Properties, deformation quantizations and bialgebras for dual pre-Poisson algebras

A dual pre-Poisson algebra is an algebraic structure that integrates a permutative algebra and a Leibniz algebra under certain compatibility conditions. As the Koszul dual notion of the pre-Poisson algebra, this structure serves as a natural generalization of the Poisson algebra. In this paper, we commence a study on dual pre-Poisson algebras from the algebraic point of view and establish a bialgebra theory for dual pre-Poisson algebras. We begin by investigating the fundamental properties of dual pre-Poisson algebras and provide several explicit constructions. In particular, we prove that the operad of dual pre-Poisson algebras is Koszul, and we compute its Hilbert-Poincar\'e series and codimension. Furthermore, we introduce the notion of diassociative formal deformations of permutative algebras and show that dual pre-Poisson algebras are the corresponding semi-classical limits. Moreover, we introduce dual pre-Poisson bialgebras, which are characterized both by Manin triples and by matched pairs of dual pre-Poisson algebras, thus developing a bialgebra theory for dual pre-Poisson algebras. Then our study leads to the permutative-Leibniz Yang-Baxter equation (PLYBE) that is composed of the permutative and the classical Leibniz Yang-Baxter equation. We conclude by introducing $\mathcal{O}$-operators and pre-dual pre-Poisson algebras, which provide a systematic method for constructing symmetric solutions to the PLYBE and, consequently, dual pre-Poisson bialgebras.