Simple weight modules for Yangian $Y({\mathfrak{sl}}_2)$

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$. A $Y(\mathfrak{g})$-module is said to be weight if it is a weight $\mathfrak{g}$-module. We give a complete classification of simple weight modules for $Y({\mathfrak{sl}}_2)$ 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 $Y({\mathfrak{sl}}_2)$ irreducible modules which have uniformly 2-dimensional weight spaces.

14 pages, 2 figures