关于退化CP ^1和Macdonald多项式的交换代数
A commutative algebra on degenerate CP^1 and Macdonald polynomials
摘要：
引入退化CP ^1上的有单位的结合代数A.证明了A是一个交换代数，其Poincar级数由分划数给出.从而我们可以把A看作是作者之一和Odesskii引入的椭圆代数的光滑退化极限。然后研究了作用在对称函数空间上的交换Macdonald差分算子族。利用A和作者之一研究的交换族的Heisenberg表示，提出了这个族的一个正则基。研究发现，Ding-Iohara代数为自由场的构造提供了一个代数框架。本文讨论了这种构造的椭圆变形，并指出了它与Drinfeld拟Hopf扭a la Babelon Bernard Billey，Ruijsenaars差分算子和Okounkov-Pandharipande算子M（q，t_1，t_2）的关系.
英文摘要：
We introduce a unital associative algebra A over degenerate CP^1. We show that A is a commutative algebra and whose Poincar'e series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using A and the Heisenberg representation of the commutative family studied by one of the authors. It is found that the Ding-Iohara algebra provides us with an algebraic framework for the free filed construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting a la Babelon Bernard Billey, the Ruijsenaars difference operator and the operator M(q,t_1,t_2) of Okounkov-Pandharipande.
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