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Infinitesimal and \(B_{\infty}\)-algebras, finite spaces, and quasi-symmetric functions. (English) Zbl 1390.16032

The authors show “that the set of finite spaces carries naturally (generalized) Hopf algebraic structures that are closely connected with usual topological constructions (such as joins or cup products) and familiar structures in topology (such as the one of cogroups in the category of associative algebras, or infinitesimal Hopf algebras, that have appeared, e.g., in the study of loop spaces of suspensions and the Bott-Samelson theorem.”
The main result of the paper are the following. “The linear span \(F\) of finite spaces carries the structure of the enveloping algebra of a \(B_{\infty}\)-algebra (Theorem 19). There is a (surjective, structure preserving) Hopf algebra morphism from \(F\) to the algebra of quasi-symmetric functions (Theorem 21).”

MSC:

16T30 Connections of Hopf algebras with combinatorics
18B30 Categories of topological spaces and continuous mappings (MSC2010)

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References:

[1] Aguiar, Marcelo; Bergeron, Nantel; Sottile, Frank, Combinatorial Hopf algebras and generalized Dehn-Sommerville relation, Compos. Math., 142, 1, 1-30 (2006) · Zbl 1092.05070
[2] Alexandroff, P., Diskrete Räume, Rec. Math. Moscou N.S., 2, 501-519 (1937), (in German) · Zbl 0018.09105
[3] Barmak, Jonathan A., Algebraic Topology of Finite Topological Spaces and Applications (2011), Springer: Springer Berlin · Zbl 1235.55001
[4] Barmak, Jonathan Ariel; Minian, Elias Gabriel, Minimal finite models, J. Homotopy Relat. Struct., 2, 1, 127-140 (2007) · Zbl 1185.55005
[5] Barmak, Jonathan Ariel; Minian, Elias Gabriel, Strong homotopy types, nerves and collapses, Discrete Comput. Geom., 47, 2, 301-328 (2012) · Zbl 1242.57019
[6] Baues, Hans J., The double bar and cobar constructions, Compos. Math., 43, 3, 331-341 (1981) · Zbl 0478.57027
[7] Berstein, Israel, On co-groups in the category of graded algebras, Trans. Am. Math. Soc., 115, 257-259 (1965) · Zbl 0134.42404
[8] Bott, Raoul; Samelson, Hans, On the Pontryagin product in spaces of paths, Comment. Math. Helv., 27, 320-337 (1953) · Zbl 0052.19301
[9] Burgunder, Emily, A symmetric version of Kontsevich graph complex and Leibniz homology, J. Lie Theory, 20, 1, 127-165 (2010) · Zbl 1197.17011
[10] Chapoton, Frédéric, Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres braces, J. Pure Appl. Algebra, 168, 1, 1-18 (2002) · Zbl 0994.18006
[11] Ebrahimi-Fard, Kurusch; Manchon, Dominique, Dendriform equations, J. Algebra, 322, 11, 4053-4079 (2009) · Zbl 1229.17001
[12] Erné, Marcel; Stege, Kurt, Counting finite posets and topologies, Order, 8, 3, 247-265 (1991) · Zbl 0752.05002
[13] Fieux, E.; Lacaze, J., Foldings in graphs and relations with simplicial complexes and posets, Discrete Math., 312, 17, 2639-2651 (2012) · Zbl 1246.05157
[14] Foissy, Loïc; Malvenuto, Claudia, The Hopf algebra of finite topologies and \(T\)-partitions, J. Algebra, 438, 130-169 (2015) · Zbl 1344.16031
[15] Foissy, Loïc; Patras, Frédéric, Natural endomorphisms of shuffle algebras, Int. J. Algebra Comput., 23, 4, 989-1009 (2013) · Zbl 1298.16021
[16] Fresse, Benoît, Algèbre des descentes et cogroupes dans les algèbres sur une opérade, Bull. Soc. Math. Fr., 126, 3, 407-433 (1998) · Zbl 0940.18004
[17] Gessel, Ira M., Multipartite \(P\)-partitions and inner products of skew Schur functions, (Combinatorics and Algebra. Combinatorics and Algebra, Boulder, CO, 1983. Combinatorics and Algebra. Combinatorics and Algebra, Boulder, CO, 1983, Contemp. Math., vol. 34 (1984), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 289-317
[18] Getzler, Ezra; Jones, John D. S., Operads, homotopy algebra and iterated integrals for double loop spaces (1994), arXiv preprint
[19] Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V., Algebras, Rings, and Modules: Lie Algebras and Hopf Algebras, Math. Surv. Monogr., vol. 168 (2010), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1211.16023
[20] Livernet, Muriel, A rigidity theorem for pre-Lie algebras, J. Pure Appl. Algebra, 207, 1, 1-18 (2006) · Zbl 1134.17001
[21] Loday, Jean-Louis; Ronco, María, On the structure of cofree Hopf algebras, J. Reine Angew. Math., 592, 123-155 (2006) · Zbl 1096.16019
[22] Malvenuto, Clauda; Reutenauer, Christophe, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177, 3, 967-982 (1995) · Zbl 0838.05100
[23] McCord, M. C., Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J., 33, 465-474 (1966) · Zbl 0142.21503
[24] McCord, M. C., Homotopy type comparison of a space with complexes associated with its open covers, Proc. Am. Math. Soc., 18, 705-708 (1967) · Zbl 0171.22001
[25] Novelli, Jean-Christophe; Patras, Frédéric; Thibon, Jean-Yves, Natural endomorphisms of quasi-shuffle Hopf algebras, Bull. Soc. Math. Fr., 141, 1, 107-130 (2013) · Zbl 1266.05175
[26] Patras, Frédéric, L’algèbre des descentes d’une bigèbre graduée, J. Algebra, 170, 2, 547-566 (1994) · Zbl 0819.16033
[27] Patras, Frédéric, A Leray theorem for the generalization to operads of Hopf algebras with divided powers, J. Algebra, 218, 2, 528-542 (1999) · Zbl 0947.16025
[28] Reutenauer, Christophe, Free Lie Algebras, London Math. Soc. Monogr. New Ser., vol. 7 (1993), Oxford University Press · Zbl 0798.17001
[29] Schützenberger, Marcel P., Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un probleme de mathématiques appliquées, Sémin. Dubreil. Algebre Théorie Nr., 12, 1, 1-23 (1958)
[30] Sharp, Henry, Cardinality of finite topologies, J. Comb. Theory, 5, 1, 82-86 (1968) · Zbl 0159.52202
[31] Sloane, N. J.A., On-line encyclopedia of integer sequences · Zbl 1044.11108
[32] Stanley, Richard P., On the number of open sets of finite topologies, J. Comb. Theory, Ser. A, 10, 1, 74-79 (1971) · Zbl 0214.21001
[33] Stanley, Richard P., Ordered Structures and Partitions, Mem. Am. Math. Soc., vol. 119 (1972), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0246.05007
[34] Stong, Robert E., Finite topological spaces, Trans. Am. Math. Soc., 123, 325-340 (1966) · Zbl 0151.29502
[35] Wright, J. A., There are 718 6-point topologies, quasi-orderings and transgraphs, Not. Am. Math. Soc., 17, 646 (1970)
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