Affinization of Zinbiel bialgebras and pre-Poisson bialgebras, infinite-dimensional Poisson bialgebras

TL;DR

This paper constructs infinite-dimensional Poisson bialgebras via affinization of pre-Poisson algebras, establishing new algebraic correspondences and extending Zinbiel bialgebra concepts to infinite dimensions.

Contribution

It introduces the affinization process for Zinbiel bialgebras and establishes novel links between solutions of the Yang-Baxter equation in pre-Poisson and Poisson bialgebras.

Findings

Constructed infinite-dimensional Poisson bialgebras from pre-Poisson algebras.

Established correspondence between Yang-Baxter solutions in pre-Poisson and Poisson algebras.

Extended Zinbiel bialgebra concepts to infinite-dimensional settings.

Abstract

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.