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arXiv · 0711.4268

Simple Lie Algebras having Extremal Elements

Arjeh M. Cohen·Gabor Ivanyos·Dan A. Roozemond
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Abstract

Let L be a simple finite-dimensional Lie algebra of characteristic distinct from 2 and from 3. Suppose that L contains an extremal element that is not a sandwich, that is, an element x such that [x, [x, L]] is equal to the linear span of x in L. In this paper we prove that, with a single exception, L is generated by extremal elements. The result is known, at least for most characteristics, but the proofs in the literature are involved. The current proof closes a gap in a geometric proof that every simple Lie algebra containing no sandwiches (that is, ad-nilpotent elements of order 2) is in fact of classical type.

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Arjeh M. Cohen, Gabor Ivanyos, Dan A. Roozemond. 2008-12-17. Simple Lie Algebras having Extremal Elements. https://doi.org/10.1016/s0019-3577(09)00003-2

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