Configurations of lines and models of Lie algebras

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dc.creatorManivel, Laurent
dc.date2005-07-06
dc.date.accessioned2026-07-07T10:10:02Z
dc.date.available2026-07-07T10:10:02Z
dc.descriptionThe 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.
dc.description31 pages
dc.identifierhttps://arxiv.org/abs/math/0507118
dc.identifierhttp://arxiv.org/abs/math/0507118
dc.identifierJournal of Algebra 304, 1 (2006) 457-486
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/171502
dc.subjectAlgebraic Geometry
dc.subjectRepresentation Theory
dc.subject14N20, 17B25, 14L40, 14M17
dc.titleConfigurations of lines and models of Lie algebras
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