Algebraic Aspects of Multiple Zeta Values

Abstract

Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves "coding" the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map \zeta: H^0 -> R, where H^0 is the graded rational vector space generated by the "admissible words" of the noncommutative polynomial algebra Q<x,y>. Now H^0 admits two (commutative) products making \zeta a homomorphism: the shuffle product and the "harmonic" product. The latter makes H^0 a subalgebra of the algebra QSym of quasi-symmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q<x,y>, and define an action of the Hopf algebra QSym on Q<x,y> that appears useful. Finally, we apply the algebraic approach to finite partial sums of multiple zeta value series.