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

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Abstract

We consider the space $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$ of functions on the Cartan subalgebra of $\mathfrak{sl}_2$ with values in the zero weight subspace $V[0]$ of a tensor product of irreducible finite-dimensional $\mathfrak{sl}_2$-modules. We consider the algebra $\mathcal B$ of commuting differential operators on $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$, constructed by Rubtsov, Silantyev, and Talalaev in 2009. We describe the relations between the action of $\mathcal B$ on $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$ and spaces of pairs of quasi-polynomials.Bibliography: 25 titles.

About the authors

Aleksandr Nikolaevich Varchenko

Department of Mathematics, University of North Carolina at Chapel Hill; Lomonosov Moscow State University, Faculty of Mechanics and Mathematics; Moscow Center for Fundamental and Applied Mathematics

Email: anv@email.unc.edu

Aleksei Mikhailovich Slinkin

Department of Mathematics, University of North Carolina at Chapel Hill; HSE University

Email: slinalex@live.unc.edu

Daniel Thompson

Department of Mathematics, University of North Carolina at Chapel Hill

Email: dthomp@email.unc.edu

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