Representations of Smith algebras which are free over the Cartan subalgebra

Vyacheslav Futorny, Samuel A. Lopes, Eduardo M. Mendonça

Published 2023-08-15Version 1

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.