Algebraic Aspects of Multiple Zeta Values

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 ζ : H0 → R from a graded rational vector space H0 generated by the “admissible words” of the noncommutative polynomial algebra Q〈x, y〉. Now H0 admits two (commutative) products making ζ a homomorphism–the shuffle product and the “harmonic” product. The latter makes H0 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 we define an action of QSym on Q〈x, y〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series.