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 &#34;coding&#34; 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 ζ: H^0 -> R, where H^0 is the graded rational vector space generated by the &#34;admissible words&#34; of the noncommutative polynomial algebra Q<x,y>. Now H^0 admits two (commutative) products making ζa homomorphism: the shuffle product and the &#34;harmonic&#34; 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.