Combinatorial Hopf algebras

Abstract

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). The choice of such an isomorphism implies the existence 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.

The second part, dealing with right-sided combinatorial Hopf algebras, has been completely modified in this new version

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Cited by in corpus (14)

• Hopf Algebras in Combinatorics
• On post-Lie algebras, Lie--Butcher series and moving frames
• Hopf Algebras and Markov Chains
• Shuffle algebras, homology, and consecutive pattern avoidance
• A group-theoretical approach to conditionally free cumulants
• Hopf algebra techniques to handle dynamical systems and numerical integrators
• Leavitt path algebras, $B_{\infty }$-algebras and Keller's conjecture for singular Hochschild cohomology
• A forest formula for the antipode in incidence Hopf algebras
• Loop of formal diffeomorphisms and F{à}a di Bruno coloop bialgebra
• On the Hopf algebra of multi-complexes
• Doubling pre-Lie algebra of rooted trees
• On the homology of the double cobar construction of a double suspension
• Left ideals in an enveloping algebra, prelie products and applications to simple complex Lie algebras
• Riemann-Hilbert Problem and Quantum Field Theory: Integrable Renormalization, Dyson-Schwinger Equations