Algebraic Aspects of Multiple Zeta Values
Algebraic Aspects of Multiple Zeta Values
复制标题
多个 Zeta 值的代数方面
DOI：
--
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
Michael E. Hoffman
中科院分区：
文献类型：
--
作者：
Michael E. Hoffman
已经用各种方法研究了多个Zeta值。在这篇文章中，我们总结了一些可以通过代数方法获得的关于它们的结果。这涉及用两个非交换变量x和y中的单项式对多个zeta值进行编码。然后，多个zet值可以被认为是从由非交换多项式代数Q<x，y>的“允许词”生成的分级有理向量空间ζ:ℌ0→R定义映射ℌ0。现在，ℌ0允许两个(可交换)乘积使ζ成为同态--混洗乘积和“调和”乘积。后者使ℌ0成为拟对称函数的代数Qsym的子代数。我们还讨论了可用Q<x，y>的导子和循环导子表示的多重Zeta值的一些结果，并定义了QSym在Q<x，y>上的一个有用的作用。最后，我们将代数方法应用于多重Zeta值级数的有限部分和之间的关系。
Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: ℌ0 → R from a graded rational vector space ℌ0 generated by the “admissible words” of the noncommutative polynomial algebra Q〈x,y〉. Now ℌ0 admits two (commutative) products making ζ a homomorphism-the shuffle product and the “harmonic” product. The latter makes ℌ0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.