Construction of simple non-weight sl(2)-modules of arbitrary rank

TL;DR

This paper classifies simple non-weight sl(2)-modules that are finitely generated over C[z], introduces stratification invariants via Smith type, and constructs new modules of arbitrary rank.

Contribution

It provides a new classification framework for simple non-weight sl(2)-modules, including stratification and duality, and constructs modules of arbitrary rank.

Findings

Modules described via semilinear endomorphisms

Smith type induces stratification and invariants

Existence of simple modules of arbitrary rank

Abstract

We study simple non-weight ${\mathfrak{sl}}(2)$-modules which are finitely generated as ${\mathbb C}[z]$-modules. We show that they are described in terms of semilinear endomorphisms and prove that the Smith type induces a stratification on the set of these ${\mathfrak{sl}}(2)$-modules, providing thus new invariants. Moreover, we show that there is a notion of duality for these type of ${\mathfrak{sl}}(2)$-modules. Finally, we show that there are simple non-weight ${\mathfrak{sl}}(2)$-modules of arbitrary rank by constructing a whole new family of them.