arXiv:1302.5391v7 [math.CO] 1 Mar 2014 Combinatorics of deformed shuffle Hopf algebras 1 G´erard H. E. Duchamp 12 Hoang Ngoc Minh 23 Christophe Tollu 12 Chiˆen B` ui 42 Hoang Nghia Nguyen 12 1 Universit´e Paris 13, 99, avenue Jean-Baptiste Cl´ement, 93430 Villetaneuse, France. 2 LIPN - UMR 7030, CNRS, 93430 Villetaneuse, France. 3 Universit´e Lille II, 1, Place D´eliot, 59024 Lille, France. Abstract In order to extend Sch¨ utzenberger’s factorization to general perturba- tions, the combinatorial aspects of the Hopf algebra of a deformed shuffle product is developed systematically in a parallel way with those of the shuffle product, with an emphasis on the Lie elements as studied by Ree. In particular, we will give an effective construction of pair of bases in duality. Contents 1 Introduction 2 2 First steps 2 3 General results on summability and duality 4 3.1 Total algebras and duality ...................... 4 3.1.1 Series and infinite sums ................... 4 3.1.2 Summable families in Hom spaces. ............. 5 3.1.3 Substitutions ......................... 7 3.2 Theorem of Cartier-Quillen-Milnor-Moore (analytic form) .... 10 3.2.1 General properties of bialgebras ............... 10 3.3 Counterexamples and discussion .................. 15 3.3.1 Counterexamples ....................... 15 3.3.2 The theorem from the point of view of summability ... 16 4 Application to the φ-deformed shuffle. 18 4.1 General results for the φ-deformed shuffle.............. 18 5 Conclusion 24 1 Version du 27-11-2021 20:56 1