Algebraic structures associated to operads (original) (raw)

We study different algebraic structures associated to an operad and their relations: to any operad mathbfP\mathbf{P}mathbfP 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 mathbfbinfty\mathbf{b\_\infty}mathbfbinfty the operad of mathbfbinfty\mathbf{b\_\infty}mathbfbinfty algebras, describing all Hopf algebra structures on a symmetric coalgebra.If there exists an operad morphism from mathbfbinfty\mathbf{b\_\infty}mathbfbinfty to mathbfP\mathbf{P}mathbfP, a pair (A,B)(A,B)(A,B) of cointeracting bialgebras is also constructed, that it to say:$B$ is a bialgebra, and AAA is a graded Hopf algebra in the category of BBB-comodules. Most examples of such pairs (on oriented graphs, posets$\ldots$) 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.