Double Shuffle Relations for Refined Symmetric Multiple Zeta Values

Abstract

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 these lifted values 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 [H. Bachmann et al., Compos. Math. 154, No. 12, 2701–2721 (2018; Zbl 1429.11161)].