Section 8.5 The Geometry of \(F_4\) and \(E_6\)
Both \(F_4\) and \(E_6\) act on the Albert algebra, which was introduced in Section 8.2. They are therefore also symmetries of the projective spaces discussed in Sections 8.3–8.4.
We will argue below that \(\ff_4\cong\su(3,\OO)\cong\su(3,\RR\otimes\OO)\) and \(\ee_{6(-26)}\cong\sl(3,\OO)\cong\su(3,\CC'\otimes\OO)\text{.}\) Before proceeding, it is worth pausing to review the properties of the corresponding elements of the \(2\times2\) magic square, namely \(\su(2,\OO)\cong\su(2,\RR\otimes\OO)\cong\so(9)\text{,}\) and \(\sl(2,\OO)\cong\su(2,\CC'\otimes\OO)\cong\so(9,1)\text{.}\) The Lorentz group \(\SO(9,1)\) in ten spacetime dimensions preserves the corresponding Lorentzian inner product—which is given by the determinant—and its subgroup \(\SO(9)\) also preserves the “\(t\)” component—which is the trace.
As suggested in Section 8.2, The Albert algebra \(\bH_3(\OO)\) can be regarded as a generalization of the vector space \(\RR^3\) to the octonions. Furthermore, the elements \(\XXX\in\bH_3(\OO)\) are Hermitian matrices, with well-defined determinants (and traces). We therefore seek transformations that preserve these properties.
To make a long story short, special linear groups preserve the determinant, and unitary groups also preserve the trace. Yes, we must verify that our octonionic generalizations of these concepts still have those properties. But the hard work was done at the \(2\times2\) level. See [1] for further details.
We can, however, shed some light on the dimensions of these groups. When constructing \(\so(9)\cong\su(2,\OO)\text{,}\) there are clearly eight off-diagonal generators. Considering the three independent ways of embedding a \(2\times2\) matrix into a \(3\times3\) matrix correctly suggests that there are 24 independent off-diagonal elements of \(\ff_4\text{.}\) How many diagonal elements are there? In the \(2\times2\) case, we can generate the rotations of \(\OO\) as a vector space, that is, of \(\RR^8\text{.}\) So there must be 28 independent \(2\times2\) diagonal elements, generating \(\so(8)\text{.}\) Over the octonions, remarkably, and as discussed briefly in Section 7.4, rearranging the positions of the diagonal elements does not lead to any new degrees of freedom. Thus, \(\ff_4\) turns out to have dimension \(24+28=52\text{.}\)
Almost the same construction enables us to determine the dimension of \(\ee_6\text{.}\) We now have twice as many (48) off-diagonal elements, and we must still have the 28 elements of \(\so(8)\text{.}\) What's missing? The elements containing \(L\text{,}\) the imaginary unit in \(\CC'\text{.}\) In this case, we are not working over the octonions, so we can count as usual. There are two independent diagonal generators proportional to \(L\text{,}\) corresponding to the Gell-Mann matrices \(\lambda_3\) and \(\lambda_8\text{.}\) And we are not working over the quaternions, so we do not need to include \(L\one\text{,}\) which commutes with everthing in \(\ee_6\text{.}\) Thus, \(\ee_6\) turns out to have dimension \(48+28+2=78\text{.}\)