A sandwich in thin Lie algebras

Mattarei, Sandro (2022) A sandwich in thin Lie algebras. Proceedings of the Edinburgh Mathematical Society . pp. 1-10. ISSN 0013-0915

Full content URL: https://doi.org/10.1017/S0013091521000845

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A sandwich in thin Lie algebras
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Abstract

A thin Lie algebra is a Lie algebra L, graded over the positive integers, with its �first homogeneous component L_1 of dimension two and generating L, and such that each nonzero ideal of L lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of L (that is, the next diamond past L_1) occurs in degree k. We prove that if k > 5, then [Lyy] = 0 for some nonzero element y of L_1. In characteristic different from two this means y is a sandwich element of L. We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

Keywords:modular Lie algebra, graded Lie algebra, thin Lie algebra, sandwich
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:47908
Deposited On:01 Feb 2022 10:12

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