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A Riemann-type duality of shuffle Hopf algebras related to multiple zeta values

Li Guo, Hongyu Xiang, Bin Zhang

TL;DR

The paper develops a rigorous Hopf-algebraic framework to model the symmetry underlying the functional equation of multiple zeta values. By constructing the extended shuffle algebra $\mathcal{H}_{\mathbb{Z}}$ and its subalgebras $\mathcal{H}_{\mathbb{Z}_{\ge 1}}$ and $\mathcal{H}_{\mathbb{Z}_{\le 0}}$, it equips them with differential Hopf structures via operators $J_{\ge 1}$ and $J_{\le 0}$. A central result is the Riemann-type dual map $\varphi$ that provides a differential Hopf algebra isomorphism between the positive-argument shuffle algebra and the graded dual of the nonpositive-argument extended shuffle algebra, with its dual $\varphi^*$ giving a corresponding coalgebra isomorphism. Together with explicit recursions for the dual coproducts and compatibility with graded duals, the work unifies the positive and nonpositive sectors of MZVs within a common algebraic framework, giving an algebraic analogue of the functional equation $\

Abstract

This paper offers a Hopf algebraic interpretation of a functional equation of multiple zeta functions, motivated by the classical symmetry of the Riemann zeta function. Starting from the extended shuffle algebra that encodes multiple zeta values (MZVs) at integer arguments, we show that its subalgebra corresponding to nonpositive arguments carries a natural differential Hopf algebra structure. This Hopf algebra is in graded linear duality with the shuffle Hopf algebra associated to MZVs at positive arguments. The resulting duality, realized through an explicit isomorphism, provides an algebraic analog of the functional equation relating $ζ(s)$ with $ζ(1-s)$ of the Riemann zeta function and unifies the positive and nonpositive sectors of multiple zeta functions within a common Hopf algebraic framework.

A Riemann-type duality of shuffle Hopf algebras related to multiple zeta values

TL;DR

The paper develops a rigorous Hopf-algebraic framework to model the symmetry underlying the functional equation of multiple zeta values. By constructing the extended shuffle algebra and its subalgebras and , it equips them with differential Hopf structures via operators and . A central result is the Riemann-type dual map that provides a differential Hopf algebra isomorphism between the positive-argument shuffle algebra and the graded dual of the nonpositive-argument extended shuffle algebra, with its dual giving a corresponding coalgebra isomorphism. Together with explicit recursions for the dual coproducts and compatibility with graded duals, the work unifies the positive and nonpositive sectors of MZVs within a common algebraic framework, giving an algebraic analogue of the functional equation $\

Abstract

This paper offers a Hopf algebraic interpretation of a functional equation of multiple zeta functions, motivated by the classical symmetry of the Riemann zeta function. Starting from the extended shuffle algebra that encodes multiple zeta values (MZVs) at integer arguments, we show that its subalgebra corresponding to nonpositive arguments carries a natural differential Hopf algebra structure. This Hopf algebra is in graded linear duality with the shuffle Hopf algebra associated to MZVs at positive arguments. The resulting duality, realized through an explicit isomorphism, provides an algebraic analog of the functional equation relating with of the Riemann zeta function and unifies the positive and nonpositive sectors of multiple zeta functions within a common Hopf algebraic framework.
Paper Structure (22 sections, 38 theorems, 182 equations)

This paper contains 22 sections, 38 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Functional equations and functional relations for multiple zeta values
  3. A duality of Hopf algebras for MZVs
  4. Hopf algebras in the study of MZVs
  5. Shuffle algebra of MZV ${{\mathcal{H}} _{\mathbb{Z} _{\ge 1}}}$
  6. Extended shuffle algebra ${{\mathcal{H}} _{\mathbb{Z} }}$ and its subalgebras
  7. Duality of Hopf algebras
  8. Outline of the paper
  9. Differential Hopf algebras
  10. The extended shuffle algebra ${{\mathcal{H}} _{\mathbb{Z} }}$ and its subalgebras
  11. The extended shuffle algebra ${{\mathcal{H}} _{\mathbb{Z} }}$
  12. The subalgebra ${{\mathcal{H}} _{\mathbb{Z} _{\ge 1}}}$
  13. The subalgebra ${{\mathcal{H}} _{\mathbb{Z} _{\le 0}}}$
  14. The differential Hopf algebra structure on ${{\mathcal{H}} _{\mathbb{Z} _{\ge 1}}}$
  15. The differential Hopf algebra structure on ${{\mathcal{H}} _{\mathbb{Z} _{\le 0}}}$
  16. ...and 7 more sections

Key Result

Theorem 2.1

GHXZ2 With respect to the depth grading, there is a unique graded associative product ${\hbox{X}}$ on ${{\mathcal{H}} _{\mathbb{Z} }}$, called the extended shuffle product, with the unit ${\bf 1}$, such that

Theorems & Definitions (66)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4: GHXZ2
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • Definition 2.7
  • ...and 56 more