Post-Poisson algebras, extended $\mathcal{O}$-operators and extended Poisson Yang-Baxter equations

Abstract

This paper introduces the extended $\mathcal{O}$-operators on Poisson algebras as a natural generalization of ordinary $\mathcal{O}$-operators, together with the extended Poisson Yang-Baxter equations. We show that $\mathcal{O}$-operators of weight $λ$ on Poisson algebras give rise to post-Poisson algebras, whose operad are the trisuccessor of the operad of Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces.Equivalent characterizations of extended $\mathcal{O}$-operators are obtained via the symmetrizer-antisymmetrizer decomposition. The generalized Poisson Yang-Baxter equations are also introduced, and their connections with coboundary Poisson bialgebras and extended $\mathcal{O}$-operators are established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended Poisson Yang-Baxter equations, which generalizes the notion of the Poisson Yang-Baxter equations. Finally, the relationships among extended $\mathcal{O}$-operators, the extended Poisson Yang-Baxter equations, and the Poisson Yang-Baxter equations are studied in the framework of quadratic Poisson algebras and semi-direct product Poisson algebras.