The Hopf algebra of finite topologies and mould composition

Fauvet, Frédéric; Foissy, Loïc; Manchon, Dominique

Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 911-945.

Nous mettons en évidence un coproduit interne sur l’algèbre de Hopf des topologies finies introduite récemment par C. Malvenuto, F. Patras et le second auteur. Ce coproduit est dual de la composition des “quasi-ormoules”, version naturelle des moules, selon la terminologie de J. Ecalle, dans ce contexte.

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.

Classification : 05E05, 06A11, 16T30
Keywords: finite topological spaces, Hopf algebras, mould calculus, posets, quasi-orders
Mot clés : espaces topologiques finis, algèbres de Hopf, calcul moulien, ensembles partiellement ordonnés, préordres

[1] Aguiar, Marcelo; Mahajan, Swapneel Monoidal functors, species and Hopf algebras, CRM Monographs Series, 29, Amer. Math. Soc., Providence, R.I., 2010, li+784 pages

[2] Aguiar, Marcelo; Mahajan, Swapneel Hopf monoids in the category of species, Proceedings of the international conference, University of Almeréa, Almeréa, Spain, July 4?8, 2011. (Contemporary Mathematics), Volume 585 (2013), pp. 17-124

[3] Aguiar, Marcelo; Santos, Walter Ferrer; Moreira, Walter The Heisenberg product: from Hopf algebras and species to symmetric functions (2015) (https://arxiv.org/abs/1504.06315)

[4] Alexandroff, Pavel Diskrete Räume, Rec. Math. Moscou, n. Ser., Volume 2 (1937) no. 3, pp. 501-519

[5] Bergeron, Nantel; Zabrocki, Mike The Hopf algebras of symmetric functions and quasi-symmetric functions in non-commutative variables are free and co-free, J. Algebra Appl., Volume 8 (2009) no. 4, pp. 581-600 | DOI

[6] Brown, Kenneth S. Semigroups, rings, and Markov chains, J. Theor. Probab., Volume 13 (2000) no. 3, pp. 871-938 | DOI

[7] Calaque, Damien; Ebrahimi-Fard, Kurusch; Manchon, Dominique Two interacting Hopf algebras of trees, Adv. Appl. Math., Volume 47 (2011) no. 2, pp. 282-308 | DOI

[8] Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Int. J. Algebra Comput., Volume 12 (2002) no. 5, pp. 671-717 | DOI

[9] Ecalle, Jean Singularités non abordables par la géométrie, Ann. Inst. Fourier, Volume 42 (1992) no. 1-2, pp. 73-164 | DOI

[10] Ecalle, Jean La trigèbre des ormoules (2010) (Private communication)

[11] Ecalle, Jean; Vallet, Bruno The arborification–coarborification transform: analytic, combinatorial, and algebraic aspects, Ann. Fac. Sci. Toulouse, Volume 13 (2004) no. 4, pp. 575-657 | DOI

[12] Fauvet, Frédéric; Menous, Frédéric 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] Foissy, Loïc Algebraic structures on double and plane posets, J. Algebr. Comb., Volume 37 (2013) no. 1, pp. 39-66 | DOI

[14] Foissy, Loïc Plane posets, special posets and permutations, Adv. Math., Volume 240 (2013), pp. 24-60 | DOI

[15] Foissy, Loïc; Malvenuto, Claudia The Hopf algebra of finite topologies and $𝒯$-partitions, J. Algebra, Volume 438 (2015), pp. 130-169 | DOI

[16] Foissy, Loïc; Malvenuto, Claudia; Patras, Frédéric $B_{\infty }$-algebras, their enveloping algebras and finite spaces, J. Pure Appl. Algebra, Volume 220 (2016) no. 6, pp. 2434-2458 | DOI

[17] Foissy, Loïc; Novelli, Jean-Christophe; Thibon, Jean-Yves Deformations of shuffles and quasi-shuffles, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 209-237 | DOI

[18] Foissy, Loïc; Novelli, Jean-Christophe; Thibon, Jean-Yves Polynomial realizations of some combinatorial Hopf algebras, J. Noncommut. Geom., Volume 8 (2104) no. 1, pp. 141-162 | DOI

[19] Gessel, Ira M. Multipartite P-partitions and inner products of skew Schur functions, Combinatorics and algebra, Boulder, Colorado, 1983 (Contemp. Math.), Volume 34, Amer. Math. Soc., Providence, R.I. (1984), pp. 289-317

[20] Hoffman, Michael E. Quasi-shuffle products, J. Algebr. Comb., Volume 11 (2000) no. 1, pp. 49-68 | DOI

[21] Malvenuto, Claudia; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | DOI

[22] Manchon, Dominique On bialgebras and Hopf algebras of oriented graphs, Confluentes Math., Volume 4 (2012) no. 1 10 pp. (electronic) | DOI

[23] Menous, Frédéric An example of local analytic $q$-difference equation: analytic classification, Ann. Fac. Sci. Toulouse, Volume 15 (2006) no. 4, pp. 773-814 | DOI

[24] Menous, Frédéric On the stability of some groups of formal diffeomorphisms by the Birkhoff decomposition, Adv. Math., Volume 216 (2007) no. 1, pp. 1-28 | DOI

[25] Novelli, Jean-Christophe; Thibon, Jean-Yves Parking functions and descent algebras, Ann. Comb., Volume 11 (2007) no. 1, pp. 59-68 | DOI

[26] Stanley, Richard P. Ordered structures and partitions, Mem. Am. Math. Soc., Volume 119 (1972) (104 pp.)

[27] Stanley, Richard P. Enumerative Combinatorics Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, 2001, xii+585 pages

[28] Stanley, Richard P. Enumerative Combinatorics Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 2011, iv+642 pages

[29] Steiner, Anne K. The lattice of topologies: structure and complementation, Trans. Am. Math. Soc., Volume 122 (1966), pp. 379-398 | DOI

[30] Vaidyanathaswamy, Ramaswamy S. Set topology, Chelsea, New-York, 1960