With cedram.org
Table of contents for this issue | Next article
Frédéric Fauvet; Loïc Foissy; Dominique Manchon
The Hopf algebra of finite topologies and mould composition
(Algèbre de Hopf des topologies finies et composition moulienne)
Annales de l'institut Fourier, 67 no. 3 (2017), p. 911-945, doi: 10.5802/aif.3100
Article PDF
Class. Math.: 05E05, 06A11, 16T30
Keywords: finite topological spaces, Hopf algebras, mould calculus, posets, quasi-orders

Résumé - Abstract

We exhibit an internal coproduct on the Hopf algebra of finite topologies recently defined by the second author, C. Malvenuto and F. Patras, dual to the composition of “quasi-ormoulds”, which are the natural version of J. Ecalle’s moulds in this setting. All these results are displayed in the linear species formalism.


[1] Marcelo Aguiar & Swapneel Mahajan, Monoidal functors, species and Hopf algebras, CRM Monographs Series 29, Amer. Math. Soc., Providence, R.I., 2010
[2] Marcelo Aguiar & Swapneel Mahajan, Hopf monoids in the category of species, in Proceedings of the international conference, University of Almeréa, Almeréa, Spain, July 4?8, 2011., Contemporary Mathematics, 2013, p. 17-124
[3] Marcelo Aguiar, Walter Ferrer Santos & Walter Moreira, “The Heisenberg product: from Hopf algebras and species to symmetric functions”, https://arxiv.org/abs/1504.06315, 2015
[4] Pavel Alexandroff, “Diskrete Räume”, Rec. Math. Moscou, n. Ser. 2 (1937) no. 3, p. 501-519
[5] Nantel Bergeron & Mike Zabrocki, “The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free”, J. Algebra Appl. 8 (2009) no. 4, p. 581-600 Article
[6] Kenneth S. Brown, “Semigroups, rings, and Markov chains”, J. Theor. Probab. 13 (2000) no. 3, p. 871-938 Article
[7] Damien Calaque, Kurusch Ebrahimi-Fard & Dominique Manchon, “Two interacting Hopf algebras of trees”, Adv. Appl. Math. 47 (2011) no. 2, p. 282-308 Article
[8] Gérard Duchamp, Florent Hivert & Jean-Yves Thibon, “Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras”, Int. J. Algebra Comput. 12 (2002) no. 5, p. 671-717 Article
[9] Jean Ecalle, “Singularités non abordables par la géométrie”, Ann. Inst. Fourier 42 (1992) no. 1-2, p. 73-164 Article
[10] Jean Ecalle, “La trigèbre des ormoules”, Private communication, 2010
[11] Jean Ecalle & Bruno Vallet, “The arborification–coarborification transform: analytic, combinatorial, and algebraic aspects”, Ann. Fac. Sci. Toulouse 13 (2004) no. 4, p. 575-657 Article
[12] Frédéric Fauvet & Frédéric Menous, “Ecalle’s arborification-coarborification transforms and Connes-Kreimer Hopf algebra”, https://arxiv.org/abs/1212.4740, to appear in Ann. Sci. Éc. Norm. Supér. (4)
[13] Loïc Foissy, “Algebraic structures on double and plane posets”, J. Algebr. Comb. 37 (2013) no. 1, p. 39-66 Article
[14] Loïc Foissy, “Plane posets, special posets and permutations”, Adv. Math. 240 (2013), p. 24-60 Article
[15] Loïc Foissy & Claudia Malvenuto, “The Hopf algebra of finite topologies and $\mathcal{T}$-partitions”, J. Algebra 438 (2015), p. 130-169 Article
[16] Loïc Foissy, Claudia Malvenuto & Frédéric Patras, “$B_\infty $-algebras, their enveloping algebras and finite spaces”, J. Pure Appl. Algebra 220 (2016) no. 6, p. 2434-2458 Article
[17] Loïc Foissy, Jean-Christophe Novelli & Jean-Yves Thibon, “Deformations of shuffles and quasi-shuffles”, Ann. Inst. Fourier 66 (2016) no. 1, p. 209-237 Article
[18] Loïc Foissy, Jean-Christophe Novelli & Jean-Yves Thibon, “Polynomial realizations of some combinatorial Hopf algebras”, J. Noncommut. Geom. 8 (2104) no. 1, p. 141-162 Article
[19] Ira M. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in Combinatorics and algebra, Boulder, Colorado, 1983, Contemp. Math., Amer. Math. Soc., Providence, R.I., 1984, p. 289-317
[20] Michael E. Hoffman, “Quasi-shuffle products”, J. Algebr. Comb. 11 (2000) no. 1, p. 49-68 Article
[21] Claudia Malvenuto & Christophe Reutenauer, “Duality between quasi-symmetric functions and the Solomon descent algebra”, J. Algebra 177 (1995) no. 3, p. 967-982 Article
[22] Dominique Manchon, “On bialgebras and Hopf algebras of oriented graphs”, Confluentes Math. 4 (2012) no. 1, 10 pp. (electronic) Article
[23] Frédéric Menous, “An example of local analytic $q$-difference equation: analytic classification”, Ann. Fac. Sci. Toulouse 15 (2006) no. 4, p. 773-814 Article
[24] Frédéric Menous, “On the stability of some groups of formal diffeomorphisms by the Birkhoff decomposition”, Adv. Math. 216 (2007) no. 1, p. 1-28 Article
[25] Jean-Christophe Novelli & Jean-Yves Thibon, “Parking functions and descent algebras”, Ann. Comb. 11 (2007) no. 1, p. 59-68 Article
[26] Richard P. Stanley, “Ordered structures and partitions”, Mem. Am. Math. Soc. 119 (1972), 104 pp.
[27] Richard P. Stanley, Enumerative Combinatorics Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, 2001
[28] Richard P. Stanley, Enumerative Combinatorics Vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 2011
[29] Anne K. Steiner, “The lattice of topologies: structure and complementation”, Trans. Am. Math. Soc. 122 (1966), p. 379-398 Article
[30] Ramaswamy S. Vaidyanathaswamy, Set topology, Chelsea, New-York, 1960