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On Hopf algebra structures over free operads. (English) Zbl 1117.16027
The author considers families of free operads \(\mathcal P\), which include the operad freely generated by a non-commutative, non-associative binary operation and the operad of Stasheff polytopes in order to deal with non-classical Hopf algebras, e.g., dendriform Hopf algebras. The main aim of the paper is to replace the operad \(As\) of associative algebras in \(\text{Prim}\,As\) by \(\mathcal P\) in order to prove Poincaré-Birkhoff Witt type theorems.

MSC:
16T05 Hopf algebras and their applications
18D50 Operads (MSC2010)
17A50 Free nonassociative algebras
18D35 Structured objects in a category (MSC2010)
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[1] Aguiar, M., Infinitesimal Hopf algebras, (), 1-29 · Zbl 0982.16028
[2] Bergman, G.M.; Hausknecht, A.O., Cogroups and co-rings in categories of associative rings, Math. surveys monogr., vol. 45, (1996), Amer. Math. Soc. · Zbl 0857.16001
[3] Bremner, M.R.; Hentzel, I.R.; Peresi, L.A., Dimension formulas for the free nonassociative algebra, Comm. algebra, 33, 4063-4081, (2005) · Zbl 1145.17300
[4] Chapoton, F., Un théorème de cartier-Milnor-Moore-Quillen pour LES bigèbres dendriformes et LES algèbres braces, J. pure appl. algebra, 168, 1-18, (2002) · Zbl 0994.18006
[5] Chapoton, F., On some anticyclic operads, Algebr. geom. topology, 5, 53-69, (2005) · Zbl 1060.18004
[6] Connes, A.; Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Comm. math. phys., 199, 203-242, (1998) · Zbl 0932.16038
[7] Deutsch, E.; Shapiro, L., A survey of the fine numbers, Discrete math., 241, 241-265, (2001) · Zbl 0992.05011
[8] Ebrahimi-Fard, K.; Guo, L., Coherent unit actions on operads and Hopf algebras, (2005), preprint
[9] Gerritzen, L., Planar rooted trees and non-associative exponential series, Adv. appl. math., 33, 2, 342-365, (2004) · Zbl 1052.05015
[10] Gerritzen, L., Planar binomial coefficients, preprint · Zbl 1052.05015
[11] Gerritzen, L.; Holtkamp, R., Hopf co-addition for free magma algebras and the non-associative Hausdorff series, J. algebra, 265, 264-284, (2003) · Zbl 1046.17001
[12] Getzler, E., Operads and moduli spaces of genus 0 Riemann surfaces, (), 199-230 · Zbl 0851.18005
[13] Griffing, G., The cofree nonassociative coalgebra, Comm. algebra, 16, 2387-2414, (1988) · Zbl 0653.17002
[14] Hofmann, K.H.; Strambach, K., Topological and analytic loops, (), 205-262 · Zbl 0747.22004
[15] Holtkamp, R., A pseudo-analyzer approach to formal group laws not of operad type, J. algebra, 237, 382-405, (2001) · Zbl 1042.14020
[16] R. Holtkamp, On Hopf algebra structures over operads, Habilitationsschrift, Bochum, 2004 · Zbl 1117.16027
[17] Loday, J.-L., Série de Hausdorff, idempotents eulériens et algèbres de Hopf, Expo. math., 12, 165-178, (1994) · Zbl 0807.17003
[18] Loday, J.-L., Dialgebras, (), 7-66 · Zbl 0999.17002
[19] Loday, J.-L., Scindement d’associativité et algèbres de Hopf, (), 155-172 · Zbl 1073.16032
[20] Loday, J.-L.; Ronco, M., Hopf algebra of the planar binary trees, Adv. math., 139, 299-309, (1998) · Zbl 0926.16032
[21] Loday, J.-L.; Ronco, M., Algèbres de Hopf colibres, C. R. acad. sci. Paris Sér. I, 337, 153-158, (2003) · Zbl 1060.16039
[22] Loday, J.-L.; Ronco, M., On the structure of cofree Hopf algebras, J. Reine Angew. Math., in press · Zbl 1096.16019
[23] Macdonald, I.G., Symmetric functions and Hall polynomials, (1995), Clarendon Press Oxford · Zbl 0487.20007
[24] Moerdijk, I., On the Connes-kreimer construction of Hopf algebras, (), 311-321 · Zbl 0987.16032
[25] Oudom, J.-M., Théorème de Leray dans la catégorie des algèbres sur une opérade, C. R. acad. sci. Paris Sér. I, 329, 101-106, (1999) · Zbl 0945.18007
[26] J.M. Pérez-Izquierdo, Algebras, hyperalgebras, nonassociative bialgebras and loops, preprint, 2004
[27] Quillen, D., Rational homotopy theory, Ann. of math. (2), 90, 205-295, (1969) · Zbl 0191.53702
[28] Reutenauer, C., Free Lie algebras, London math. soc. monogr., (1993), Oxford Univ. Press New York · Zbl 0798.17001
[29] Ronco, M., Primitive elements in a free dendriform algebra, (), 245-263 · Zbl 0974.16035
[30] Ronco, M., A Milnor-Moore theorem for dendriform Hopf algebras, C. R. acad. sci. Paris Sér. I, 332, 109-114, (2000) · Zbl 0978.16031
[31] Ronco, M., Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. algebra, 254, 1, 152-172, (2002) · Zbl 1017.16033
[32] ()
[33] Stanley, R.P., Enumerative combinatorics, vol. 1, (1997), Cambridge Univ. Press Cambridge · Zbl 0889.05001
[34] Stanley, R.P., Enumerative combinatorics, vol. 2, (1999), Cambridge Univ. Press Cambridge · Zbl 0928.05001
[35] Stasheff, J.D., From operads to physically inspired theories, () · Zbl 0872.55010
[36] Shestakov, I.P.; Umirbaev, U.U., Free akivis algebras, primitive elements, and hyperalgebras, J. algebra, 250, 2, 533-548, (2002) · Zbl 0993.17002
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