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Archiv der Mathematik

, Volume 80, Issue 4, pp 368–383 | Cite as

Comparison of Hopf algebras on trees

  • R. Holtkamp
Original paper

Abstract.

Several Hopf algebra structures on vector spaces of trees can be found in the literature (cf. [10], [8], [2]). In this paper, we compare the corresponding notions of trees, the multiplications and comultiplications. The Hopf algebras are connected graded or, equivalently, complete Hopf algebras. The Hopf algebra structure on planar binary trees introduced by Loday and Ronco [10] is noncommutative and not cocommutative. We show that this Hopf algebra is isomorphic to the noncommutative version of the Hopf algebra of Connes and Kreimer [3]. We compute its first Lie algebra structure constants in the sense of [7], and show that there is no cogroup structure compatible with the Hopf algebra on planar binary trees.

Mathematics Subject Classification (2000): 16W30, 16W60, 17A50, 18D35. 

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Copyright information

© Birkhäuser Verlag Basel, 2003

Authors and Affiliations

  • R. Holtkamp
    • 1
  1. 1.Ruhr-Universität Bochum, Fakultät für Mathematik, NA2/32, D-44801 Bochum, Germany,¶e-mail: ralf.holtkamp@ruhr-uni-bochum.deDE

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