The Hopf algebras of decorated rooted trees. I (Q1599930)

The purpose of this paper is to introduce and study the Hopf algebra \({\mathcal H}^{\mathcal D}_{P,R}\) of decorated planar binary trees, a noncommutative version of the Hopf algebra \({\mathcal H}^{\mathcal D}_R\) studied by \textit{A. Connes} and \textit{D. Kreimer} [Commun. Math. Phys. 199, No. 1, 203-242 (1998; Zbl 0932.16038)]. The base field \(K\) has characteristic 0. As an algebra, \({\mathcal H}^{\mathcal D}_{P,R}\) is the free algebra on the set \({\mathcal T}^{\mathcal D}_{P,R}\) of planar rooted trees decorated by a set \(\mathcal D\). Thus, a basis of \({\mathcal H}^{\mathcal D}_{P,R}\) is the set \(\mathcal F\) of forests of planar rooted trees decorated by \(\mathcal D\). The comultiplication is defined recursively in terms of cuts of trees. It is then shown that \({\mathcal H}^{\mathcal D}_{P,R}\) is universal with respect to a property expressed in terms of Hochschild cohomology of coalgebras. Next, it is proved that \({\mathcal H}^{\mathcal D}_{P,R}\) is isomorphic to its own graded dual; as consequences, first a Hopf algebra duality \((\mid)\colon{\mathcal H}^{\mathcal D}_{P,R}\times{\mathcal H}^{\mathcal D}_{P,R}\to K\), and second, a coalgebra isomorphism from \({\mathcal H}^{\mathcal D}_{P,R}\) to the tensor coalgebra on the planar rooted trees decorated by \(\mathcal D\), are obtained. Let \((e_F)_{F\in{\mathcal F}}\) be the basis in \({\mathcal H}^{\mathcal D}_{P,R}\) dual to \(\mathcal F\) with respect to \((\mid)\); it is shown that \((e_F)_{F\in{\mathcal T}^{\mathcal D}_{P,R}}\) is the basis of the space of primitive elements of \({\mathcal H}^{\mathcal D}_{P,R}\). Let \({\mathcal A}^{\mathcal D}_{P,R}\) be the \(\mathbb{Z}\)-subalgebra of \({\mathcal H}^{\mathcal D}_{P,R}\) generated by \(\mathcal F\); it is a \(\mathbb{Z}\)-form of \({\mathcal H}^{\mathcal D}_{P,R}\) and \((e_F)_{F\in{\mathcal F}}\) is actually another \(\mathbb{Z}\)-basis of \({\mathcal A}^{\mathcal D}_{P,R}\). Among other results, the Hopf algebra endomorphisms of \({\mathcal H}^{\mathcal D}_{P,R}\) are determined. Finally, it is shown that there is a Hopf algebra epimorphism from \({\mathcal H}^{\mathcal D}_{P,R}\) to \({\mathcal H}^{\mathcal D}_R\), and the space of primitive elements of \({\mathcal H}^{\mathcal D}_R\) is then computed. See also the following review of Part II [\textit{L. Foissy}, Bull. Sci. Math. 126, No. 4, 249-288 (2002; Zbl 1013.16027)].