Dynamical ${\frak{sl}}_2$ Bethe algebra and functions on pairs of quasi-polynomials

TL;DR

This paper explores the algebra of commuting differential operators acting on functions related to  representations and establishes connections with spaces of pairs of quasi-polynomials, advancing understanding of the dynamical  Bethe algebra.

Contribution

It provides a detailed description of the relations between the  Bethe algebra's action and spaces of pairs of quasi-polynomials, linking algebraic and functional perspectives.

Findings

Describes the structure of the  Bethe algebra on function spaces.

Establishes a correspondence between algebra actions and quasi-polynomial pairs.

Enhances understanding of the algebraic structure underlying  integrable systems.

Abstract

We consider the space $\text{Fun}_{\frak{sl}_2}V[0]$ of functions on the Cartan subalgebra of $\frak{sl}_2$ with values in the zero weight subspace $V[0]$ of a tensor product of irreducible finite-dimensional $\frak{sl}_2$-modules. We consider the algebra $\mathcal B$ of commuting differential operators on $\text{Fun}_{\frak{sl}_2}\,V[0]$, constructed by V.Rubtsov, A.Silantyev, D.Talalaev in 2009. We describe the relations between the action of $\mathcal B$ on $\text{Fun}_{\frak{sl}_2}V[0]$ and spaces of pairs of quasi-polynomials.