Isomorphism between Hopf algebras for multiple zeta values

TL;DR

This paper demonstrates an isomorphism between the quasi-shuffle and shuffle Hopf algebras for multiple zeta values using quasi-symmetric functions, connecting two important algebraic structures in number theory.

Contribution

It establishes a new isomorphism between two Hopf algebras for multiple zeta values, extending known relationships via quasi-symmetric functions.

Findings

The quasi-shuffle and shuffle Hopf algebras are isomorphic.

The isomorphism is constructed using quasi-symmetric functions.

Comparison with Hoffman, Newman, and Radford's isomorphism is provided.

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 utilizing quasi-symmetric functions. This Hopf algebra isomorphism is compared with with the well-known isomorphism between the shuffle Hopf algebra and quasi-shuffle Hopf algebra of Hoffman, Newman and Radford.