A uniform construction of the root lattices \(E_ 6,E_ 7,E_ 8\) and their dual lattices
A simple construction of the root lattice \(E_r\) and its dual lattices \(E^*_r\) is given, which works uniformly with respect to the rank \(r= 6, 7, 8\). This construction is inspired by the classical theory of del Pezzo surfaces and the viewpoint of Mordell-Weil lattices. Let \(L_r\) be a free \(\mathbb{Z}\)-module of rank \(r= 6, 7, 8\), generated by \(r\) elements \(u_1, \dots, u_r\), and define a symmetric bilinear pairing on \(L_r\) by the rule \((u_i, u_j)= \delta_{ij}+ {1\over {9-r}}\). Let \(\widetilde {L}_r\) be the \(\mathbb{Z}\)-submodule of \(L_r \otimes Q\) generated by \(L_r\) and element \(v_0= {1\over 3} \sum^r_{i=1} u_i\). Let \(L^0_r\) be the sublattice of \(\widetilde {L}_r\) generated by \(r\) elements \(\alpha_i= u_i- u_j\) \((i=1, \dots, r-1)\) and \(\beta= v_0- u_1- u_2- u_3\). Then the lattice \(L^0_r\) is isomorphic to the root lattice \(E_r\) and \(\widetilde {L}_r\) is isomorphic to the dual lattice \(E^*_r\) of \(E_r\). The set \(\{\alpha_i, \beta\}\) forms a basis of the root system of type \(E_r\). An advantage of this construction is that the description of the minimal vectors in \(E_r\) and \(E^*_r\) is reasonably concise.
scientific article; zbMATH DE number 897671