Affinization of Zinbiel bialgebras and pre-Poisson bialgebras, infinite-dimensional Poisson bialgebras
The purpose of this paper is to construct infinite-dimensional Poisson bialgebras by the affinization of pre-Poisson algebras. There is a natural Poisson algebra structure on the tensor product of a pre-Poisson algebra and a perm algebra, and the Poisson algebra structure on the tensor product of a pre-Poisson algebra and a special perm algebra characterizes the pre-Poisson algebra. We extend such correspondences to the context of bialgebras, that is, there is a Poisson bialgebra structure on the tensor product of a pre-Poisson bialgebra and a quadratic $\bz$-graded perm algebra.In this process, we provide the affinization of Zinbiel bialgebras, and give a correspondence between symmetric solutions of the Yang-Baxter equation in pre-Poisson algebras and certain skew-symmetric solutions of the Yang-Baxter equation in the induced infinite-dimensional Poisson algebras. The similar correspondences for the related triangular bialgebra structures and $\mathcal{O}$-operators are given.