On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives

TL;DR

This paper explores the potential connection between Goncharov's groups and the co-Lie algebra of mixed Tate motives, supported by results assuming Beilinson and Soulé's conjecture on K-groups.

Contribution

It proposes a possible linear map linking Goncharov's groups to the co-Lie algebra of mixed Tate motives, advancing understanding of motivic polylogarithms.

Findings

Results support the possibility of defining the linear map under certain conjectural assumptions.

Provides evidence linking Goncharov's groups to the structure of mixed Tate motives.

Abstract

Goncharov defined for each field $F$ and an integer $n$ greater than 1 a certain group $B_n(F)$. We consider the possibility of defining a linear map from $B_n(F)$ to the co-Lie algebra of the category of mixed Tate motives defined by Bloch and Kriz, in terms of motivic polylogarithms. We give results which support this possibility assuming part of the conjecture by Beilinson and Soul\'{e} on vanishing of $K$-groups of fields.