Double Shuffle Relations for Refined Symmetric Multiple Zeta Values
Double Shuffle Relations for Refined Symmetric Multiple Zeta Values
复制标题
精细对称多 Zeta 值的双重洗牌关系
DOI：
10.4171/dm/750
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
M. Hirose
中科院分区：
文献类型：
--
作者：
M. Hirose
对称多重Zeta值(SMZV)是Kaneko-Zagier作为有限多个Zeta值的对等物，模于由$\zeta(2)$生成的理想的所有多重Zeta值的环中的元素。众所周知，对称的多个Zeta值满足双重洗牌关系和对偶关系。本文将所有多个Zeta值和$2pi$生成的环上的SMZV的提升构造为沿某一闭路在$\mathbb{P}^{1}\setminus\{0，1，\inty$上的某些迭代积分.我们将这种提升的值称为精化对称多重Zeta值(RSMZV)。我们证明了RSMZV的双重洗牌关系和对偶关系。这些关系是SMZV的双重洗牌关系和对偶关系的提炼。此外，我们将RSMZV与SMZV的其他变种进行了比较。特别地，我们证明了RSMZV与Bachmann-Takeyama-Tasaka的$\xi$-值重合。
Symmetric multiple zeta values (SMZVs) are elements in the ring of all multiple zeta values modulo the ideal generated by $\zeta(2)$ introduced by Kaneko-Zagier as counterparts of finite multiple zeta values. It is known that symmetric multiple zeta values satisfy double shuffle relations and duality relations. In this paper, we construct certain lifts of SMZVs which live in the ring generated by all multiple zeta values and $2\pi i$ as certain iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty\}$ along a certain closed path. We call this lifted values as refined symmetric multiple zeta values (RSMZVs). We show double shuffle relations and duality relations for RSMZVs. These relations are refinements of the double shuffle relations and the duality relations of SMZVs. Furthermore we compare RSMZVs to other variants of lifts of SMZVs. Especially, we prove that RSMZVs coincide with Bachmann-Takeyama-Tasaka's $\xi$-values.
影响因子：
1.8
作者：
K. Ihara
通讯作者：
K. Ihara