论文标题
非交换双光谱代数及其演示文稿
Noncommutative Bispectral Algebras and their Presentations
论文作者
论文摘要
我们证明了有限生成的代数的介绍的一般结果，并将其应用以获取在基质双光谱问题中引起的某些非共同代数的良好演示文稿。通过“好的演示”，我们的意思是表现出尽可能少的定义关系。反过来，这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。在前两个中，我们认为物理方程中具有特征值的双光谱代数为标量值，价格为$ 2 \ times 2 $和$ 3 \ times 3 $矩阵值得值。在第三部分中，我们假设物理方程中的特征值是矩阵值的，并与旋转的Calogero-Moser系统建立了重要的联系。在所有情况下，我们都表明这些代数是有限提出的。作为副产品，我们对这些代数的F. 〜A.grünbaum的猜想积极回答。
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.
