Combinatorial Hopf algebras

By Jean-Louis Loday, Maria O. Ronco. 2008

99 citations recorded by the source corpus.

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). We show that the choice of such an isomorphism implies the existence of a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them.

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

View Paper: https://www.semanticscholar.org/paper/f1b5a159924983125050ac84c8090853f9f1a050

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Citations in the corpus

Recent Trends in Quasisymmetric Functions (2018, background): “See Hivert [66] for an analogue in NSym and its QSym companion, and see Novelli, Thibon, and Williams [97] for a different noncommutative analogue.”
A Lift of the Schur and Hall–Littlewood Bases to Non-commutative Symmetric Functions (2012, background): “It should be noted that many different versions of a non-commutative Hall–Littlewood symmetric function already exist [5, 16, 23, 28, 35], but as far as we are aware, none of these project for all partitions to the classical Hall–Littlewood functions (ours do!).”
Generalized bialgebras and triples of operads (2006, background): “C A P ≬ reference Com Com V ect Hopf [6] Com Parastat NLie Hopf [50] Com As Lie Hopf [12, 64] Com ?? As Hopf+? conjectural Com SGV PreLie Hopf+ ad hoc [55] Com ComAs SMB Hopf+ad hoc [55] Com Mag Sabinin Hopf [69, 35] Com Dipt PrimComDipt Hopf see 4.”