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

Abstract

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 $\mathfrak{g}=⟨x⟩⋉\mathfrak{a}$ 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.