Duality of Shuffle Hopf algebras related to Multiple zeta values

Abstract: The symmetry between of Riemann $\zeta$-function at positive integers and nonpositive integers is generalized to the symmetry of positive integer points and nonpositive integer points of multiple zeta functions algebraically, in terms of the duality of Hopf algebras. As we know, the shuffle product in the algebra of positive integer points can be extended to all integer points, and there is a Hopf algebra structure in the algebra of positive integer points. It is showed in this paper that the algebra of nonpositive integer points also has a Hopf algebra structure for the extended shuffle product. By characterizing the coproducts in the Hopf algebras of positive integer points and nonpositive integer points, we prove that the Riemann dual maps gives duality between the Hopf algebras of positive integer points and nonpositive integer points.