The connection between the symmetric space \(E_ 6/SO(10)\cdot SO(2)\) and projective planes
Let \({\mathcal U}^{(i)}\), \(i=1,2\) be real composition algebras with canonical involution and norm, let \(M^{(3)}\) denote a \(3\times 3\) real matrix algebra and let \({\mathcal M}\) be the space of skew symmetric elements of trace zero in \({\mathcal U}^{(1)}\otimes M^{(3)}\otimes {\mathcal U}^{(2)}\). The author first recalls that if Der \({\mathcal U}^{(i)}\) is the Lie algebra of inner derivations of \(U^{(i)}\) and \(U^{(1)}\) is the Cayley algebra \({\mathfrak C}\) then \(L(U^{(1)},M^{(3)},U^{(2)})=Der U^{(1)}\oplus {\mathcal M}\oplus Der U^{(2)}\) can be made into a Lie algebra and furthermore \(L(U^{(1)},M^{(3)},U^{(2)})\) is of type \(F_ 4\), \(E_ 6\), \(E_ 7\) or \(E_ 8\) according to whether \(U^{(2)}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}}\) or \({\mathfrak C}\) [see the author, Kodai Math. J. 6, 122-133 (1983; Zbl 0521.17006)]. If \({\mathfrak g}\) is a real semisimple Lie algebra and \({\mathcal X}=\{x: ad^ 2_ x(ad^ 2_ x+1)(ad^ 2_ x+4)=0\}\) then the connected component of any x in \({\mathcal X}\) is a reductive homogeneous space. If \({\mathfrak g}=L({\mathfrak C},M^ 3,U^{(2)})\) and \[ x= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix} \] the dimension of the connected component of x is 30, 48, 84 or 156 depending on \({\mathfrak g}\) being of type \(F_ 4\), \(E_ 6\), \(E_ 7\) or \(E_ 8\). If G is the adjoint group of \({\mathfrak g}\) and \(P(x)=-(4/3) ad^ 2_ x -(1/3) ad^ 2_ x\) then \(\Pi =\{gP(x): g\in G\}\) is a compact connected globally symmetric space of dimension 16, 32, 64 or 128 according to the type of \({\mathfrak g}.\) The author studies the structure of maximal flat tori in \(\Pi\) when \({\mathfrak g}\) is of type \(E_ 6\). Also for such a \({\mathfrak g}\), the structure of \(\Pi\), points and lines, is analyzed from the viewpoint of projective geometry showing that \(\Pi\) is a projective plane in the wider sense.
scientific article; zbMATH DE number 3962672