Realizations of Hopf algebras of graphs by alphabets

Abstract

We here give polynomial realizations of various Hopf algebras or bialgebras on Feynman graphs, graphs, posets or quasi-posets, that it to say injections of these objects into polynomial algebras generated by an alphabet. The alphabet here considered are totally quasi-ordered. The coproducts are given by doubling the alphabets; a second coproduct is defined by squaring the alphabets, and we obtain cointeracting bialgebras in the commutative case.

Nonsymmetric Operads

“…Property (iii) is known as the alphabet doubling trick. This property, enjoyed by polynomial realizations of a large number of Hopf algebras, allows us to rephrase their coproduct by such alphabet transformations [3,17,12,8,7,6].…”

“…A polynomial realization (abbreviated as PR) of a Hopf algebra consists of interpreting its elements as polynomials, either commutative or not, in such a way that its product translates as polynomial multiplication and the coproduct translates as a simple transformation of the alphabet of variables. A great portion of combinatorial Hopf algebras appearing in combinatorics admit PRs [3,17,7,6]. It is striking to note that Hopf algebras involving a variety of different families of combinatorial objects and operations on them can be translated and understood in a common manner through adequate PRs.…”

“…The particularity of our approach is that we consider a PR based on variables belonging to alphabets endowed with several unary and binary relations in order to capture the particularities of the coproduct of N • O. This approach, using what we call related alphabets, generalizes the previous approaches using totally ordered alphabets [3,17,12], quasi-ordered alphabets [6], or alphabets endowed with a single binary relation [7,8].…”