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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.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 05C05 Trees 81T15 Perturbative methods of renormalization applied to problems in quantum field theory
##### Keywords:
Hopf algebras; decorated rooted trees; dendriform algebras
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##### References:
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