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

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 [10] 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 [11]. In a series of recent talks, Hirose and Sato proposed a conjectural family of relations coming from a new generalised shuffle structure, lifting a set of graded relations established by the author [13] to conjectural genuine 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 suggesting several more which can be verified numerically, hence providing further evidence for the conjecture of Hirose and Sato.

Block shuffle identities for multiple zeta values

“…We denote by dec : X → X ⊗ X the deconcatenation coproduct. Then (X, , dec) is a commutative Hopf algebra (see [18]). Let R = H X 1 , X 2 , .…”