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

TL;DR

This paper classifies linked modules over a class of solvable Lie algebras, focusing on modules with a specific structure, and identifies which among them are uniserial, expanding understanding of module structures in Lie algebra theory.

Contribution

The paper provides a complete classification of linked and uniserial modules for a specific class of solvable Lie algebras with diagonalizable actions.

Findings

Classified all linked modules for the given Lie algebra structure.

Identified and characterized all uniserial modules within these linked modules.

Established relationships between linked, uniserial, and indecomposable modules.

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 ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$ 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.