论文标题
准剃管代数的概括和对多个Zeta值的应用
A generalisation of quasi-shuffle algebras and an application to multiple zeta values
论文作者
论文摘要
可以使用Shuffle和Quasi-shuffle代数的组合来描述多个Zeta值之间的大型关系。尽管一段时间以来，人们已经对洗牌代数的结构进行了充分的了解，但仅最近对准代数的代数进行了正式研究。特别是，霍夫曼对代数结构进行了详尽的讨论，包括选择代数基础，并将其结果应用于多个Zeta值及其概括之间的关系家庭。在最近的预印本中，Hirose和Sato建立了来自新的广义洗牌结构的一系列关系，提升了作者建立的一系列分级关系，以实现真正的未分级关系。在本文中，我们在可计数字母中的非交通多项式空间上定义了一个交换代数结构，从而概括了Hirose和Sato的混洗结构。我们表明，在有理数的数字上，这种广义的准剃须代数对标准的随机代数是同构的，使我们能够在Quasi-shuffle代数上复制大多数霍夫曼的结果。然后，我们将这些结果应用于多个Zeta值的情况下，再现了几个已知的结果系列并建立了多个。
A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras have been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalisations. In a recent preprint, Hirose and Sato establish a family of relations coming from a new generalised shuffle structure, lifting a set of graded relations established by the author to genuine ungraded relations. In this paper, we define a commutative algebra structure on the space of non-commutative polynomials in a countable alphabet, generalising the shuffle-like structure of Hirose and Sato. We show that, over the rational numbers, this generalised quasi-shuffle algebra is isomorphic to the standard shuffle algebra, allowing us to reproduce most of Hoffman's results on quasi-shuffle algebras. We then apply these results to the case of multiple zeta values, reproducing several known families of results and establishing several more.
