THE GEOMETRY OF THE EXCEPTIONAL LIE GROUPS

Section 7.1 The \(2\times2\) Magic Square

In Section 6.8 we constructed \(4\times4\) gamma matrices over any pair of (possibly split) division algebras. In the “half-split” case of one division algebra \(\KK\) and one split division algebra \(\KK'\text{,}\) with dimensions \(\kappa\) and \(\kappa'\text{,}\) respectively, these gamma matrices generate \(\Cl(\kappa+\frac{\kappa'}2,\frac{\kappa'}2)\text{,}\) which in turn generate \(\so(\kappa+\frac{\kappa'}2,\frac{\kappa'}2)\) as the elements of rank \(2\text{.}\) These orthogonal algebras, parameterized by \(\KK\) and \(\KK'\text{,}\) make up the half-split \(\bmit{2\times2}\) magic square, as shown in Table 7.1.1.

Many of the algebras shown in Table 7.1.1 have alternate names. First of all, there is the obvious isomorphism \(\so(3)\cong\su(2)\text{.}\) We also have \(\so(2,1)\cong\sl(2,\RR)\text{,}\) which is a real form of \(\su(2)\text{,}\) and the Lorentz Lie algebra satisfies \(\so(3,1)\cong\sl(2,\CC)\text{.}\) Finally, since \(\bb_2\cong\cc_2\text{,}\) we have \(\so(5,1)\cong\sp(2)\cong\su(2,\HH)\text{,}\) and it is also true that \(\so(4,2)\cong\su(2,2)\cong\sp(2,\CC)\text{.}\) ¹ (Extending the definitions of the special linear (\(\sl\)) and symplectic (\(\sp\)) algebras in a reasonably straightforward fashion, the first three rows of the half-split \(2\times2\) magic square can be described as shown in Table 7.1.2. However, the fourth row does not appear to admit analogous alternate names.

If instead two division algebras are used, we obtain the compact \(\bmit{2\times2}\) magic square, as shown in Table 7.1.3, and if two split division algebras are used, we get the double-split \(\bmit{2\times2}\) magic square, as shown in Table 7.1.4.

The orthogonal Lie algebras in these magic squares act on both spinors (column vectors) and vectors (matrices \(X\) of the form (6.8.7)). As shown in Section 6.8, a basis for \(\so(p,q)\) can be obtained in the form \(\sigma_m\sigma_n\text{,}\) using the generalized Pauli matrices defined in (6.8.9). How many such basis elements are there, and what do they look like?

Consider first some simple examples. We work throughout with the half-split magic square, but only minor changes are needed to handle the other magic squares. The Lorentz Lie algebra \(\so(3,1)\) is generated by \(\{\sigma_L,\sigma_1,\sigma_i,\sigma_U\}\text{,}\) which are new names for our old friends \(\{L\one,\sigma_x,\sigma_y,\sigma_z\}\text{.}\) There are six independent products of these four matrices, namely the \(i\sigma_m\) and \(L\sigma_m\text{,}\) with \(m=x,y,z\text{.}\) All of these products are anti-Hermitian.

Moving up to the quaternions with \(\so(5,1)\text{,}\) we add \(\{\sigma_j,\sigma_k\}\) to our basis, resulting in the additional products \(\{L\sigma_m,m\sigma_x,m\sigma_z\}\) with \(m=j,k\text{,}\) as well as new products of the form \(q\one\text{,}\) where \(q=i,j,k\text{.}\) Again, all of these matrices are anti-Hermitian. Writing out these 15 matrices, there are \(3+1=4\) traceless diagonal matrices (one for each imaginary unit element in \(\CC'\oplus\HH\)), along with three imaginary multiples of the identity matrix (one for each imaginary unit element in \(\HH\)), as well as eight off-diagonal matrices (corresponding to \(\CC'\otimes\HH\)). ²

This pattern continues: We get \(\kappa\kappa'\) off-diagonal anti-Hermitian matrices, corresponding to \(\KK\otimes\KK'\text{,}\) along with \((\kappa-1)+(\kappa'-1)\) traceless diagonal matrices, together with three imaginary diagonal matrices for each copy of \(\HH\) or \(\HH'\) that is present. What about copies of \(\OO\) or \(\OO'\text{?}\) As discussed in Section 6.9, in this case (only) we must treat the product of Pauli matrices as composition, that is, we must nest. Thus, rather than seven imaginary multiples of the identity matrix, we get \({7\choose2}=21\) pairs of imaginary multiples of the identity matrix—just enough to handle the \(\so(7)\) rotations of the imaginary units.

We will revisit this counting of basis elements in Section 7.4, verifying that we have just enough anti-Hermitian matrices to serve as a basis for \(\so(p,q)\) in each case. This construction justifies referring to the elements of all three magic squares as \(\su(2,\KK'\otimes\KK)\) for suitable choices of \(\KK'\) and \(\KK\text{.}\)