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Algebraic Aspects of Multiple Zeta Values

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Zeta Functions, Topology and Quantum Physics

Part of the book series: Developments in Mathematics ((DEVM,volume 14))

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 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.

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, U. S. Naval Academy, Annapolis, MD, 21402

    Michael E. Hoffman

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  1. Michael E. Hoffman
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Editor information

Editors and Affiliations

  1. Kinki University, Japan

    Takashi Aoki , Shigeru Kanemitsu , Mikio Nakahara  & Yasuo Ohno , ,  & 

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Hoffman, M.E. (2005). Algebraic Aspects of Multiple Zeta Values. In: Aoki, T., Kanemitsu, S., Nakahara, M., Ohno, Y. (eds) Zeta Functions, Topology and Quantum Physics. Developments in Mathematics, vol 14. Springer, Boston, MA. https://doi.org/10.1007/0-387-24981-8_4

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