A generalisation of quasi-shuffle algebras and an application to multiple zeta values

TL;DR

This paper introduces a generalized quasi-shuffle algebra structure, demonstrating its isomorphism with the standard shuffle algebra over rationals, and applies it to derive new relations among multiple zeta values.

Contribution

It defines a new commutative algebra structure generalizing quasi-shuffle algebras and shows its isomorphism with shuffle algebras, enabling new insights into multiple zeta values.

Findings

Reproduces most of Hoffman's results on quasi-shuffle algebras

Establishes several new relations among multiple zeta values

Shows the algebraic structure is isomorphic to the standard shuffle algebra over rationals

Abstract

A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras have been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalisations. In a recent preprint, Hirose and Sato establish a family of relations coming from a new generalised shuffle structure, lifting a set of graded relations established by the author to genuine ungraded relations. In this paper, we define a commutative algebra structure on the space of non-commutative polynomials in a countable alphabet, generalising the shuffle-like structure of Hirose and Sato. We show that, over the rational numbers, this generalised quasi-shuffle algebra is isomorphic to the standard shuffle algebra, allowing us to reproduce most of Hoffman's results on quasi-shuffle algebras. We then apply these results to the case of multiple zeta values, reproducing several known families of results and establishing several more.