## Advanced Studies in Pure Mathematics

### Geometric interpretation of double shuffle relation for multiple $L$-values

Hidekazu Furusho

#### Abstract

This paper gives a geometric interpretation of the generalized (including the regularization relation) double shuffle relation for multiple $L$-values. Precisely it is proved that Enriquez' mixed pentagon equation implies the relations. As a corollary, an embedding from his cyclotomic analogue of the Grothendieck–Teichmüller group into Racinet's cyclotomic double shuffle group is obtained. It cyclotomically extends the result of our previous paper [F3] and the project of Deligne and Terasoma which are the special case $N=1$ of our result.

#### Article information

Dates
Revised: 24 February 2012
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/ euclid.aspm/1540417818

Digital Object Identifier
doi:10.2969/aspm/06310163

Mathematical Reviews number (MathSciNet)
MR3051243

Zentralblatt MATH identifier
1321.11089

#### Citation

Furusho, Hidekazu. Geometric interpretation of double shuffle relation for multiple $L$-values. Galois–Teichmüller Theory and Arithmetic Geometry, 163--187, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310163. https://projecteuclid.org/euclid.aspm/1540417818