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.

💡 Research Summary

This paper presents a systematic framework for constructing infinite-dimensional Poisson bialgebras through a process called “affinization” of pre-Poisson bialgebras. The core idea is to tensor a finite-dimensional pre-Poisson bialgebra with a specific infinite-dimensional algebra (a quadratic Z-graded perm algebra) to generate a new, infinite-dimensional bialgebra with a Poisson structure.

The work is built upon several key steps and results:

1. Algebraic Foundation: It first establishes that the tensor product of a pre-Poisson algebra and a perm algebra naturally carries a Poisson algebra structure. A crucial “reverse property” is shown: when using a special perm algebra (like the Laurent polynomial algebra), the Poisson structure on the tensor product uniquely characterizes the original pre-Poisson algebra.

2. Affinization of Zinbiel Bialgebras: As a pre-Poisson algebra contains a Zinbiel algebra structure, the paper independently develops the affinization for Zinbiel bialgebras. This involves defining a “completed” coalgebra structure to handle infinite dimensions properly. The authors prove that the tensor product of a Zinbiel bialgebra and a quadratic Z-graded perm algebra results in a completed infinitesimal bialgebra.

3. Main Theorem on Pre-Poisson Affinization: By combining the affinization of the Zinbiel part and the pre-Lie part (the latter relying on existing work), the paper achieves its main goal. Theorem 4.11 states that given a finite-dimensional pre-Poisson bialgebra (A, ∗, ◦, ϑ, θ) and a quadratic Z-graded perm algebra (B, ⋄, ω), one can define a completed Poisson bialgebra structure on the tensor product A⊗B. Importantly, when B is a specific algebra akin to Laurent polynomials, A⊗B being a completed Poisson bialgebra is equivalent to (A, ∗, ◦, ϑ, θ) being a pre-Poisson bialgebra. This establishes affinization as both a constructive and a characterizing tool.

4. Correspondence for Related Structures: The framework extends to related mathematical objects:

   • Yang-Baxter Equations: A symmetric solution ‘r’ to the Pre-Poisson Yang-Baxter Equation (PPYBE) in a pre-Poisson algebra can be “lifted” to a skew-symmetric solution ‘R’ to the Poisson Yang-Baxter Equation (PYBE) in the affinized infinite-dimensional Poisson algebra. Consequently, if the original pre-Poisson bialgebra is triangular (induced by ‘r’), the resulting Poisson bialgebra is also triangular (induced by ‘R’).
   • Quasi-Frobenius Structures: The affinization method is also applied to quasi-Frobenius pre-Poisson algebras, providing a construction path for quasi-Frobenius structures on the resulting infinite-dimensional Poisson algebras.

In summary, this research provides a unified and rigorous method for generating infinite-dimensional Poisson bialgebras from finite-dimensional pre-Poisson data. It deepens the understanding of the relationships between finite and infinite dimensions, Poisson and pre-Poisson algebras, and connects various structures like bialgebras, Yang-Baxter equations, and O-operators within this affinization paradigm. The results have potential implications for areas involving infinite-dimensional algebraic structures, such as integrable systems and quantum field theory.