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The Lie coalgebra of multiple polylogarithms

Zachary Greenberg, Dani Kaufman, Haoran Li, Christian K. Zickert

Abstract

We use Goncharov's coproduct of multiple polylogarithms to define a Lie coalgebra over an arbitrary field. It is generated by symbols subject to inductively defined relations, which we think of as functional relations for multiple polylogarithms. In particular, we have inversion relations and shuffle relations. We relate our definition to Goncharov's Bloch groups, and to the concrete model in weight less than 5 by Goncharov and Rudenko.

The Lie coalgebra of multiple polylogarithms

Abstract

We use Goncharov's coproduct of multiple polylogarithms to define a Lie coalgebra over an arbitrary field. It is generated by symbols subject to inductively defined relations, which we think of as functional relations for multiple polylogarithms. In particular, we have inversion relations and shuffle relations. We relate our definition to Goncharov's Bloch groups, and to the concrete model in weight less than 5 by Goncharov and Rudenko.
Paper Structure (16 sections, 11 theorems, 63 equations)

This paper contains 16 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The motivic Lie coalgebra
  3. Multiple polylogarithms and Goncharov's coproduct
  4. Real single valued polylogarithms
  5. Structure of the paper
  6. A purely symbolic Lie coalgebra
  7. The cobracket on $\mathbb L^{\mathop{\mathrm{symb}}\nolimits}(F)$
  8. Terms involving zero or infinity
  9. The relations
  10. Basic examples of relations
  11. The issue of arbitrary symbols
  12. Conjectures and speculation
  13. Inversion relations
  14. A coalgebra without inverted terms
  15. Shuffle relations
  16. ...and 1 more sections

Key Result

Theorem 1.1

The coproduct $\Delta$ of $\mathop{\mathrm{Li}}\nolimits^{\mathscr C}(x_1,\dots,x_d|t_1,\dots,t_d)$ is given by The sum is over all $k=0,\dots, d$, and all sequences $\{i_\alpha\}_{\alpha=0}^{k+1}$ and $\{j_\alpha\}_{\alpha=0}^k$ with and by definition we have $x_{i\to j}=\prod_{r=i}^{j-1}x_r$(note that this convension differs from Goncharov's, but works better in this paper) and $\mathop{\mathr

Theorems & Definitions (45)

  • Theorem 1.1: Goncharov_GaloisSymmetriesOfFundamentalGroupoidsAndNoncommutativeGeometry
  • Example 1.2
  • Corollary 1.3
  • proof
  • Theorem 1.4: ZagierDedekindZeta
  • Remark 1.5
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem 2.4
  • ...and 35 more