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.

加权无穷小unit双代数的概念是非齐次联想Yang–Baxter方程的代数含义。在本文中，我们为装饰性的平面根系森林的空间配备了一个乘积，使其成为加权的无穷小unit双代数。此外，我们在Loday和Ronco的意义上，在装饰的平面根系森林上构造了一个无穷小的unit霍夫代数。然后，我们介绍对称的1-cocycle条件的概念，该概念是从Hochschild谐函数的对偶派生而来的。我们研究了带有对称的1-cocycle的装饰性平面生根森林空间的普遍性质，从而得出加权\（\ Omega \）的表示法-cocycle无穷小ary双代数。作为一种应用，我们获得了未装饰的平面根森林上的自由循环无穷小unit双代数类别中的初始对象，这是著名的非交换Connes–Kreimer Hopf代数中研究的对象。最后，我们在经过装饰的平面根系森林上构建了一个前李代数。