Simple Lie Algebras having Extremal Elements

Details Simple Lie Algebras having Extremal Elements Simple Lie Algebras having Extremal Elements

2007-11-27

arxiv.org/abs/0711.4268v3

Mathematics

Rings and Algebras

11 pages

Scientific paper

10.1016/S0019-3577(09)00003-2

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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