The geometry of extremal elements in a Lie algebra

A.M. Cohen

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Abstract

Let L be a simple finite-dimensional Lie algebra over an algebraically closed field of characteristic distinct from 2 and from 3. Then L contains an extremal element, that is, an element x such that [x, [x, L]] is contained in the linear span of x in L. Suppose that L contains no sandwich, that is, no element x such that [x, [x, L]] = 0. Then, up to very few exceptions in characteristic 5, the Lie algebra L is generated by extremal elements and we can construct a building of irreducible and spherical type on the set of extremal elements of L. Therefore, by Tits’ classification of such buildings, L is determined by a known shadow space of a building. This gives a geometric alternative to the classical classification of finite-dimensional simple Lie algebras over the complex numbers and of classical finite-dimensional simple modular Lie algebras over algebraically closed fields of characteristic = 5. This paper surveys developments pertaining to this kind of approach to classical Lie algebras.
Original languageEnglish
Title of host publicationBuildings, finite geometries and groups
Subtitle of host publicationproceedings of a Satellite Conference, International Congress of Mathematicians, Hyderabad, India, 2010
EditorsN.S. Narasimha Sastry
Place of PublicationNew York
PublisherSpringer
Chapter2
Pages15-35
Number of pages21
ISBN (Electronic)978-1-4614-0709-6
ISBN (Print)978-1-4614-0708-9
DOIs
Publication statusPublished - 2012

Publication series

NameSpringer Proceedings in Mathematics (PROM)
Volume10
ISSN (Print)2190-5614

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