Advertisement

Journal of Algebraic Combinatorics

, Volume 24, Issue 1, pp 83–101 | Cite as

Mixable shuffles, quasi-shuffles and Hopf algebras

  • Kurusch Ebrahimi-FardEmail author
  • Li Guo
Article

Abstract

The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota–Baxter algebras.

Keywords

Hopf Algebra Commutative Ring Algebr Comb Admissible Pair Baxter Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Download to read the full article text

References

  1. 1.
    M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Applied Algebra, 191, (2004), 205–221. arXiv:math.QA/03090171.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G. E. Andrews, L. Guo, W. Keigher and K. Ono, Baxter algebras and Hopf algebras, Trans. AMS, 355 (2003), no. 11, 4639–4656.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10 (1960), 731–742.zbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Bowman and D. M. Bradley, The algebra and combinatorics of shuffles and multiple zeta values, J. Combinatorial Theory Ser. A, 97 (1) (2002), 43–61. arXiv:math.CO/0310082MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D. M. Bradley, Multiple q-zeta values, J. Algebra, 283 (2005), 752–798. arXiv:math.QA/0402093zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    P. Cartier, On the structure of free Baxter algebras, Adv. in Math., 9 (1972), 253–265.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    K.T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math., 65 (1957), 163–178.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 210(1) (2000), 249–273. arXiv:hep-th/9912092Google Scholar
  9. 9.
    A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The thβ-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 216(1) (2001), 215–241. arXiv:hep-th/0003188Google Scholar
  10. 10.
    K. Ebrahimi-Fard, Loday-type algebras and the Rota—Baxter relation, Letters in Mathematical Physics, 61(2) (2002), 139–147.zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    K. Ebrahimi-Fard, On the associative Nijenhuis algebras, The Electronic Journal of Combinatorics, Volume 11(1), R38, (2004). arXiv:math-ph/0302062zbMATHMathSciNetGoogle Scholar
  12. 12.
    K. Ebrahimi-Fard and L. Guo, On products and duality of binary, quadratic regular operads, J. Pure Applied Algebra, 200 (2005), 293–317. arXiv:math.RA/0407162MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    K. Ebrahimi-Fard and L. Guo, Rota-Baxter Algebras, Dendriform Algebras and Poincaré—Birkhoff—Witt Theorem, preprint, arXiv:math.RA/0503342.Google Scholar
  14. 14.
    K. Ebrahimi-Fard and L. Guo, Rota—Baxter algebras and multiple zeta values, preprint, 2005, http://newark.rutgers.edu/liguo.
  15. 15.
    K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable Renormalization I: the ladder case, J. Math. Phys., 45 (2004), 3758–2769. arXiv:hep-th/0402095MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable Renormalization II: the general case, Annales Henri Poincaré, 6 (2005), 369–395. arXiv:hep-th/0403118MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    K. Ebrahim-Fard, L. Guo and D. Kreimer, Spitzer’s Identity and the Algebraic Birkhoff Decomposition in pQFT, J. Phys. A: Math. Gen., 37 (2004), 11037–11052. arXiv:hep-th/0407082CrossRefGoogle Scholar
  18. 18.
    F. Fares, Quelques constructions d’algébres et de coalgébres’, Thesis, Université du Québec á Montréal.Google Scholar
  19. 19.
    A. B. Goncharov, Periods and mixed motives, preprint, arXiv:math.AG/0202154.Google Scholar
  20. 20.
    L. Guo, Baxter algebra and differential algebra, in: Differential Algebra and Related Topics, World Scientific Publishing Company, (2002), 281–305. arXiv:math.RA/0407180Google Scholar
  21. 21.
    L. Guo, Baxter algebras and the umbral calculus, Adv. in Appl. Math., 27 (2–3) (2001), 405–426.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    L. Guo, Baxter algebras, Stirling numbers and partitions, J. Algebra and Its Appl., 4 (2005), 153–164. arXiv:math.AC/0402348zbMATHCrossRefGoogle Scholar
  23. 23.
    L. Guo and W. Keigher, Free Baxter algebras and shuffle products, Adv. in Math., 150 (2000), 117–149. arXiv:math.RA/0407155MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    L. Guo and W. Keigher, On Baxter algebras: completions and the internal construction, Adv. in Math., 151 (2000), 101–127.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra, 194(2), (1997), 477–495.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin., 11(1), (2000), 49–68.zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    M. E. Hoffman, Algebraic aspects of multiple zeta values, to appear in Zeta functions, topology and quantum physics, Springer-Verlag, 2005. arXiv: math.QA/0309425.Google Scholar
  28. 28.
    M. E. Hoffman, Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra, 262 (2003), 332–347.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., 2 (1998), 303–334. arXiv:q-alg/9707029zbMATHMathSciNetGoogle Scholar
  30. 30.
    D. Kreimer, Chen’s iterated integral represents the operator product expansion, Adv. Theor. Math. Phys., 3 (1999), 627–670. arXiv:hep-th/9901099zbMATHMathSciNetGoogle Scholar
  31. 31.
    P. Leroux, Ennea-algebras, J. Algebra, 281 (2004), 287–302. arXiv:math.QA/0309213.zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Z. Lin and D. Nakano, Representations of Hopf algebras arising from Lie algebras of Cartan type, J. Algebra, 189 (1997), 529–567.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    J.-L. Loday and M. Ronco, Trialgebras and families of polytopes, in “Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory” Contemporary Mathematics 346 (2004), 369–398, arXiv:math.AT/0205043.MathSciNetGoogle Scholar
  34. 34.
    S. Montgomery, Hopf Algebras and Their Actions on Rings, American Mathematical Society, Providence, 1993.Google Scholar
  35. 35.
    C. Reutenauer, Free Lie Algebras, Oxford University Press, Oxford, 1993.zbMATHGoogle Scholar
  36. 36.
    G.-C. Rota, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc., 75 (1969), 325–329.zbMATHMathSciNetGoogle Scholar
  37. 37.
    G.-C. Rota, Baxter algebras and combinatorial identities II, Bull. Amer. Math. Soc., 75 (1969), 330–334.zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    G.-C. Rota and D. A. Smith, Fluctuation theory and Baxter algebras, Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971), pp. 179–201. Academic Press, London, (1972).Google Scholar
  39. 39.
    G.-C. Rota, Baxter operators, an introduction, In: Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries, Joseph P.S. Kung, Editor, Birkhäuser, Boston, 1995.Google Scholar
  40. 40.
    G.-C. Rota, Ten mathematics problems I will never solve, Invited address at the joint meeting of the American Mathematical Society and the Mexican Mathematical Society, Oaxaca, Mexico, December 6, 1997. DMV Mittellungen Heft 2, 1998, 45–52.Google Scholar
  41. 41.
    F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc., 82 (1956), 323–339.zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    M. A. Semenov-Tian-Shansky, Classical r-matrices, Lax equations, Poisson Lie groups and dressing transformations, In: Field theory, quantum gravity and strings, II (Meudon/Paris, 1985/1986), 174–214, Lecture Notes in Phys., 280 Springer, Berlin, (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    M. Sweedler, Hopf Algebras, Benjamin, New York, 1969.Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Universität Bonn - Physikalisches InstitutBonnGermany
  2. 2.Department of Mathematics and Computer ScienceRutgers UniversityNewarkUSA

Personalised recommendations