Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras

Abstract

The concept of weighted infinitesimal unitary bialgebra is an algebraic meaning of the nonhomogenous associative Yang–Baxter equation. In this paper, we equip the space of decorated planar rooted forests with a coproduct which makes it a weighted infinitesimal unitary bialgebra. Further, we construct an infinitesimal unitary Hopf algebra on decorated planar rooted forests in the sense of Loday and Ronco. We then introduce the concept of symmetric 1-cocycle condition, which is derived from the dual of the Hochschild cohomology. We study the universal properties of the space of decorated planar rooted forests with the symmetric 1-cocycle, leading to the notation of a weighted $$\Omega $$ -cocycle infinitesimal unitary bialgebra. As an application, we obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes–Kreimer Hopf algebra. Finally, we construct a pre-Lie algebra on decorated planar rooted forests.