关于Gaudin模$p$模型的注记
Remarks on the Gaudin model modulo $p$
摘要：
讨论了Gaudin模型中有限维$sl_2$ -模在有$p$元素的域$F_p$上张量积的Bethe映射，其中$p$是素数.定义了Bethe不等式方程，证明了若$(t^0_1,\dots,t^0_k)$是Bethe不等式方程的解，则相应的Bethe向量是Gaudin Hamilton算子的特征向量.我们将Bethe不等式方程的解$(t^0_1,\dots,t^0_k)$刻画为多项式空间$F_p[x]$的某些二维子空间。我们考虑当参数$k$的数目等于1时的情况。在这种情况下，我们表明，贝特代数，产生的Gaudin Hamilton，是同构的代数的功能上的计划所定义的贝特代数方程。如果$k=1$和另外的张量积是向量表示的乘积，则Bethe代数也同构于合适的Wronski映射纤维上的函数代数。
英文摘要：
We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In that case we show that the Bethe algebra, generated by the Gaudin Hamiltonians, is isomorphic to the algebra of functions on the scheme defined by the Bethe ansatz equation. If $k=1$ and in addition the tensor product is the product of vector representations, then the Bethe algebra is also isomorphic to the algebra of functions on the fiber of a suitable Wronski map.
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