Cocommutative Hopf Algebras of Permutations and Trees

By Marcelo Aguiar, Frank Sottile. 2004

47 citations recorded by the source corpus.

Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.

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

View Paper: https://www.semanticscholar.org/paper/acbc9fa25562a8ce5e6e88c65ccaa561d434ecc3

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Citations in the corpus

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”
On products and duality of binary, quadratic, regular operads (2004, background): “In the next few years, their properties were studied by several authors in areas related to operads [35], homology [17, 18], Hopf algebras [7, 26, 42, 39], combinatorics [16, 37, 2, 3], arithmetic [34] and quantum field theory [16].”
Commutative combinatorial Hopf algebras (2006, background): “We show that the algebras based on permutations and planar binary trees are isomorphic (in a nontrivial way) to those of [1] and study some generalizations such as endofunctions, parking functions, set partitions, trees, forests, and so on.”
Monoidal Functors, Species, and Hopf Algebras (2010, methodology): “N. Bergeron and F. Sottile, Hopf algebras and edge-labeled posets, J. Algebra 216 (1999), no. 2, 641–651.”