Averaging pre-Lie bialgebras and the related admissible classical Yang-Baxter equations

In this paper, we initiate the representation theory for averaging pre-Lie algebras, and establish the intrinsic equivalence among matched pairs, Manin triples, and bialgebra structures of averaging pre-Lie algebras under appropriate hypotheses. Furthermore, by introducing averaging operators on quadratic Rota-Baxter pre-Lie algebras, we obtain a canonical construction of averaging pre-Lie bialgebras. Moreover, we define admissible classical Yang-Baxter equations and relative Rota-Baxter operators in the setting of averaging pre-Lie algebras, and prove that relative Rota-Baxter operators yield symmetric solutions of the admissible classical Yang-Baxter equations and such symmetric solutions further give rise to a pre-Lie bialgebra structure. Finally, we reveal that every averaging pre-Lie bialgebra induces an averaging Lie bialgebra within our unified algebraic framework.