Simple $\mathfrak{sl}_2$-modules that are torsion free $U(\mathfrak{h})$-modules of rank $1$

TL;DR

This paper classifies simple modules over certain Lie algebras that are torsion free of rank 1 over the Cartan subalgebra, extending results to the Weyl algebra and Lie superalgebra sp(1|2).

Contribution

It provides an explicit classification of all such modules for sl_2, the Weyl algebra, and osp(1|2), revealing new structural insights.

Findings

Complete classification of simple sl_2-modules that are torsion free of rank 1.

Extension of classification results to the Weyl algebra.

Extension of classification results to the Lie superalgebra osp(1|2).

Abstract

We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.