Representations of Smith algebras which are free over the Cartan subalgebra

TL;DR

This paper investigates modules over Smith algebras that are free of finite rank over the polynomial subalgebra generated by the Cartan element, providing classifications, criteria, and algorithms for their structure.

Contribution

It offers a comprehensive classification of rank-one modules, a simplicity criterion, and an algorithm for composition series, advancing understanding of Smith algebra representations.

Findings

Full description of rank-one module isomorphism classes

Simplicity criterion for modules

Algorithm for constructing composition series

Abstract

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element $h$ and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity.