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Combinatorics of rooted trees and Hopf algebras

Author: Michael E. Hoffman
Journal: Trans. Amer. Math. Soc. 355 (2003), 3795-3811
MSC (2000): Primary 05C05, 16W30; Secondary 81T15
Published electronically: May 15, 2003
MathSciNet review: 1990174
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Abstract: We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities.

Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

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Additional Information

Michael E. Hoffman
Affiliation: Department of Mathematics, United States Naval Academy, Annapolis, Maryland 21402

Keywords: Rooted tree, Hopf algebra, differential poset
Received by editor(s): June 25, 2002
Received by editor(s) in revised form: February 24, 2003
Published electronically: May 15, 2003
Additional Notes: The author was partially supported by a grant from the Naval Academy Research Council
Some of the results of this paper were presented to an AMS Special Session on Combinatorial Hopf Algebras on May 4, 2002.
Article copyright: © Copyright 2003 American Mathematical Society

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