Coassociative magmatic bialgebras and the Fine numbers
Ralf Holtkamp1
, Jean-Louis Loday2
and María Ronco3
1Ruhr-Universität Fakultät für Mathematik 44780 Bochum Germany
2CNRS et Université Louis Pasteur Institut de Recherche Mathématique Avancée 7 rue R. Descartes 67084 Strasbourg Cedex France
3Facultad de Ciencias, Universidad de Valparaiso Departamento de Matematicas Avda. Gran Bretana 1091 Valparaiso Chile
2CNRS et Université Louis Pasteur Institut de Recherche Mathématique Avancée 7 rue R. Descartes 67084 Strasbourg Cedex France
3Facultad de Ciencias, Universidad de Valparaiso Departamento de Matematicas Avda. Gran Bretana 1091 Valparaiso Chile
DOI: 10.1007/s10801-007-0089-9
Abstract
We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n - 2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F n - 1. In short, the triple of operads ( As, Mag, MagFine) is good.
Pages: 97–114
Keywords: keywords bialgebra; generalized bialgebra; Hopf algebra; cartier-Milnor-Moore; Poincaré-Birkhoff-Witt; magmatic; operad; fine number; pre-Lie algebra
Full Text: PDF
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67. Cambridge University Press, Cambridge (1998) (xx+457 pp. Translated from the 1994 French original by M. Readdy, with a foreword by G.-C. Rota)
3. Burgunder, E.: Infinite magmatic bialgebras. Adv. Appl. Math. (2007, to appear)
4. Cartier, P.: Hyperalgèbres et groupes de Lie formels. Séminaire Sophus Lie, 2e année: 1955/56. Faculté des Sciences de Paris
5. Deutsch, E., Shapiro, L.: A survey of the Fine numbers. Selected papers in honor of Helge Tverberg. Discrete Math. 241(1-3), 241-265 (2001)
6. Ginzburg, V., Kapranov, M.: Koszul duality for operads. Duke Math. J. 76(1), 203-272 (1994)
7. Hofmann, K.H., Strambach, K.: Topological and analytic loops. In: Chein, O., et al. (eds.) Quasigroups and Loops: Theory and Applications, pp. 205-262. Heldermann, Berlin (1990)
8. Holtkamp, R.: Comparison of Hopf algebras on trees. Arch. Math. (Basel) 80(4), 368-383 (2003)
9. Holtkamp, R.: On Hopf algebra structures over free operads. Adv. Math. 207(2), 544-565 (2006)
10. Loday, J.-L.: Generalized bialgebras and triples of operads.