Post-Poisson algebras and Poisson Yang-Baxter equation via bimodule algebra deformations

A fundamental construction of Poisson algebras is as the quasiclassical limits (QCLs) of associative algebra deformations of commutative associative algebras. This paper extends this construction to the relative context with the notion of (bi)module algebras over another algebra for a given algebraic structure. In this language, a module Poisson algebras can be realized as the QCLs of bimodule associative deformations of module commutative associative algebras. Moreover, the notion of the scalar deformation of an $\mathcal O$-operator is introduced so that the process of bimodule algebras deformations to QCLs is endowed with $\mathcal O$-operators in a consistent manner. As an explicit illustration of this process, post-Poisson algebras are realized as the QCLs of bimodule associative deformations of module commutative associative algebras with the identity maps as $\mathcal O$-operators, recovering the known fact that post-Poisson algebras are the QCLs of tridendriform algebra deformations of commutative tridendriform algebras. Furthermore, the notion of scalar deformations of solutions of the associative Yang-Baxter equation (AYBE) is applied to realize solutions of the Poisson Yang-Baxter equation (PYBE) in Poisson algebras as solutions of the AYBE in commutative associative algebras, giving a YBE version of Poisson algebras as the QCLs of associative deformations of the commutative associative algebras. Finally, concrete solutions of the PYBE are obtained from the aforementioned tridendriform deformation-to-QCLs process.