Configurations of lines and models of Lie algebras

Laurent Manivel (1)

1 IF - Institut Fourier

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of $E_6$ and $E_7$. We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For ${\mathfrak e}_7$ and ${\mathfrak e}_8$ we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras. We also interpret the group of the bitangents as a group of transformations of the triangles in the Fano plane, and show how this allows to realize the isomorphism $PSL(3,F_2)\simeq PSL(2,F_7)$ in terms of harmonic cubes.

Mots clés: lines configuration; exceptional Lie algebra; Fano plane; octonion; Klein group; polytope; homogeneous space

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

Laurent Manivel. Configurations of lines and models of Lie algebras. Journal of Algebra, 2006, 304 (1), pp.457-486. ⟨hal-00005848⟩