Mixable shuffles, quasi-shuffles and Hopf algebras
Kurusch Ebrahimi-Fard1
and Li Guo2
1Universit\ddot at Bonn - Physikalisches Institut, Nussallee 12, D-53115 Bonn, Germany
2Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA
2Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, USA
DOI: 10.1007/s10801-006-9103-x
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.
Pages: 83–101
Full Text: PDF
References
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2. G. E. Andrews, L. Guo, W. Keigher and K. Ono, Baxter algebras and Hopf algebras, Trans. AMS, 355 (2003), no. 11, 4639-4656.
3. G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10 (1960), 731-742.
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/0310082
5. D. M. Bradley, Multiple q-zeta values, J. Algebra, 283 (2005), 752-798. arXiv:math.QA/0402093
6. P. Cartier, On the structure of free Baxter algebras, Adv. in Math., 9 (1972), 253-265.
7. K.T. Chen, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. of Math., 65 (1957), 163-178.
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/9912092
9. A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 216(1) (2001), 215-241. arXiv:hep-th/0003188
10. K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Letters in Mathematical Physics, 61(2) (2002), 139-147.
11. K. Ebrahimi-Fard, On the associative Nijenhuis algebras, The Electronic Journal of Combinatorics, Volume 11(1), R38, (2004). arXiv:math-ph/0302062
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/0407162
13. K. Ebrahimi-Fard and L. Guo, Rota-Baxter Algebras, Dendriform Algebras and Poincaré-Birkhoff- Witt Theorem, preprint, arXiv:math.RA/0503342.
14. K. Ebrahimi-Fard and L. Guo, Rota-Baxter algebras and multiple zeta values, preprint, 2005,