A uniform construction of the root lattices $E_6, E_7, E_8$ and their dual lattices
Tetsuji Shioda · Proceedings of the Japan Academy Series A Mathematical Sciences · 1995
A simple construction of the root lattice Er and its dual lattice E* is given, which works uni- formly with respect to the rank r 6,7,8.An advantage of this construction is that the descrip- tion of the minimal vectors in E r and E* is reasonably concise; for instance, the 240 roots in E 8 can be enumerated in a few lines.(Compare with standard references such as [1], [2, Ch.4,8], [3, Ch. 4]).Our construction is inspired by 1) the clas- sical theory of del Pezzo surfaces, according to which a del Pezzo surface of degree d 1,2,3 is a blowing up of r 9-d points from the pro- jective plane P (cf.[3]), and 2) the viewpoint of Mordell-Weil lattices (cf.[4], [5], [61, [7]).