On generating series of finitely presented operads
On generating series of finitely presented operads
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DOI：
10.1016/j.jalgebra.2014.12.012
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发表时间：
2012-02
影响因子：
0.9
通讯作者：
A. Khoroshkin;D. Piontkovski
中科院分区：
文献类型：
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作者：
A. Khoroshkin;D. Piontkovski
给定一个具有有限Gröbner关系基的算子P，研究了其分次分支P（n）的维数的生成函数.在适当的假设下，证明了序列{dim P（n）}的指数母函数是微分代数的，如果P是一个非对称算子的对称化，则它是代数的.此外，如果P（n）的维数的增长以n的指数为界（或在非对称情况下以n的多项式为界），则上述序列{dim P（n）}的普通生成函数是有理的。我们给出了一些计算的例子，并讨论了更一般的运算类的上述生成函数。
Given an operad P with a finite Gröbner basis of relations, we study the generating functions for the dimensions of its graded components P (n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P (n)} is differential algebraic, and in fact algebraic if P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions of P (n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, moreover, the ordinary generating function for the above sequence {dim P (n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.