The combinatorial Hopf algebra of motivic dissection polylogarithms

TL;DR

This paper introduces dissection polylogarithms, a new family of motivic periods linked to combinatorial dissection diagrams, and encodes their coproduct structure via a Hopf algebra, generalizing existing formulas for iterated integrals.

Contribution

It develops a combinatorial Hopf algebra framework for motivic dissection polylogarithms, extending Goncharov's coproduct formula to a broader class of periods.

Findings

Defined dissection polylogarithms indexed by dissection diagrams

Established a Hopf algebra structure encoding their motivic coproduct

Generalized Goncharov's formula for iterated integrals

Abstract

We introduce a family of periods of mixed Tate motives called dissection polylogarithms, that are indexed by combinatorial objects called dissection diagrams. The motivic coproduct on the former is encoded by a combinatorial Hopf algebra structure on the latter. This generalizes Goncharov's formula for the motivic coproduct on (generic) iterated integrals. Our main tool is the study of the relative cohomology group corresponding to a bi-arrangement of hyperplanes.