Free 2-step nilpotent Lie algebras and indecomposable modules

TL;DR

This paper classifies all uniserial representations of a specific solvable Lie algebra constructed from free 2-step nilpotent Lie algebras, focusing on cases where an element acts via a Jordan block.

Contribution

It provides a complete classification of uniserial modules for a class of solvable Lie algebras involving free 2-step nilpotent structures and Jordan block actions.

Findings

Classification of uniserial modules for the given Lie algebra

Explicit description of module structures in the Jordan block case

Extension of representation theory for 2-step nilpotent Lie algebras

Abstract

Given an algebraically closed field $F$ of characteristic 0 and an $F$-vector space $V$, let $L(V)=V\oplus\Lambda^2(V)$ denote the free 2-step nilpotent Lie algebra associated to $V$. In this paper, we classify all uniserial representations of the solvable Lie algebra $\mathfrak g=\langle x\rangle\ltimes L(V)$, where $x$ acts on $V$ via an arbitrary invertible Jordan block.