Thin subalgebras of Lie algebras of maximal class

Abstract

For every field $\mathbb{F}$ which has a quadratic extension $\mathbb{E}$ 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 $\mathbb{F}$-subalgebras of Lie algebras M of maximal class over $\mathbb{E}$. We characterise the thin Lie $\mathbb{F}$-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 $\mathbb{F}$ can be obtained in this way. We also characterise the 2-generator $\mathbb{F}$-subalgebras of a Lie algebra of maximal class over $\mathbb{E}$ which are ideally r-constrained for a positive integer r.