Skip to main content
SpringerLink
Log in

Polylogarithms and multiple zeta values from free Rota-Baxter algebras

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series. We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aguiar M. Infinitesimal bialgebras, pre-Lie and dendriform algebras. (11/15/02) In: Hopf Algebras. Lecture Notes in Pure and Applied Mathematics, vol. 237. New York-Basel: Marcel Dekker, Inc, 2004, 1–33

    Google Scholar 

  2. André Y. Une Introduction aux Motifs. Société Mathématique de France, Paris, 2004

    MATH  Google Scholar 

  3. Bai C M. A unified algebraic approach to classical Yang-Baxter equation. J Phy A: Math Theor, 2007, 40: 11073–11082

    Article  MATH  Google Scholar 

  4. Bai C, Guo L, Ni X. Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras. Comm Math Phys, 2010, 297: 553–596

    Article  MATH  Google Scholar 

  5. Baxter G. An analytic problem whose solution follows from a simple algebraic identity. Pacific J Math, 1960, 10: 731–742

    MathSciNet  Google Scholar 

  6. Borwein J M, Broadhurst D J, Bradley D M, et al. Special values of multiple polylogarithms. Trans Amer Math Soc, 2001, 353: 907–941

    Article  MATH  MathSciNet  Google Scholar 

  7. Broadhurst D J, Kreimer D. Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys Lett B, 1997, 393: 403–412

    Article  MATH  MathSciNet  Google Scholar 

  8. Cartier P. On the structure of free Baxter algebras. Adv Math, 1972, 9: 253–265

    Article  MATH  MathSciNet  Google Scholar 

  9. Cartier P. Fonctions polylogarithmes, nombres polyzetas et groupes pro-unipotents. Astérisque, 2002, 282: 137–173

    MathSciNet  Google Scholar 

  10. Connes A, Kreimer D. Hopf algebras, Renormalization and Noncommutative Geometry. Comm Math Phys, 1988, 199: 203–242

    Article  MathSciNet  Google Scholar 

  11. Connes A, Kreimer D. Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Comm Math Phys, 2000, 210: 249–273

    Article  MATH  MathSciNet  Google Scholar 

  12. Ebrahimi-Fard K, Guo L. Rota-Baxter algebras and dendriform algebras. J Pure Appl Algebra, 2008, 212: 320–339

    Article  MATH  MathSciNet  Google Scholar 

  13. Ebrahimi-Fard K, Guo L. Quasi-shuffles, mixable shuffles and Hopf algebras. J Algebraic Combin, 2006, 24: 83–101, arXiv:math.RA/0506418

    Article  MATH  MathSciNet  Google Scholar 

  14. Ebrahimi-Fard K, Guo L, Kreimer D. Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. J Phys A: Math Gen, 2004, 37: 11037–11052, arXiv:hep-th/0407082

    Article  MATH  MathSciNet  Google Scholar 

  15. Ebrahimi-Fard K, Guo L, Manchon D. Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion. Comm Math Phys, 2006, 267: 821–845, arXiv: math-ph/0602004

    Article  MATH  MathSciNet  Google Scholar 

  16. Goncharov A. Multiple polylogarithms, cyclotomy and modular complexes. Math Res Lett, 1998, 5: 497–516

    MATH  MathSciNet  Google Scholar 

  17. Goncharov A, Manin Y. Multiple ζ-motives and moduli spaces . Comp Math, 2004, 140: 1–14

    Article  MATH  MathSciNet  Google Scholar 

  18. Guo L. Algebraic Birkhoff decomposition and its applications. In: Automorphic Forms and Langlands Program. Hong Kong: International Press, 2008, 283–323

    Google Scholar 

  19. Guo L. What is a Rota-Baxter algebra. Notice Amer Math Soc, 2009, 56: 1436–1437

    MATH  Google Scholar 

  20. Guo L, Keigher W. Free Baxter algebras and shuffle products. Adv Math, 2000, 150: 117–149

    Article  MATH  MathSciNet  Google Scholar 

  21. Guo L, Keigher W. On free Baxter algebras: completions and the internal construction. Adv Math, 2000, 151: 101–127, arXiv:math/0407156

    Article  MATH  MathSciNet  Google Scholar 

  22. Guo L, Paycha S, Xie B, et al. Double shuffle relations and renormalization of multiple zeta values. Proceedings of the Conference on Algebraic Cycles, to appear, arXiv:0906.0092[math.NT]

  23. Guo L, Xie B. Explicit double shuffle relations and a generalization of Euler’s decomposition formula, arXiv:0808.2618[math.NT]

  24. Guo L, Zhang B. Renormalization of multiple zeta values. J Algebra, 2008, 319: 3770–3809, arXiv:math.NT/0606076

    Article  MATH  MathSciNet  Google Scholar 

  25. Guo L, Zhang B. Differential Algebraic Birkhoff Decomposition and renormalization of multiple zeta values. J Number Theory, 2008, 128: 2318–2339, arXiv:0710.0432[math.NT]

    Article  MATH  MathSciNet  Google Scholar 

  26. Hoffman M E. Multiple harmonic series. Pacific J Math, 1992, 152: 275–290

    MATH  MathSciNet  Google Scholar 

  27. Hoffman M E. The algebra of multiple harmonic series. J Algebra, 1997, 194: 477–495

    Article  MATH  MathSciNet  Google Scholar 

  28. Hoffman M E. Quasi-shuffle products. J Algebraic Combin, 2000, 11: 49–68

    Article  MATH  MathSciNet  Google Scholar 

  29. Ihara K, Kaneko M, Zagier D. Derivation and double shuffle relations for multiple zeta values. Compos Math, 2006, 142: 307–338

    Article  MATH  MathSciNet  Google Scholar 

  30. Le T Q T, Murakami J. Kontsevich s integral for the Homfly polynomial and relations between values of the multiple zeta functions. Topology Appl, 1995, 62: 193–206

    Article  MathSciNet  Google Scholar 

  31. Manchon D, Paycha S. Renormalized Chen integrals for symbols on ℝn and renormlized polyzeta functions. arXiv:math.NT/0702135

  32. Racinet G. Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Pub Math IHES, 2002, 95: 185–231

    MATH  MathSciNet  Google Scholar 

  33. Rota G C. Baxter operators, an introduction. In: “Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries”, Joseph P.S. Kung, Editor, Birkhäuser, Boston, 1995

    Google Scholar 

  34. Semenov-Tyan-Shanskii M A. What is a classical R-matrix? Funct Anal Appl, 1983, 17: 259–272

    Article  MathSciNet  Google Scholar 

  35. Terasoma T. Mixed Tate motives and multiple zeta values. Invent Math, 2002, 149: 339–369

    Article  MATH  MathSciNet  Google Scholar 

  36. Zagier D. Values of zeta functions and their applications. In: First European Congress of Mathematics, Vol. II, Progress Math, 120, Basel: Birkhäuser, 1994, 497–512

    Google Scholar 

  37. Zhao J. Analytic continuation of multiple zeta functions. Proc Amer Math Soc, 2000, 128: 1275–1283

    Article  MATH  MathSciNet  Google Scholar 

  38. Zudilin W. Algebraic relations for multiple zeta values (in Russian). Uspekhi Mat Nauk, 2003, 58: 3–32; Translation in Russian Math Survey, 2003, 58: 1–29

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Li Guo

  2. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, USA

    Li Guo

  3. Yangtze Center of Mathematics, Sichuan University, Chengdu, 610064, China

    Bin Zhang

Authors
  1. Li Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Zhang.

Additional information

Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, L., Zhang, B. Polylogarithms and multiple zeta values from free Rota-Baxter algebras. Sci. China Math. 53, 2239–2258 (2010). https://doi.org/10.1007/s11425-010-4044-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-010-4044-1

Keywords

MSC(2000)

Search

Navigation