Free ideals of one-relator graded Lie algebras

Abstract:

In this paper we show that a one-relator graded Lie algebra $\mathfrak {g} = L/(r)$, over a principal ideal domain $K$, has a homogeneous ideal $\mathfrak {h}$ with $\mathfrak {g}/\mathfrak {h}$ a free $K$-module of finite rank if the relator $r$ is not a proper multiple of another element in the free Lie algebra $L$. As an application, we deduce that the center of a one-relator Lie algebra over $K$ is trivial if the rank of $L$ is greater than two. As another application, we find a new class of one-relator pro-$p$-groups which are of cohomological dimension $2$.