Hopf algebras for the shuffle algebra and fractions from multiple zeta values

Abstract: The algebra of multiple zeta values (MZVs) is encoded as a stuffle (quasi-shuffle) algebra and a shuffle algebra. The MZV stuffle algebra has a natural Hopf algebra structure. This paper equips a Hopf algebra structure to the MZV shuffle algebra. The needed coproduct is defined by a recursion through a family of weight-increasing linear operators. To verify the Hopf algebra axioms, we make use of a family of fractions, called Chen fractions, that have been used to study MZVs and also serve as the function model for the MZV shuffle algebra. Applying natural derivations on functions and working in the context of locality, a locality Hopf algebra structure is established on the linear span of Chen fractions. This locality Hopf algebra is then shown to descend to a Hopf algebra on the MZV shuffle algebra, whose coproduct satisfies the same recursion as the first-defined coproduct. Thus the two coproducts coincide, establishing the needed Hopf algebra axioms on the MZV shuffle algebra.