Generalized Connes-Kreimer Hopf algebras on decorated rooted forests by weighted cocycles

Abstract

The Connes-Kreimer Hopf algebra of rooted trees is an operated Hopf algebra whose coproduct satisfies the classical Hochschild 1-cocycle condition. In this paper, we extend the setting from rooted trees to the space $H_{\rm RT}(X,\Omega)$ of $(X,\Omega)$-rooted trees, in which internal vertices are decorated by a set $\Omega$ and leafs are decorated by $X \cup \Omega$. We introduce a new coalgebra structure on $H_{\rm RT}(X,\Omega)$ whose coproduct satisfies a weighted Hochschild 1-cocycle condition involving multiple operators, thereby generalizing the classical condition. A combinatorial interpretation of this coproduct is also provided. We then endow $H_{\rm RT}(X,\Omega)$ with a Hopf algebra structure. Finally, we define weighted $\Omega$-cocycle Hopf algebras, characterized by a Hochschild 1-cocycle condition with weights, and show that $H_{\rm RT}(X,\Omega)$ is the free object in the category of $\Omega$-cocycle Hopf algebras.