Poles of finite-dimensional representations of Yangians

TL;DR

This paper characterizes the poles of rational currents in finite-dimensional Yangian representations, linking them to Drinfeld polynomials and the $q$-Cartan matrix, and applies this to classify irreducible modules and tensor product properties.

Contribution

It provides a uniform description of pole sets in terms of Drinfeld polynomials and the $q$-Cartan matrix, and classifies finite-dimensional irreducible Yangian double representations.

Findings

Describes poles of rational currents via Drinfeld polynomials and $q$-Cartan matrix.

Provides conditions for cyclicity and simplicity of tensor products.

Classifies finite-dimensional irreducible representations of the Yangian double.

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $Y_{\hbar}(\mathfrak{g})$ be the Yangian of $\mathfrak{g}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_{\hbar}(\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. Using a weak, rational version of Frenkel and Hernandez' Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of $V$ and the inverse of the $q$-Cartan matrix of $\mathfrak{g}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.