Thin subalgebras of Lie algebras of maximal class

TL;DR

This paper constructs and characterizes certain infinite-dimensional thin graded Lie algebras over fields with quadratic extensions, revealing new structures within Lie algebras of maximal class.

Contribution

It introduces non-metabelian thin graded Lie algebras with specific component dimensions, constructed as subalgebras of maximal class Lie algebras over quadratic extensions, and characterizes their properties.

Findings

Existence of non-metabelian thin Lie algebras with specified dimensions

Construction of these algebras as subalgebras of maximal class Lie algebras

Characterization of thin Lie subalgebras generated in degree 1

Abstract

For every field $F$ which has a quadratic extension $E$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension $2$. We construct such Lie algebras as $F$-subalgebras of Lie algebras $M$ of maximal class over $E$. We characterise the thin Lie $F$-subalgebras of $M$ generated in degree $1$. Moreover we show that every thin Lie algebra $L$ whose ring of graded endomorphisms of degree zero of $L^3$ is a quadratic extension of $F$ can be obtained in this Lie algebra of maximal class over $E$ which are ideally $r$-constrained for a positive integer $r$.