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arXiv 0705.4114math.QAmath.AGmath.CA
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具有多项式解的二阶微分算子空间上的Bethe代数与函数代数

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

E. Mukhin, V. Tarasov, A. Varchenko

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摘要：

我们证明了以下两个代数是同构的。第一类是$\C$上的一元线性二阶微分算子概型上的函数代数$A_P$，它在$z_1,..., z_n, \infty$上具有规定的正则奇点，在奇点上具有规定的指数$\La^{(1)}, ..., \La^{(n)}, \La^{(\infty)}$，并且核仅由多项式组成.第二个是交换线性算子的Bethe代数，作用在具有最高权值$\La^{(1)},..., \La^{(n)}$的有限维多项式$gl_2$ -模的张量积中权值为$\La^{(\infty)}$的奇异向量的向量空间$\Sing L_{\La^{(1)}} \otimes ... \otimes L_{\La^{(n)}}[\La^{(\infty)}]$上。

英文摘要：

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 $z_1,..., z_n, \infty$, prescribed exponents $\La^{(1)}, ..., \La^{(n)}, \La^{(\infty)}$ 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 $\Sing L_{\La^{(1)}} \otimes ... \otimes L_{\La^{(n)}}[\La^{(\infty)}]$ of singular vectors of weight $\La^{(\infty)}$ in the tensor product of finite dimensional polynomial $gl_2$-modules with highest weights $\La^{(1)},..., \La^{(n)}$.

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