Extended Rota-Baxter Leibniz bialgebras and Leibniz Yang-Baxter equations

In this paper, we first introduce the notions of extended Leibniz-dendriform algebras and extended post-Leibniz algebras, and investigate their intrinsic relationship with extended Rota-Baxter Leibniz algebras. Next, we propose the concepts of extended Rota-Baxter Leibniz bialgebras and admissible quadruples of extended Rota-Baxter Leibniz algebras. Furthermore, we define the Manin triple and the matched pair of extended Rota-Baxter Leibniz algebras. We then prove that extended Rota-Baxter Leibniz bialgebras can be equivalently characterized by matched pairs and Manin triples of extended Rota-Baxter Leibniz algebras. Finally, we construct several classes of extended Rota-Baxter Leibniz bialgebras via the admissible classical Leibniz Yang-Baxter equation (cLYBe) and 𝒪-operators.