Weighted infinitesimal bialgebras

As an algebraic meaning of the nonhomogenous associative Yang-Baxter equation, weighted infinitesimal bialgebras play an important role in mathematics and mathematical physics. In this paper, we introduce the concept of weighted infinitesimal Hopf modules and show that any module carries a natural structure of weighted infinitesimal unitary Hopf module over a weighted quasitriangular infinitesimal unitary bialgebra. We decorate planar rooted forests in a new way, and prove that the space of rooted forests, together with a coproduct and a family of grafting operations, is the free $\Omega$-cocycle infinitesimal unitary bialgebra of weight zero on a set. A combinatorial description of the coproduct is given. As applications, we obtain the initial object in the category of cocycle infinitesimal unitary bialgebras on undecorated planar rooted forests, which is the object studied in the (noncommutative) Connes-Kreimer Hopf algebra. Finally, we derive two pre-Lie algebras from an arbitrary weighted infinitesimal bialgebra and weighted commutative infinitesimal bialgebra, respectively. The second construction generalizes the Gel'fand-Dorfman Theorem on Novikov algebras.