A sandwich in thin Lie algebras

TL;DR

This paper studies the structure of thin Lie algebras, showing that in infinite-dimensional cases with a certain diamond structure, specific commutator relations vanish, revealing new structural constraints.

Contribution

It establishes a new condition on the vanishing of certain brackets in infinite-dimensional thin Lie algebras with a specific diamond configuration.

Findings

If the second diamond occurs at position k>5, then [Lyy]=0 for some nonzero y in L_1.

Provides structural constraints on infinite-dimensional thin Lie algebras.

Enhances understanding of the lower central series in thin Lie algebras.

Abstract

A thin Lie algebras 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 its homogeneous components have dimension one or two, and the two-dimensional components are called diamonds. We prove that if the next diamond past $L_1$ of an infinite-dimensional thin Lie algebra $L$ is $L_k$, with $k>5$, then $[Lyy]=0$ for some nonzero element $y$ of $L_1$.