Weighted infinitesimal unitary bialgebras on free monoid algebras
The concept of a weighted infinitesimal unitary bialgebra is an algebraic abstraction of the non-homogeneous associative classical Yang–Baxter equation. In this paper, we equip the free monoid algebra with a suitable coproduct which makes it a weighted infinitesimal unitary bialgebra. Furthermore, by exploring the relationship between weighted infinitesimal unitary bialgebras and pre-Lie algebras, we construct a pre-Lie algebra on free monoid algebras. Finally, we show that the weighted infinitesimal unitary bialgebra on free monoid algebras has an infinitesimal unitary Hopf algebra structure in the sense of Aguiar.