A family of simple $U(\mathfrak{h})$-free modules of rank 2 over $\mathfrak{sl} (2)$

TL;DR

This paper classifies simple rank 2 modules over rak{sl}(2) that are free over a Cartan subalgebra, providing explicit criteria for their isomorphism and reducing the problem to twisted conjugacy classes in C[h].

Contribution

It offers the first explicit classification of scalar-type simple rak{sl}(2)-modules of rank 2, including criteria for module isomorphism.

Findings

Classification of scalar-type simple modules of rank 2 over rak{sl}(2)

Reduction of isomorphism problem to twisted conjugacy classes in C[h]

Use of Cohn's standard form in the classification process

Abstract

We study simple $\mathfrak{sl}(2)$-modules over $\mathbb C$ that are free of finite rank as $U(\mathfrak h)$-modules, where $\mathfrak h$ is a Cartan subalgebra of $\mathfrak{sl}(2)$. Our main result is an explicit classification of the scalar-type simple modules of rank $2$. We also give a criterion for when two such modules are isomorphic. Both the classification and the isomorphism problem reduce to twisted conjugacy classes in $\mbox{GL}_2({\mathbb C}[h])$ and rely on Cohn's standard form.