Combinatorics of rooted trees and Hopf algebras

By Michael E. Hoffman. 2002

148 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3b93d46badc8d7cc453908ab737289310581737a

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Citations in the corpus

Ramification of rough paths (2006, background): “Literature on combinatorial and algebraic properties of rooted trees is quite large, we prefer to single out the work of Hoffman [22] and the two papers of Foissy [14, 13] on labeled rooted trees.”
Structure of the Loday–Ronco Hopf algebra of trees (2004, background): “by Hoffman [ 18 ], this is the (cocommutative) Hopf algebra of rooted trees constructed by Grossman and Larson in [14].”
Combinatorial Hopf algebras (2008, unknown): “[49, 26]).”
Cocommutative Hopf Algebras of Permutations and Trees (2004, background): “Recent independent work of Foissy [6, 7] and Hoffman [14] showed that the Hopf algebra of Connes-Kreimer is dual to the Hopf algebra of rooted trees of GrossmanLarson.”
On the Lie envelopping algebra of a pre-Lie algebra (2004, result): “This gives another proof of the duality between the Connes-Kreimer and Grossman-Larson Hopf algebras [6] [19] [ 12 ].”
Finite dimensional comodules over the Hopf algebra of rooted trees (2001, background): “It is shown in [8, 12] that U(L(1)) and the Grossman-Larson Hopf algebra on rooted trees are isomorphic (see [6, 7]).”