A weighted infinitesimal unitary bialgebra is a module \(A\), which is simultaneously an algebra (possibly without a unit) and a coalgebra (possibly without a counit) such that the coproduct \(\Delta\) is a weighted derivation of \(A\): \[ \Delta(ab)=a\Delta(b)+\Delta(a)b+\lambda a\otimes b,\quad \lambda\in k \] Take an associative classical Young-Baxter equation \[ r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}=\lambda r_{13} \] Then any solution of this equarion in an algebra \(A\) endows \(A\) with an infinitesimal unitary bialgebra of weight zero. If one takes an nonhomogenous associative classical Yang-Baxter equation \[ r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}=\lambda r_{13} \] then the coproduct \[ \Delta_r(a)=a\dot r-r\dot a-\lambda (a\otimes 1) \] is weighter derivation [O. Ogievetsky and T. Popov, Adv. Theor. Math. Phys. 14, No. 2, 439–505 (2010; Zbl 1208.81118)].
Also such algebras appear naturally in combinatorics [S. A. Joni and G. C. Rota, Stud. Appl. Math. 61, 93–139 (1979; Zbl 0471.05020)]. The authors “introduce the concept of symmetric \(1\)-cocycle condition, which is derived from the dual of the Hochschild cohomology. Also they 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, they 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, they construct a pre-Lie algebra on decorated planar rooted forests”.