Averaging Zinbiel bialgebras

In this paper, we introduce and study averaging Zinbiel bialgebras. We first define representations of averaging Zinbiel algebras and prove a semi-direct product characterization. We then introduce matched pairs and Manin triples of averaging Zinbiel algebras, and characterize averaging Zinbiel bialgebras by these two structures. We show that every averaging Zinbiel algebra induces a pre-Lie algebra, and that every averaging Zinbiel bialgebra induces a pre-Lie bialgebra. We also study coboundary averaging Zinbiel bialgebras and introduce the \(S\) admissible Zinbiel Yang-Baxter equation. Its symmetric solutions give rise to averaging Zinbiel bialgebras. Finally, we investigate averaging operators on quadratic Rota-Baxter Zinbiel algebras and relative Rota-Baxter operators on averaging Zinbiel algebras.