Simple Lie algebras having extremal elements |
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Authors: | Arjeh M Cohen |
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Institution: | a Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands b Informatics Research Laboratory, Computer and Automation Institute, Hungarian Academy of Sciences, Lágymányosi u. 11, H-1111, Budapest, Hungary |
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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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