Algebraic structures associated to operads

Abstract

We study different algebraic structures associated to an operad and their relations: to any operad $\mathbf{P}$ is attached a bialgebra,the monoid of characters of this bialgebra, the underlying pre-Lie algebra and its enveloping algebra; all of them can be explicitely describedwith the help of the operadic composition. non-commutative versions are also given. We denote by $b\_\infty$ the operad of $b\_\infty$ algebras, describing all Hopf algebra structures on a symmetric coalgebra.If there exists an operad morphism from $b\_\infty$ to $\mathbf{P}$, a pair $(A,B)$ of cointeracting bialgebras is also constructed, that it to say:$B$ is a bialgebra, and $A$ is a graded Hopf algebra in the category of $B$-comodules. Most examples of such pairs (on oriented graphs, posets$\dots$) known in the literature are shown to be obtained from an operad; colored versions of these examples andother ones, based on Feynman graphs, are introduced and compared.

85 pages

References in corpus (7)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• Operads, homotopy algebra and iterated integrals for double loop spaces
• Homotopy G-algebras and moduli space operad
• Two interacting Hopf algebras of trees
• Pre-Lie algebras and the rooted trees operad
• Relating two Hopf algebras built from an operad
• Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

Cited by in corpus (5)

• Algebraic structures on typed decorated rooted trees
• Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations
• The incidence comodule bialgebra of the Baez-Dolan construction
• Translations of rough paths in combinatorial Hopf algebras
• Generalized prelie and permutative algebras