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.
Subject Classifications: AMS classification: 17B05, 17B45, 51B25
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Cohen, A.M. (2012). The Geometry of Extremal Elements in a Lie Algebra. In: Sastry, N. (eds) Buildings, Finite Geometries and Groups. Springer Proceedings in Mathematics, vol 10. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0709-6_2
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DOI: https://doi.org/10.1007/978-1-4614-0709-6_2
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