2017/02/17 by Loïc Foissy, Foissy, Loïc · 3 citations
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1702.05344
openalex publication_date 2017/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study different algebraic structures associated to an operad and their relations: to any operad 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_∞ the operad of b_∞ algebras, describing all Hopf algebra structures on a symmetric coalgebra.If there exists an operad morphism from b_∞ to 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…) 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.