Configurations of lines and models of Lie algebras
By Laurent Manivel. 2005
57 citations recorded by the source corpus.
Abstract 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 e 7 and 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 ) ≃ PSL ( 2 , F 7 ) in terms of harmonic cubes.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0e1cc217d23ed964d541387ba099d978fbcfead3
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Black Holes, Qubits and Octonions (2008, background): “The remarkable fact that E7 has a natural structure of an O-graded algebra [28, 29], compatible with its action on the minimal 56-dimensional representation, provides a new octonionic view of M -theory compactified on T , and hence a new quaternionic view of string theory compactified on T .”