WITH 27 RECTILINEAR GENERATORS AND 108 TRISECANTS AND ITS CONNECTION WITH THE MAXIMAL EXCEPTIONAL SIMPLE LIE GROUP

Abstract. A 2-dimensional algebraic variety in 4-dimensional projective space determining a regular configuration is considered and its connection with sim-ple exceptional Lie group E8 is found. 1. Galois and Weyl groups connected with simple Lie groups With every complex and real compact simple Lie group two finite groups called “Galois and Weyl groups ” are connected. Galois group Γ of a simple Lie group is the Galois group of characteristic equation of this group, Weyl group W of this Lie group is the group generated by reflections determined by root vectors of this Lie group (see [1, pp. 63 and 69]). É. Cartan [2] has proved that for simple Lie groups Bn, Cn, G2, F4, E7 and E8 groups V and W are isomorphic, and for simple Lie groups An, Dn, and E6 group W is an invariant subgroup of group Γ, and the quotient group Γ/W for group D4 consists of 3! elements, for other groups An, Dn, and E6 consists of 2 elements. Note that the Dynkin diagrams for simple Lie groups with isomorphic Γ and W have no symmetry, for group D4 has 3-lateral symmetry, for other groups An, Dn, and E6 have 2-lateral symmetries (see [1, pp. 70 and 72]). Weyl groups of simple Lie groups An, Bn, Cn, Dn, G2, and F4 are isomor-phic to the groups of symmetries of n-dimensional regular simplex, and the n-dimensional cube and cocube, and n-dimensional semicube, regular plane hexagon, and 4-dimensional bicube, respectively (see [1, p. 71]). É. Cartan [3, pp. 35–43 and 43–50] has established that the Galois group of simple Lie group E6 is isomorphic to the group of symmetries of a smooth cubi