Abstract
For any commutative algebra R the shuffle product on the tensor module T(R) can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted T q(R). We show that if R is the polynomial algebra, then T q(R) is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CTD, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasi-shuffle algebra in the noncommutative setting.
Similar content being viewed by others
References
Aguiar M. (2000). Pre-Poisson algebras. Lett. Math. Phys. 54(4): 263–277
Brouder Ch., Frabetti A. and Krattenthaler C. (2006). Non-commutative Hopf algebra of formal diffeomorphisms. Adv. Math. 200(2): 479–524
Brouder Ch. and Schmitt W. (2007). Renormalization as a functor on bialgebras. J. Pure Appl. Alg. 209: 477–495
Burgunder, E.: Bigèbre magmatique et bigèbre associative-Zinbiel, Mémoire de Mastère, Strasbourg (2005)
Cartier P. (1972). On the structure of free Baxter algebras. Adv. Math. 9: 253–265
Cartier P.: Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents. Séminaire Bourbaki, vol. 2000/2001. Astérisque No. 282, Exp. No. 885, viii, 137–173 (2002)
Ebrahimi-Fard K. and Guo L. (2005). Integrable renormalization. II. The general case. Ann. Henri Poincaré 6(2): 369–395
Guo L. and Keigher W. (2000). Baxter algebras and shuffle products. Adv. Math. 150(1): 117–149
Hoffman M. (2000). Quasi-shuffle products. J. Algebraic Combin. 11(1): 49–68
Ihara K., Kaneko M. and Zagier D. (2006). Derivation and double shuffle relations for Multiple Zeta Values. Compos. Math. 142(2): 307–338
Loday, J.-L.: Dialgebras, in “Dialgebras and related operads”, Springer Lecture Notes in Math. 1763, pp. 7–66 (2001)
Loday, J.-L.: Scindement d’associativité et algèbres de Hopf, Proceedings of the Conference in honor of Jean Leray, Nantes 2002, Séminaire et Congrès (SMF) 9, 155–172 (2004)
Loday, J.-L.: Generalized bialgebras and triples of operads. Preprint 2006, ArXiv:math. QA/0611885
Loday, J.-L., Ronco, M.: Trialgebras and families of polytopes. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory. Contemp. Math. 346, 369–398 (2004)
Loday J.-L. and Ronco M. (2006). On the structure of cofree Hopf algebras. J. reine angew. Math. 592: 123–155
Manchon D. (1997). L’algèbre de Hopf bitensorielle. Comm. Algebra 25(5): 1537–1551
Milnor J.W. and Moore J.C. (1965). On the structure of Hopf algebras. Ann. Math. 81(2): 211–264
Quillen D. (1969). Rational homotopy theory. Ann. Math. 90(2): 205–295
Ronco, M.: Primitive elements in a free dendriform algebra, New trends in Hopf algebra theory (La Falda, 1999), 245–263, Contemp. Math., 267, Amer. Math. Soc., Providence, RI (2000)
Ronco M. (2002). Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras. J. Algebra 254(1): 152–172
Vallette B. (2007). Homology of generalized partition posets. J. Pure Appl. Algebra 208: 699–725