Simple weight modules for Yangian $\operatorname{Y}(\mathfrak{sl}_{2})$

TL;DR

This paper classifies simple weight modules with one-dimensional weight spaces for the Yangian of sl2, revealing four classes and identifying new modules with two-dimensional weight spaces unlike classical sl2 theory.

Contribution

It provides a complete classification of simple weight modules for Y(sl2) with one-dimensional weight spaces, including new modules with two-dimensional weight spaces.

Findings

Four classes of modules: finite, highest, lowest, dense

Existence of irreducible modules with two-dimensional weight spaces

Complete classification of simple weight modules for Y(sl2)

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$. A $\operatorname{Y}(\mathfrak{g})$-module is said to be weight if it is a weight $\mathfrak{g}$-module. We give a complete classification of simple weight modules for $\operatorname{Y}(\mathfrak{sl}_2)$ which admits a one-dimensional weight space. We prove that there are four classes of such modules: finite, highest weight, lowest weight and dense modules. Different from the classical $\mathfrak{sl}_{2}$ representation theory, we show that there exist a class of $\operatorname{Y}(\mathfrak{sl}_{2})$ irreducible modules which have uniformly 2-dimensional weight spaces.