \Cl (1) = \CC \oplus \CC \). \(\Cl (2)\) is easily seen to be \(\End (\CC ^2)\) by sending an
orthonormal basis to \(\begin{pmatrix}i & 0 \\ 0 & -i\end{pmatrix}, \begin{pmatrix}0 & i \\ i & 0\end{pmatrix}\). These cases suffice to compute the rest of \(\Cl (n)\). If \(n \geq 2\), we there is a map from \(\Cl (n+2)\) to \(\Cl (n) \otimes \Cl (2)\)
produced in the following way. Let \(e_1,\dots e_{n+2}\) be the orthonormal primitive elements of \(\Cl (n)\). and similarly let \(e'_1,\dots e'_n\)
and \(e'_{n+1}, e'_{n+2}\) be the same for \(\Cl (n), \Cl (2)\) respectively. Then we can send \(e_i, 1 \leq i \leq n\) to \(i e'_i\otimes e'_{n+1}e'_{n+2}\) and \(e_i, n< i\) to \(1 \otimes e_i\). Via the universal property
one sees that the relations are satisfied to extend to an algebra homomorphism. Since
it is injective on generators, it is injective, and for dimension reasons, it must be an
isomorphism.
Thus we inductively get that \(\Cl (n) \cong \End (\CC ^{2^{\frac n 2}})\) for even \(n\) and \(\End (\CC ^{2^{\frac{n-1} 2}}) \oplus \End (\CC ^{2^{\frac{n-1} 2}})\) for odd \(n\). There is a periodicity here with period \(2\): it
alternates between being a matrix algebra and a sum of them. One way to think about the
periodicity, is to say that \(\Cl (n)\) is Morita equivalent to \(\Cl (n+2)\), meaning that they have the same categories of
left modules. Indeed for any division ring \(R\), an equivalence between left modules on \(R\) and \(\End (R^n)\) is given
by the functor sending a left module \(M\) to \(M \otimes _R R^n\) where \(R^n\) is a left \(\End{(R^n)}\) module, but also a \(R\)-module via the
natural homomorphism \(R \to \End{(R^n)}\). Thus categorical properties of complex Clifford modules depend on the
parity of the dimension.
Theorem 0.1. For even \(n\), \(\Cl (n)\) is Morita equivalent to \(\CC \), and for odd \(n\), \(\Cl (n)\) is Morita equivalent to \(\CC \oplus \CC \).
The same kind of Bott periodicity result holds for real Clifford algebras, but it works mod \(8\). To
prove it we can produce relations among the real Clifford algebras similarly to the way it was
proven for complex Clifford algebras. If \(e_1,\dots , e_{n+2}\) is an orthonormal basis for \(\Cl (n+2,0)\), and \(e'_1,\dots ,e'_n\), \(e'_{n+1}, e'_{n+2}\) orthonormal bases for \(\Cl (2,0), \Cl (0,n)\)
respectively, then by sending \(e_i, 1 \leq i \leq n\) to \(e'_i \otimes e'_{n+1}e'_{n+2}\) and \(e_i, n< i\) to \(1 \otimes e'_i\) we get an algebra isomorphism \(\Cl (n+2,0) \cong \Cl (0,n) \otimes \Cl (2,0)\). The exact same map
also gives isomorphisms \(\Cl (0,n+2) \cong \Cl (n,0) \otimes \Cl (0,2)\) and \(\Cl (p+1,q+1) \cong \Cl (p,q) \otimes \Cl (1,1)\). \(\Cl (1,1)\) is seen to be \(\End (\RR ^2)\) by sending an orthonormal basis to \(\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}, \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\). Since this is
Morita equivalent to \(\RR \), we already see that the Morita equivalence class is only dependent on the
signature \(\sigma = p-q\). Moreover, \(\Cl (0,2)\) is also \(\End (\RR ^2)\) via \(\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}, \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}\) and \(\Cl (2,0)\) is \(\HH \). Since \(\HH \) is the nontrivial element in the Brauer group of \(\RR \),
the relations give \(\Cl (n+8,0) \cong \Cl (n,0) \otimes \End (\RR ^4) \otimes \HH \otimes \HH = \Cl (n,0) \otimes \End (\RR ^{16})\) and similarly for \(\Cl (0,n)\), so the Morita equivalence class for positive and
negative signatures depends only on \(\sigma \) mod \(8\). To compute it, we can begin with the basic
examples \(\Cl (0,0) = \RR , \Cl (0,1) = \RR \oplus \RR , \Cl (1,0) = \CC \), and use the relations proven above and the facts that \(\CC \otimes _\RR \HH = \End (\CC ^2), \CC \otimes _\RR \CC = \CC \oplus \CC \). The result is the
following:
Theorem 0.2. The Morita equivalence class of \(\Cl (p,q)\) depends only on \(\sigma = p-q\) mod \(8\). It is given by:
- \(\RR , \sigma \equiv 0\)
- \(\CC , \sigma \equiv 1\)
- \(\HH , \sigma \equiv 2\)
- \(\HH \oplus \HH , \sigma \equiv 3\)
- \(\HH , \sigma \equiv 4\)
- \(\CC , \sigma \equiv 5\)
- \(\RR , \sigma \equiv 6\)
- \(\RR \oplus \RR , \sigma \equiv 7\)
This can actually be used to compute what the Clifford algebra is since we know the
dimension. For example, \(\Cl (4,7)\) has Morita equivalence class \(\CC \) and dimension \(2^{11}\) so it must be
\(\End (\CC ^{2^5})\).