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Bethe algebra and the algebra of functions on the space of differential operators of order two with polynomial kernel

Authors:	E Mukhin V Tarasov A Varchenko

Institution:	(1) Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, 402 N. Blackford St, Indianapolis, IN 46202-3216, USA;(2) St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia;(3) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Abstract:	We show that the algebra of functions on the scheme of monic linear second-order ordinary differential operators with prescribed n + 1 regular singular points, prescribed exponents $\Lambda^{(1)},\ldots,\Lambda^{(n)},\Lambda^{(\infty)}$ at the singular points, and having the kernel consisting of polynomials only, is isomorphic to the Bethe algebra of the Gaudin model acting on the vector space Sing $L_{\Lambda^{(1)}}\,\otimes\,\cdots\,\otimes L_{\Lambda^{(n)}}\Lambda^{(\infty)}]$ of singular vectors of weight Λ^(∞) in the tensor product of finite-dimensional polynomial ${\mathfrak{g}}{\mathfrak{l}}_2$ -modules with highest weights $\Lambda^{(1)},\ldots,\Lambda^{(n)}$ .

Keywords:	" target="_blank"> Bethe algebra Bethe ansatz Gaudin model differential operators with polynomial kernel separation of variables
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