Comparison of Hopf algebras on trees.
type: Item
instance of: scholarly article
title: Comparison of Hopf algebras on trees.
zbMATH Open document ID: 1056.16030
author: Ralf Holtkamp
published in: Archiv der Mathematik
publication date: 2003-08-18T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81T17
Mathematics Subject Classification ID: 81T18
Mathematics Subject Classification ID: 81V10
zbMATH DE Number: 1966461
zbMATH Keywords: Hopf algebras
zbMATH Keywords: planar binary trees
zbMATH Keywords: Feynman diagrams
zbMATH Keywords: renormalization
zbMATH Keywords: formal quantum group laws
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: s00013-003-0796-y
OpenAlex ID: W1571192943
DOI: 10.1007/S00013-003-0796-Y
description: scientific article; zbMATH DE number 1966461