Weak stuffle algebras

Year 2022, Volume: 32 Issue: 32, 1 - 45, 16.07.2022

Cecile Mammez

https://doi.org/10.24330/ieja.1060709

Abstract

Motivated by $q$-shuffle products determined by Singer from $q$-analogues of multiple zeta values, we build in this article a generalisation of the shuffle and stuffle products in terms of weak shuffle and stuffle products. Then, we characterise weak shuffle products and give as examples the case of an alphabet of cardinality two or three. We focus on a comparison between algebraic structures respected in the classical case and in the weak case. As in the classical case, each weak shuffle product can be equipped with a dendriform structure. However, they have another behaviour towards the quadri-algebra and the Hopf algebra structure. We give some relations satisfied by weak stuffle products.

Keywords

Shuffle algebra, stuffle algebra, dendriform algebra, quadri-algebra, Hopf algebra

References

• M. Aguiar and J-L. Loday, Quadri-algebras, J. Pure Appl. Algebra, 191(3) (2004), 205-221.
• D. M. Bradley, Multiple q-zeta values, J. Algebra, 283(2) (2005), 752-798.
• F. Chapoton, Un theoreme de Cartier-Milnor-Moore-Quillen pour les bigebres dendriformes et les algebres braces, J. Pure Appl. Algebra, 168(1) (2002), 1-18.
• G. Duchamp, F. Hivert, J.-C. Novelli and J.-Y. Thibon, Noncommutative symmetric functions VII: Free quasi-symmetric functions revisited, Ann. Comb., 15 (2011), 655-673.
• G. Duchamp, F. Hivert and J-Y. Thibon, Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput., 12(5) (2002), 671-717.
• K. Ebrahimi-Fard and L. Guo, Mixable shuffles, quasi-shuffles and Hopf algebras, J. Algebraic Combin., 24(1) (2006), 83-101.
• K. Ebrahimi-Fard, D. Manchon and J. Singer, Duality and q-multiple zeta values, Adv. Math., 298 (2016), 254-285.
• K. Ebrahimi-Fard, D. Manchon and J. Singer, The Hopf algebra of q-multiple polylogarithms with non-positive arguments, Int. Math. Res. Not., 16 (2017), 4882-4922.
• L. Foissy, Les algebres de Hopf des arbres enracines decores. II, Bull. Sci.Math., 126(4) (2002), 249-288.
• L. Foissy, Bidendriform bialgebras, trees and free quasi-symmetric functions, J. Pure Appl. Algebra, 209(2) (2007), 439-459.
• L. Foissy, Free quadri-algebras and dual quadri-algebras, Comm. Algebra, 48(12) (2020), 5123-5141.
• L. Foissy, F. Patras and J.-Y. Thibon, Deformations of shuffles and quasi-shuffles, Ann. Inst. Fourier, 66(1) (2016), 209-237.
• I. M. Gessel, Multipartite p-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., (1983)), Contemp. Math., 34, Amer. Math. Soc., Providence, RI, (1984), 289-317.
• L. Guo and W. Keigher, Baxter Algebras and Shuffle Products, Adv. Math., 150(1) (2000), 117-149.
• M. E. Hoffman, The algebra of multiple harmonic series, J. Algebra, 194(2) (1997), 477-495.
• M. E. Hoffman, Quasi-shuffle products, J. Algebraic Combin., 11(1) (2000), 49-68.
• M. E. Hoffman, Quasi-shuffle algebras and applications, In: Chapoton, F., et al. (eds.) Algebraic Combinatorics, Resurgence, Moulds and Applications, Vol. 2 (IRMA Lectures in Mathematics and Theoretical Physics vol. 32), European Math. Soc. Publ. House, Berlin (2020), 327-348.
• M. E. Hoffman and K. Ihara, Quasi-shuffle products revisited, J. Algebra, 481 (2017), 293-326.
• M. E. Hoffman and Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra, 262(2) (2003), 332-347.
• R-Q. Jian, Quantum quasi-shuffle algebras II, J. Algebra, 472 (2017), 480-506.
• R-Q. Jian, M. Rosso and J. Zhang, Quantum Quasi-Shuffle Algebras, Lett. Math. Phys., 92(1) (2010), 1-16.
• J-L. Loday, Dialgebras, in Dialgebras and related operads, Lecture Notes in Math., Springer, Berlin, 1763 (2001), 7-66.
• J-L. Loday and M. Ronco, Hopf algebra of the planar binary trees, Adv. Math., 139(2) (1998), 293-309.
• C. Malvenuto, Produits et Coproduits des Fonctions Quasi-Symetriques etde lalgebre des Descentes, Ph.D. thesis, Universite du Quebec a Montreal, Laboratoire de Combinatoire et d’Informatique Mathematique, 1994.
• C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177(3) (1995), 967-982.
• C. Mammez, A propos de lalgebre de Hopf des mots tasses WMat, Bull. Sci.Math., 145 (2018), 53-96.
• M. Ronco, A Milnor-Moore theorem for dendriform Hopf algebras, C. R. Acad.Sci. Paris Ser. I Math., 332(2) (2001), 109-114.
• K.-G. Schlesinger, Some remarks on q-deformed multiple polylogarithms, arXiv:math/0111022, (2001).
• J. Singer, On q-analogues of multiple zeta values, Funct. Approx. Comment. Math., 53(1) (2015), 135-165.
• J. Singer, On Bradley’s q-MZVs and a generalized Euler decomposition formula, J. Algebra, 454 (2016), 92-122.
• J. Singer, q-Analogues of Multiple Zeta Values and Their Application in Renormalization, Ph.D. thesis, Der Naturwissenschaftlichen Fakultat, der Friedrich-Alexander-Univedrsitat, 2016.
• B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math., 620 (2008), 105-164.
• Y. Vargas, Hopf algebra of permutation pattern functions, 26th International Conference on Formal Power Series and Algebraic Combinatorics, Chicago, United States, (2014), 839-850.
• D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992), vol. 120 of Progr. Math., Birkhauser, Basel, (1994), 497-512.
• J. Zhao, Multiple q-zeta functions and multiple q-polylogarithms, Ramanujan J., 14(2) (2007), 189-221.
• V. V. Zudilin, Algebraic relations for multiple zeta values, Russian Math. Surveys, 58(1) (2003), 1-29.