Recovering the Lie algebra from its extremal geometry

H. Cuypers, K. Roberts, S. Shpectorov

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Abstract

An element x of a Lie algebra L over the field F is extremal if [. x, [. x, L]] = Fx. Under minor assumptions, it is known that, for a simple Lie algebra L, the extremal geometry E(L) is a subspace of the projective geometry of L and either has no lines or is the root shadow space of an irreducible spherical building δ. We prove that if δ is of simply-laced type, then L is a quotient of a Chevalley algebra of the same type.

Original languageEnglish
Pages (from-to)196-215
Number of pages20
JournalJournal of Algebra
Volume441
Issue numberNovember 2015
DOIs
Publication statusPublished - 1 Nov 2015

Keywords

  • Buildings
  • Lie algebras
  • Root groups
  • Shadow spaces

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