Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the concept of $\mathrm{\Omega }$-cocycle infinitesimal bialgebras of weight $\lambda$ and then prove that the space of decorated planar rooted forests $H_{\mathrm{R}\mathrm{T}}(X,\mathrm{\Omega })$, together with a set of grafting operations $\{B_{\omega }^{+}\mid \omega \in \mathrm{\Omega }\}$, is the free $\mathrm{\Omega }$-cocycle infinitesimal unitary bialgebra of weight $\lambda$ on a set $X$, involving a weighted version of a Hochschild 1-cocycle condition. As an application, we equip a free cocycle infinitesimal unitary bialgebraic structure on the undecorated planar rooted forests, which is the object studied in the well-known (noncommutative) Connes-Kreimer Hopf algebra. Finally, we construct a new pre-Lie algebraic structure on decorated planar rooted forests.