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

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.

• The paper systematically constructs infinite-dimensional Poisson bialgebras by affinizing finite-dimensional pre-Poisson bialgebras using quadratic graded perm algebras.
• It demonstrates how symmetric PPYBE solutions lift to skew-symmetric structures in infinite dimensions through explicit tensor constructions and Yang-Baxter equations.
• The work establishes dualities between Zinbiel, pre-Lie, and perm operads, paving the way for applications in quantization and integrable systems.

Affinization of Zinbiel and Pre-Poisson Bialgebras: Construction of Infinite-Dimensional Poisson Bialgebras

Introduction and Problem Setting

This work systematically constructs infinite-dimensional Poisson bialgebras from the affinization of finite-dimensional pre-Poisson bialgebras, establishing both algebraic and coalgebraic lifts of the pre-Poisson structure. It unifies and extends several categorical correspondences: pre-Lie and perm operad duality, Zinbiel and (co)perm (co)algebra tensor product mechanisms, and relations between symmetric solutions of the pre-Poisson Yang-Baxter equation (PPYBE) and skew-symmetric solutions of the Poisson Yang-Baxter equation (PYBE). Central to the framework are concrete recipes for tensor-induced Poisson bialgebras via quadratic $-graded perm algebras, establishing dualities, explicit coalgebraic completions, and universal constructions. ## Algebraic Preliminaries and Definitions The foundation draws on classical structures: - **Zinbiel algebra:** A vector space $A$with a binary operation satisfying$a_1 \ast (a_2 \ast a_3) = (a_1 \ast a_2) \ast a_3 + (a_2 \ast a_1) \ast a_3$ and their dual algebra/coalgbera (the co-Zinbiel objects). - **Pre-Lie algebra:** Bilinear multiplication with the pre-Lie identity, underlying the subadjacent Lie algebra via commutator. - **Pre-Poisson algebra:** A vector space with compatible Zinbiel and pre-Lie operations. - **Perm algebra:** Bilinear product with the "perm" identity, and its completed graded variants with invariant, nondegenerate, antisymmetric bilinear forms. Pre-Poisson bialgebras are endowed simultaneously with Zinbiel bialgebra and pre-Lie bialgebra structures and compatibility via several cohomological identities. The work employs both completed and ordinary tensor products, with completion essential for infinite-dimensionality and algebraic control. ## Affinization of Zinbiel (Co)Bialgebras and Construction of Infinitesimal Bialgebras The principal algebraic construction realizes infinite-dimensional infinitesimal bialgebras from the tensor product $A \otimes B$, where$A$is a finite-dimensional Zinbiel bialgebra and$B$$is a quadratic graded perm algebra. - **Algebra structure:** Defining the product as$$(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = (a_1 \ast a_2) \otimes (b_1 \diamond b_2) + (a_2 \ast a_1) \otimes (b_2 \diamond b_1)$$yields a commutative associative structure if and only if$$A$is Zinbiel and$B$$is perm. - **Coalgebra structure:** Employing completed tensor products, the comultiplication$$\Delta$on$A \otimes B$is defined via the Zinbiel coproduct$\vartheta$and the perm coalgebra structure$\nu$$, both potentially infinite (completed) sums. - **Bialgebra compatibility:** Leveraging the invariance of the quadratic form on$$B$$, the induced structure satisfies infinitesimal bialgebra compatibility, i.e.,$$ \Delta((a_1\otimes b_1)(a_2\otimes b_2)) = (L_{a_2\otimes b_2} \hat{\otimes} \mathrm{id})\Delta(a_1\otimes b_1) + (\mathrm{id} \hat{\otimes} L_{a_1\otimes b_1})\Delta(a_2\otimes b_2) $$.

Yang-Baxter Equations, $\mathcal{O}$-Operators, and Triangular Structures

A highlight of the framework is the interplay between solutions of the (pre-)Poisson/Yang-Baxter equations in the finite-dimensional setting and their affinized, infinite-dimensional counterparts.

• Zinbiel Yang-Baxter equation (ZYBE): Symmetric solutions $r$ in $A\otimes A$ correspond to triangular (coboundary) Zinbiel bialgebra structures.
• Associative Yang-Baxter equation (AYBE): The paper shows how a symmetric ZYBE solution in $A$ lifts, via the affinization and dual basis induced by the perm quadratic form, to a skew-symmetric completed AYBE solution in the completed commutative associative algebra $A\otimes B$.
• Classical Yang-Baxter equation (CYBE): Similarly, the lifting propagates to the Lie algebra structure induced by the affinization.
• Triangular structures: The construction ensures that triangularity propagates; if the underlying finite-dimensional structure is triangular, so is its affinization.
• $\mathcal{O}$-operators: The correspondence is established between $\mathcal{O}$-operators on the finite-dimensional Zinbiel algebra and on the affinized algebra, explicitly via the tensor product of the operator and a duality-induced isomorphism.

Affinization of Pre-Poisson Bialgebras and Infinite-Dimensional Poisson Bialgebras

The core of the work details how, given a finite-dimensional pre-Poisson bialgebra $(A, \ast, \circ, \vartheta, \theta)$ and a quadratic graded perm algebra $$0, the affinized object $$1 acquires a natural infinite-dimensional graded Poisson bialgebra structure.

• Poisson algebra: Multiplication and Lie bracket are defined via tensor rules combining the Zinbiel and pre-Lie operations of $$2 with the perm structure on $$3.
• Coalgebra: The completed coalgebra structures combine the Zinbiel $$4 and pre-Lie $$5 via the perm coalgebra $$6, with precise compatibility ensuring the full pre-Poisson coalgebra structure.
• Bialgebra compatibility: The required bialgebra and Poisson compatibility conditions are checked in detail, showing the induced structure is a genuine Poisson bialgebra in the completed, infinite-dimensional setting.
• Characterization: Importantly, for special choices of $$7 (as in explicit examples), the existence of a Poisson bialgebra structure on $$8 is equivalent to the pre-Poisson bialgebra structure on $$9.

Explicit Solutions, Dualities, and Categorical Correspondences

• Yang-Baxter and operator lifting: For a symmetric solution $\mathcal{O}$0 of the PPYBE in $\mathcal{O}$1, it is shown that its affinized version $\mathcal{O}$2 is a skew-symmetric solution of the PYBE in the constructed Poisson algebra, providing a canonical procedure for constructing triangular infinite-dimensional Poisson bialgebras.
• $\mathcal{O}$3-operators: The notion of operator forms of the Yang-Baxter equations and their lifting—$\mathcal{O}$4-operators—are handled by explicit correspondences, ensuring that classical operator interpretations persist under affinization.
• Quasi-Frobenius structures: The construction is shown to preserve and reflect quasi-Frobenius structures and Connes cocycles, both in the associative and Poisson settings, via the induced bilinear forms.

Examples and Demonstration

A suite of explicit, low-dimensional examples for two-dimensional Zinbiel and pre-Poisson (bi)algebras is provided. These illustrate:

• Construction of the affinized algebra and coalgebra structures,
• Explicit calculation of the Yang-Baxter solutions and corresponding bialgebra maps,
• Verification of triangular Poisson bialgebra structures,
• The explicit form of induced $\mathcal{O}$5-operators and corresponding duality maps.

Theoretical and Practical Implications

The constructed infinite-dimensional Poisson bialgebras are parametrized by quadratic graded perm algebras, linking their structure theory to duality results in operad theory (Koszul duality between pre-Lie and perm, etc.). The tensor construction approach brings a new perspective for building rich infinite-dimensional algebraic systems from finite ingredients, with direct control over compatibility identities and cohomological properties.

Furthermore, the lifting of Yang-Baxter solutions and $\mathcal{O}$6-operators through explicit affinization schemes provides a pathway for generating and classifying infinite-dimensional triangular (coboundary) Poisson and pre-Poisson bialgebras, potentially feeding into the broader program of explicit quantization, moduli of solutions, and the study of algebraic structures in classical and quantum integrable systems.

Conclusion

The paper presents a coherent and rigorous algebraic machinery for constructing infinite-dimensional Poisson bialgebras by affinizing finite-dimensional pre-Poisson bialgebras via quadratic graded perm algebras. The results unify several operadic duality relationships, provide concrete realization and characterization theorems, and establish a canonical passage from symmetric Yang-Baxter solutions in the finite-dimensional context (Zinbiel, pre-Poisson) to skew-symmetric solutions in the infinite-dimensional Poisson setup. The theory is well-posed for further applications in deformation and quantization problems, as well as in the explicit algebraic modeling of infinite-dimensional Hamiltonian and integrable systems.