Structure of the Loday–Ronco Hopf algebra of trees

By Marcelo Aguiar, Frank Sottile. 2004

87 citations recorded by the source corpus.

Abstract Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto–Reutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopf-algebraic structure. In the dual basis for the graded dual Hopf algebra, our formula for the coproduct gives an explicit isomorphism with a free associative algebra. We also obtain a transparent proof of its isomorphism with the non-commutative Connes–Kreimer Hopf algebra of Foissy, and show that this algebra is related to non-commutative symmetric functions as the (commutative) Connes–Kreimer Hopf algebra is related to symmetric functions.

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

View Paper: https://www.semanticscholar.org/paper/75a01747068b499ae6ec7f22236a73c5d0027ac4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Polynomial realizations of some trialgebras (2006, background): “As in the case of the permutohedron, one can take sums of Mu, over all the elements upper or lower than u in the lattice, or restricted to elements belonging to the same “class” as u (see [5, 1] for examples of such bases).”
Generalized bialgebras and triples of operads (2006, background): “So we have δj(Id− ω([2]))(x) = δj(x) − δj(x) + ∑ δi ◦ higher terms(x).”
Rota–Baxter algebras and dendriform algebras (2005, methodology): “in 1995 with motivation from algebraic K-theory, and was further studied in connection with several areas in mathematics and physics, including operads [54], homology [34, 35], Hopf algebras [13, 47, 58, 65, 32], Lie and Leibniz algebras [35], combinatorics [6, 7, 33, 57], arithmetic [55] and quantu”