# Graded torsion-free ${\mathfrak{sl}_2(\mathbb{C})}$-modules of rank 2

## TL;DR

This paper investigates the structure of graded, simple infinite-dimensional modules over the Lie algebra ${\mathfrak{sl}_2(\mathbb{C})}$, focusing on modules of rank 2 and their grading properties.

## Contribution

It introduces a new perspective on grading structures of infinite-dimensional simple modules over ${\mathfrak{sl}_2(\mathbb{C})}$, extending previous finite-dimensional studies.

## Key findings

- Characterization of graded structures on infinite-dimensional modules
- Extension of grading concepts from finite to infinite dimensions
- Identification of conditions for rank 2 modules to admit gradings

## Abstract

In this paper we explore the possibility of endowing simple infinite-dimensional ${\mathfrak{sl}_2(\mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].