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.
Reviewer: Goutam Mukherjee (Kolkata)
MSC:
| 16T05 | Hopf algebras and their applications |
| 18D50 | Operads (MSC2010) |
| 17A50 | Free nonassociative algebras |
| 18D35 | Structured objects in a category (MSC2010) |
Software:
OEISReferences:
| [1] | Aguiar, M., Infinitesimal Hopf algebras, (Andruskiewitsch, N.; etal., New Trends in Hopf Algebra Theory. New Trends in Hopf Algebra Theory, Contemp. Math., vol. 267 (2000), Amer. Math. Soc.), 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, (Progr. Math., vol. 129 (1995), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 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, (Chein, O.; etal., Quasigroups and Loops: Theory and Applications (1990), Heldermann Verlag: Heldermann Verlag Berlin), 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; 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, (Dialgebras and Related Operads. Dialgebras and Related Operads, Lecture Notes in Math., vol. 1763 (2001), Springer), 7-66 · Zbl 0999.17002 |
| [19] | Loday, J.-L., Scindement d’associativité et algèbres de Hopf, (Actes des journées mathématiques à la mémoire de Jean Leray. Actes des journées mathématiques à la mémoire de Jean Leray, Nantes, 2002. Actes des journées mathématiques à la mémoire de Jean Leray. Actes des journées mathématiques à la mémoire de Jean Leray, Nantes, 2002, Séminaire et Congrès, vol. 9 (2004), SMF), 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: Clarendon Press Oxford · Zbl 0487.20007 |
| [24] | Moerdijk, I., On the Connes-Kreimer construction of Hopf algebras, (Contemp. Math., vol. 271 (2001)), 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; 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: Oxford Univ. Press New York · Zbl 0798.17001 |
| [29] | Ronco, M., Primitive elements in a free dendriform algebra, (Andruskiewitsch, N.; etal., New Trends in Hopf Algebra Theory. New Trends in Hopf Algebra Theory, Contemp. Math., vol. 267 (2000), Amer. Math. Soc.), 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] | (Sloane, N., The On-Line Encyclopedia of Integer Sequences (2004)) · Zbl 1044.11108 |
| [33] | Stanley, R. P., Enumerative Combinatorics, vol. 1 (1997), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0889.05001 |
| [34] | Stanley, R. P., Enumerative Combinatorics, vol. 2 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0928.05001 |
| [35] | Stasheff, J. D., From operads to physically inspired theories, (Loday, J.-L.; etal., Operads, Proceeding of Renaissance Conferences (1997), Amer. Math. Soc.) · 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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