The Hopf algebra structure of multiple harmonic sums
Michael E. Hoffman
TL;DR
The paper develops a Hopf-algebraic framework for multiple harmonic sums with roots of unity, unifying the $A_I$ and $S_I$ families and revealing a duality via compositions and Möbius inversion. It introduces the Euler algebra $\mathcal{E}_r$, proves it forms a graded Hopf algebra generated by Lyndon words, and provides antipode and reversal structures that underpin algebraic manipulations of these sums. A key bridge is the map $\rho_n$ that specializes words to $A_w(n)$ and $S_w(n)$, enabling explicit relations between the two sum types and the expression of symmetric sums in terms of ordinary harmonic sums. These results extend the $r=1$ case (QSym), link to multiple zeta values and polylogarithms, and supply algebraic tools for perturbative quantum field theory computations involving polylogarithmic sums.
Abstract
Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of their properties that can be exploited in calculations.