Solving the Poisson Yang-Baxter equation via deformation-to-quasiclassical-limits

A fundamental construction of Poisson algebras is to derive them as the quasiclassical limits (QCLs) of associative algebra deformations of commutative associative algebras. This paper lifts this process to the level of classical Yang-Baxter type equations. Solutions of the Poisson Yang-Baxter equation (PYBE) in Poisson algebras is obtained by scalarly deforming solutions of the associative Yang-Baxter equation (AYBE) in commutative associative algebras. Inspired by the characterization of solutions of various classical Yang-Baxter type equations by $\mathcal{O}$-operators, we introduce the notions of deformations of (bi)module algebras and scalar deformations of the corresponding $\mathcal{O}$-operators, from which the QCLs give $\mathcal{O}$-operators for Poisson algebras, which in turn provide solutions of the PYBE. Furthermore, as the QCLs of tridendriform algebra deformations of commutative tridendriform algebras, post-Poisson algebras produce deformations-QCLs of $\mathcal{O}$-operators for Poisson algebras, thus offering explicit solutions of the PYBE.