Comparison of Hopf algebras on trees.
In recent years, various Hopf algebras involving certain trees have been constructed and studied. \textit{A. Connes} and \textit{D. Kreimer} [Commun. Math. Phys. 199, No. 1, 203-242 (1998; Zbl 0932.16038)] and \textit{D. Kreimer} [Adv. Theor. Math. Phys. 2, No. 2, 303-334 (1998; Zbl 1041.81087)], motivated by renormalization, constructed a commutative Hopf algebra based on Feynman diagrams. Replacing Feynman diagrams by planar binary trees, \textit{C. Brouder} and \textit{A. Frabetti} [J. Algebra 267, No. 1, 298-322 (2003; see Zbl 1056.16026 above)] constructed a noncommutative Hopf algebra for renormalization. \textit{J.-L. Loday} and \textit{M. O. Ronco} [Adv. Math. 139, No. 2, 293-309 (1998; Zbl 0926.16032)] also introduced a noncommutative Hopf algebra \(\mathbb{Q}(Y_\infty)\) based on planar binary trees. The author of the paper under review replaces abstract trees by planar trees to get a noncommutative version \(\mathbb{Q}(X_\infty)\) of the commutative Hopf algebra of \textit{A. Connes} and \textit{D. Kreimer} [op. cit]. He proves that \(\mathbb{Q}(X_\infty)\) and \(\mathbb{Q}(Y_\infty)\) are isomorphic as Hopf algebras. A comparison of these Hopf algebras by different methods was done by \textit{L. Foissy} [Les algèbres de Hopf des arbres enracinés décorés, Thèse de doctorat, Université de Reims (2002); see also Bull. Sci. Math. 126, No. 3, 193-239 (2002; Zbl 1013.16026) and ibid. 126, No. 4, 249-288 (2002; Zbl 1013.16027)] who also related these to the Hopf algebra of \textit{C. Brouder} and \textit{A. Frabetti} [op. cit.]. Completing these Hopf algebras with respect to the augmentation ideal yields a formal quantum group law. The author computes the first Lie algebra structure constants of this complete Hopf algebra. He shows that there exists no cogroup structure which has the complete Hopf algebra structure as associative formal group law. However, there does exist a cogroup structure which works ``locally'', i.e., for the sub-Hopf algebra generated by trees of a certain maximum number of vertices.
scientific article; zbMATH DE number 1966461