非交换双谱代数及其表示
Noncommutative Bispectral Algebras and their Presentations
摘要：
我们证明了有限生成代数表示的一个一般结果，并将其应用于矩阵双谱问题中某些非交换代数的表示。通过“好的呈现”，我们指的是具有尽可能少的定义关系的呈现。这反过来又在计算机代数实现和示例中具有潜在的应用。我们的结果可以分为三个部分。在前两章中，我们考虑了双谱代数的特征值为$2\times 2$和$3\times 3$矩阵值特征函数的标量值。在第三部分中，我们假设物理方程中的本征值是矩阵值的，并将其与自旋Calogero-Moser系统联系起来。在所有的情况下，我们证明了这些代数是双线性代数。作为副产品，我们肯定地回答了F.~ A.~ Grünbaum关于这些代数。
英文摘要：
We prove a general result on presentations of finitely-generated algebras and apply it to obtain nice presentations for some noncommutative algebras arising in the matrix bispectral problem. By "nice presentation" we mean a presentation that has as few as possible defining relations. This in turn, has potential applications in computer algebra implementations and examples. Our results can be divided into three parts. In the first two, we consider bispectral algebras with the eigenvalue in the physical equation to be scalar-valued for $2\times 2$ and $3\times 3$ matrix-valued eigenfunctions. In the third part, we assume the eigenvalue in the physical equation to be matrix-valued and draw an important connection with spin Calogero-Moser systems. In all cases, we show that these algebras are finitely presented. As a byproduct, we answer positively a conjecture of F.~A.~Grünbaum about these algebras.
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