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

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