Geometric interpretation of double shuffle relation for multiple L-values

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-Teichmuller group into Racinet’s cyclotomic double shuffle group is obtained. It cyclotomically extends the result of our previous paper and the project of Deligne and Terasoma which are the special case N=1 of our result.