Structure of BiHom-pre-Poisson algebras

In the current research paper, we define and investigate the structure of a BiHom-pre-Poisson algebra. This algebraic structure is defined by two products "Λ", "◊" and two linear maps f, g on A. In particular, (A, Λ, f, g) is a BiHom-Zinbiel algebra and (A, ◊, f, g) is a BiHom-pre-Lie algebra. Additionally two compatibility conditions between Λ and ◊ are verified. Our first main results are devoted to demonstrating that if A is a BiHom-pre-Lie algebra, then a tensorial algebra of A has a structure of a BiHom-pre-Poisson algebra. Furthermore, we prove that any BiHom-Poisson algebra together with a Rota–Baxter operator defines a BiHom-pre-Poisson algebra. Finally, we define the structure of a dual BiHom-pre-Poisson algebra and we demonstrate that an averaging operator on a BiHom-Poisson algebra gives rise to a dual BiHom-pre-Poisson algebra.