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Section 8.8 From \(E_6\) to \(E_7\)

We outline the argument that \(E_7\) can be interpreted as the conformal group associated with \(E_6\text{.}\) Further details can be found in [1].

Recall that the minimal representation of \(\ee_6\) is the Albert algebra \(\bH_3(\OO)\text{,}\) and that

\begin{equation} |\ee_6| = 78, \qquad |\bH_3(\OO)| = 27 .\tag{8.8.1} \end{equation}

Thus, whereas \(\so(3,1)\) acts on a 4-dimensional vector space, \(\ee_6\) acts on a 27-dimensional space. Applying the same “conformalization” construction as above, we expect to obtain a new \(\so(1,1)\) symmetry, together with 27 translations and 27 conformal translations. Sure enough,

\begin{equation} 78 + 27 + 27 + 1 = 133 = |\ee_7| ,\tag{8.8.2} \end{equation}

and this construction does indeed turn out to generate \(\ee_7\) from \(\ee_6\text{.}\) This Lie algebra decomposition of \(\ee_7\) is represented symbolically by writing

\begin{equation} \ee_{7(-25)} = \ee_{6(-26)} + 2\times27 + \so(1,1) ,\tag{8.8.3} \end{equation}

where we have identified the real forms in our construction as belonging to the half-split magic square.

However, there is an important difference between “conformalization” as applied to the \(2\times2\) and \(3\times3\) magic squares, which we here merely summarize. One way to see this difference is to rewrite the construction explicitly in terms of two division algebras, noting that conformalization corresponds in both cases to the transition from the second row to the third. It is however more common to use larger matrices rather than explicit split quaternions. In the \(2\times2\) case, we can combine a vector \(\XX\) and our new degrees of freedom \((p,q)\) into a \(4\times4\) matrix of the form

\begin{equation} \PP = \begin{pmatrix} p+q \amp \XX \\ -\tilde\XX \amp p-q \\ \end{pmatrix} ,\tag{8.8.4} \end{equation}

where \(\tilde\XX=\XX-\tr\XX\) as usual denotes trace reversal. There is a duality operation that allows \(\PP\) to be rewritten as a completely antisymmetric matrix. The elements of the conformalized algebra (e.g. \(\so(4,2)\)) can also be expressed as \(4\times4\) matrices, of the general form

\begin{equation} \MM = \begin{pmatrix} \Phi-\frac12\rho \amp A \\ B \amp -\Phi^\dagger+\frac12\rho \\ \end{pmatrix} ,\tag{8.8.5} \end{equation}

where \(\Phi\) is in the original algebra (e.g. \(\so(3,1)\)), so that \(\tr(\Phi)=0\text{,}\) \(\rho\in\RR\) is the dilation, and the Hermitian matrices \(A\) and \(B\) are the translations and conformal translations, respectively. Although (8.8.5) generalizes directly to the \(3\times3\) case, with \(\phi\in\ee_6\text{,}\) \(\rho\in\RR\text{,}\) and \(A,B\in\bH_3(\OO)\text{,}\) (8.8.4) does not.

Figure 8.8.1. The block structure of a \(4\times4\) antisymmetric matrix in terms of \(2\times2\) blocks is shown on the left; the block structure of a \(6\times6\times6\) antisymmetric tensor in terms of \(3\times3\times3\) blocks is shown on the right. Blocks with similar shading contain equivalent information.

The issue is that, in the \(2\times2\) magic square, \(\PP\) is (reinterpreted as) a completely antisymmetric matrix, a rank 2 tensor. In the \(3\times3\) magic square, we must replace \(\PP\) by a completely antisymmetric rank 3 tensor. We can think of this process as replacing a square by a cube, as shown in Figure 8.8.1. The square corresponds to \(\PP\text{,}\) with two real degrees of freedom on the diagonal (\(p\pm q\)), and one vector degree of freedom on the off diagonal (\(\XX\)). In a cube, we still have two real degrees of freedom on the main diagonal, but there are now two independent sets of vectors, in three copies each, filling out the cube. This geometric description correctly explains the fact that the minimal representation of \(\ee_7\) is given by a pair of real numbers \(p\pm q\text{,}\) together with two elements \(\XXX,\YYY\in\bH_3(\OO)\text{.}\) The collection \((p+q,p-q,\XXX,\YYY)\) is called a Freudenthal triple system, and has dimension \(1+1+27+27=56\) as a vector space.