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

TL;DR

This paper classifies all indecomposable, admissible, and uniserial modules over a specific class of 2-step solvable Lie algebras in arbitrary characteristic, extending understanding of their module structure.

Contribution

It provides a complete classification of indecomposable admissible uniserial modules for a class of 2-step solvable Lie algebras in any characteristic.

Findings

Classification of all such modules achieved

Results hold in arbitrary characteristic

Enhances understanding of module structures over solvable Lie algebras

Abstract

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, 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.