Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Isomorphism between Hopf algebras for multiple zeta values

Li Guo, Hongyu Xiang, Bin Zhang

TL;DR

The paper addresses the problem of relating the shuffle Hopf algebra and the quasi-shuffle Hopf algebra that encode multiple zeta values ($\mathrm{MZVs}$). It builds a combinatorial framework using the Hopf-algebraic theory of quasisymmetric functions and a character-based universal map $\Psi_\chi$ to connect the two pictures, establishing an order on tensors to obtain upper-triangularity and a bijection between nonvanishing characters and isomorphisms. A key contribution is the explicit construction of a Hopf algebra isomorphism via a special character $\chi_0$ with $\chi_0([n])=1/n!$, yielding concrete formulas like $\Psi_{\chi_0}([n])=[n]/n!$ and $\Psi_{\chi_0}([1,1])=\tfrac{1}{2}[2]+[1,1]$, thereby concretely linking the shuffle and quasi-shuffle structures. This work advances the understanding of the algebraic underpinnings of $\mathrm{MZVs}$ and has potential implications for the study of relations among MZVs and renormalization techniques in quantum field theory.

Abstract

The classical quasi-shuffle algebra for multiple zeta values have a well-known Hopf algebra structure. Recently, the shuffle algebra for multiple zeta values are also equipped with a Hopf algebra structure. This paper shows that these two Hopf algebras are isomorphic.

Isomorphism between Hopf algebras for multiple zeta values

TL;DR

The paper addresses the problem of relating the shuffle Hopf algebra and the quasi-shuffle Hopf algebra that encode multiple zeta values (). It builds a combinatorial framework using the Hopf-algebraic theory of quasisymmetric functions and a character-based universal map to connect the two pictures, establishing an order on tensors to obtain upper-triangularity and a bijection between nonvanishing characters and isomorphisms. A key contribution is the explicit construction of a Hopf algebra isomorphism via a special character with , yielding concrete formulas like and , thereby concretely linking the shuffle and quasi-shuffle structures. This work advances the understanding of the algebraic underpinnings of and has potential implications for the study of relations among MZVs and renormalization techniques in quantum field theory.

Abstract

The classical quasi-shuffle algebra for multiple zeta values have a well-known Hopf algebra structure. Recently, the shuffle algebra for multiple zeta values are also equipped with a Hopf algebra structure. This paper shows that these two Hopf algebras are isomorphic.
Paper Structure (11 sections, 16 theorems, 106 equations)

This paper contains 11 sections, 16 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Shuffle products for MZVs
  3. Hopf algebra structures on MZV shuffle and quasi-shuffle algebras
  4. The shuffle Hopf algebra and quasi-shuffle Hopf algebra for MZVs
  5. The shuffle Hopf algebra for MZVs
  6. Quasi-symmetric functions
  7. The quasi-shuffle algebra for MZVs
  8. Isomorphisms between Hopf algebras for MZVs
  9. An order for tensors
  10. Induced Hopf algebra homomorphisms
  11. A special isomorphism between the Hopf algebras

Key Result

Proposition 2.1

GZ3 The shuffle product on ${{\mathcal{H}} _{\mathbb{Z} _{\ge 1}}}$ is the restriction of the unique commutative multiplication ${\hbox{X}}$ on ${{\mathcal{H}} _{\mathbb{Z} _{\ge 0}}}$ that makes $I$ a Rota-Baxter operator of weight $0$ on ${\mathcal{H}}^+_{\mathbb{Z} _{\ge 0}}$. More precisely, ${\ and

Theorems & Definitions (28)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Example 3.1
  • Lemma 3.2
  • Definition 3.3
  • Example 3.4
  • ...and 18 more