Indecomposable modules of 2-step solvable Lie algebras in arbitrary characteristic

Abstract

Let $F$ be an algebraically closed field and consider the Lie algebra $\mathfrak{g}=⟨x⟩⋉\mathfrak{a}$, where $\mathrm{ad}x$ acts diagonalizably on the abelian Lie algebra $\mathfrak{a}$. Refer to a $\mathfrak{g}$-module as admissible if $[\mathfrak{g},\mathfrak{g}]$ acts via nilpotent operators on it, which is automatic if $\mathrm{char}(F)=0$. In this paper we classify all indecomposable $\mathfrak{g}$-modules $U$ which are admissible as well as uniserial, in the sense that $U$ has a unique composition series.