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

2007/05/28 by E. Mukhin, Mukhin, E., V. Tarasov +4
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Polynomial and algebraic computation #Quantum Algebra (math.QA) #math.AG #math.CA #math.QA

paper · pdf · doi:10.48550/arxiv.0705.4114

arxiv created 2007/05/28 · openalex publication_date 2007/05/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the following two algebras are isomorphic. The first is the algebra A_P of functions on the scheme of monic linear second-order differential operators on \C with prescribed regular singular points at z₁,..., z_n, ∞, prescribed exponents \La⁽¹⁾, ..., \La⁽ⁿ⁾, \La^(∞) at the singular points, and having the kernel consisting of polynomials only. The second is the Bethe algebra of commuting linear operators, acting on the vector space \Sing L_\La⁽¹⁾ ⊗ ... ⊗ L_\La⁽ⁿ⁾[\La^(∞)] of singular vectors of weight \La^(∞) in the tensor product of finite dimensional polynomial gl₂-modules with highest weights \La⁽¹⁾,..., \La⁽ⁿ⁾.

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