Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

Let ${\mathfrak g}$ be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical ${\mathfrak r}$ and nilpotent radical ${\mathfrak n}=[{\mathfrak g},{\mathfrak r}]$. Given a finite dimensional ${\mathfrak g}$-module $U$, its nilpotency series $ 0\subset U({\mathfrak n}^1)\subset\cdots\subset U({\mathfrak n}^m)=U$ is defined so that $U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U$, $U({\mathfrak n}^2)/U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U/U({\mathfrak n}^1)$, and so on. We say that $U$ is linked if each factor of its nilpotency series is a uniserial ${\mathfrak g}/{\mathfrak n}$-module, i.e., its ${\mathfrak g}/{\mathfrak n}$-submodules form a chain. Every uniserial ${\mathfrak g}$-module is linked, every linked ${\mathfrak g}$-module is indecomposable with irreducible socle, and both converse fail. In this paper we classify all linked ${\mathfrak g}$-modules when ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$ and $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Moreover, we identify and classify all uniserial ${\mathfrak g}$-module amongst them.