The Hopf algebra of (q)multiple polylogarithms with non-positive arguments

TL;DR

This paper constructs a Hopf algebra framework for multiple polylogarithms at non-positive integers, applying algebraic renormalization techniques and extending results to q-analogues, thus advancing the algebraic understanding of these functions.

Contribution

It introduces a Hopf algebra structure for multiple polylogarithms at non-positive integers and applies Birkhoff decomposition for their renormalization, including q-analogues.

Findings

Established a connected graded Hopf algebra for these polylogarithms

Applied algebraic Birkhoff decomposition for renormalization

Extended results to q-analogues and compared with classical case

Abstract

We consider multiple polylogarithms in a single variable at non-positive integers. Defining a connected graded Hopf algebra, we apply Connes' and Kreimer's algebraic Birkhoff decomposition to renormalize multiple polylogarithms at non-positive integer arguments, which satisfy the shuffle relation. The q-analogue of this result is as well presented, and compared to the classical case.