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The Hopf algebras of decorated rooted trees. II. (Les algèbres de Hopf des arbres enracinés décorés. II.) (French) Zbl 1013.16027
The paper under review is Part II of [L. Foissy, Bull. Sci. Math. 126, No. 3, 193-239 (2002; see the preceding review Zbl 1013.16026)]. In Part I, the Hopf algebra \({\mathcal H}^{\mathcal D}_{P,R}\) of planar binary trees was introduced and studied; \({\mathcal H}^{\mathcal D}_{P,R}\) is a noncommutative version of the Hopf algebra \({\mathcal H}^{\mathcal D}_R\) studied by A. Connes and D. Kreimer [Commun. Math. Phys. 199, No. 1, 203-242 (1998; Zbl 0932.16038)].
In this second part, \({\mathcal H}^{\mathcal D}_{P,R}\) is related to several Hopf algebras of trees. Namely, \({\mathcal H}^{\mathcal D}_{P,R}\) is shown to be isomorphic to the Hopf algebra \({\mathcal H}^\gamma\) introduced by C. Brouder and A. Frabetti [Noncommutative renormalization for massless QED, hep-th/0011161]. Also, if the set \(\mathcal D\) has one element, \({\mathcal H}^{\mathcal D}_{P,R}\) is isomorphic to the dendriform algebra free in one generator \({\mathcal H}_L\) introduced by J.-L. Loday and M. O. Ronco [Adv. Math. 139, No. 2, 293-309 (1998; Zbl 0926.16032)]; and to the two parameter Hopf algebra studied by P. van der Laan [Some Hopf algebras of trees, preprint (2001)] and I. Moerdijk [Contemp. Math. 271, 311-321 (2001; Zbl 0987.16032)]. Further relations with the Hopf algebras studied by R. Grossman and R. G. Larson [J. Algebra 126, No. 1, 184-210 (1989; Zbl 0717.16029)] and A. Connes and H. Moscovici [Commun. Math. Phys. 198, No. 1, 199-246 (1998; Zbl 0940.58005)] are established. Other interesting results are also included.

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
05C05 Trees
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
Full Text: DOI
[1] Brouder, Ch., Runge-Kutta methods and renormalization, Eur. phys. J. C, 12, 521-534, (2000)
[2] Brouder, Ch.; Frabetti, A., Noncommutative renormalization for massless QED, 2000
[3] Chapoton, F., Un théorème de cartier-Milnor-Moore-Quillen pour LES bigèbres dendriformes et LES algèbres braces, 2000
[4] Connes, A.; Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Comm. math. phys., 199, 203, (1998) · Zbl 0932.16038
[5] Connes, A.; Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, IHES/M/98/37 · Zbl 0940.58005
[6] L. Foissy Les algèbres de Hopf des arbres enracinés décorés, I Bull. Sci. Math., à paraitre
[7] Grossman, R.; Larson, R.G., Hopf-algebraic structure of combinatorial objects and differential operators, Israel J. math., 72, 1-2, 109-117, (1990) · Zbl 0780.16029
[8] Grossman, R.; Larson, R.G., Hopf-algebraic structure of families of trees, J. algebra, 126, 1, 184-210, (1989) · Zbl 0717.16029
[9] R. Holtkamp, Comparison of Hopf algebras on trees, Preprint, 2001 · Zbl 1056.16030
[10] M. Kontsevich, On the algebraic structure of the Hochschild complex, Séminaire Groupes Quantiques, Ecole Polytechnique, 27 octobre 1998
[11] Kreimer, D., On the Hopf algebra structure of pertubative quantum field theories, 1998
[12] Kreimer, D., On overlapping divergences, 1999 · Zbl 0977.81091
[13] Kreimer, D., Chen’s iterated integral represents the operator product expansion, 1999 · Zbl 0971.81093
[14] Kreimer, D., Combinatorics of (pertubative) quantum field theory, 2000
[15] P. van der Laan, Some Hopf algebras of trees, Preprint, 2001
[16] Loday, J.L.; Ronco, M.O., Hopf algebra of the planar binary trees, Adv. math., 139, 2, 293-309, (1998) · Zbl 0926.16032
[17] Moerdijk, I., On the Connes-kreimer construction of Hopf algebras, Contemp. math., 271, 311-321, (2001) · Zbl 0987.16032
[18] Panaite, F., Relating the Connes-kreimer and grossman-larson Hopf algebras built on rooted trees, 2000 · Zbl 0959.16023
[19] Ronco, M.O., On the primitive elements of a free dendriform algebra, Contemp. math., 267, (2000) · Zbl 0974.16035
[20] Ronco, M.O., A Milnor-Moore theorem for dendriform Hopf algebras, C. R. acad. sci. Paris, Sér. I math., 332, 109-114, (2000) · Zbl 0978.16031
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