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Zinbiel bialgebras, relative Rota-Baxter operators and the related Yang-Baxter Equation

You Wang

TL;DR

The paper develops a comprehensive bialgebra framework for Zinbiel algebras, introducing matched pairs, Manin triples, and coboundary constructions that lead to a Zinbiel Yang-Baxter equation $\llbracket r,r\rrbracket=0$. It then defines quasi-triangular and factorizable Zinbiel bialgebras, showing how quasi-triangular structures yield relative Rota-Baxter operators of weight $-1$ on the coregular representation and how factorizability enables a canonical factorization of the underlying Zinbiel algebra. A central contribution is the Rota-Baxter characterization of factorizable Zinbiel bialgebras, proving a one-to-one correspondence with quadratic Rota-Baxter Zinbiel algebras of nonzero weight (and with Rota-Baxter commutative algebras equipped with Connes cocycles). The results unify bialgebra, Yang-Baxter, and Rota-Baxter perspectives in the Zinbiel setting, with the Zinbiel double providing a natural and factorizable extension, and establish explicit equivalences and constructions that may impact integrable systems and operad theory. The work thus extends classical Lie bialgebra concepts to Zinbiel structures, enriching the algebraic toolbox for dual Leibniz-type theories.

Abstract

In this paper, we first introduce the notion of a Zinbiel bialgebra and show that Zinbiel bialgebras, matched pairs of Zinbiel algebras and Manin triples of Zinbiel algebras are equivalent. Then we study the coboundary Zinbiel bialgebras, which leads to an analogue of the classical Yang-Baxter equation. Moreover, we introduce the notions of quasi-triangular and factorizable Zinbiel bialgebras as special cases. A quasi-triangular Zinbiel bialgebra can give rise to a relative Rota-Baxter operator of weight $-1$. A factorizable Zinbiel bialgebra can give a factorization of the underlying Zinbiel algebra. As an example, we define the Zinbiel double of a Zinbiel bialgebra, which enjoys a natural factorizable Zinbiel bialgebra structure. Finally, we introduce the notion of quadratic Rota-Baxter Zinbiel algebras, as the Rota-Baxter characterization of factorizable Zinbiel bialgebras. We show that there is a one-to-one correspondence between quadratic Rota-Baxter Zinbiel algebras and factorizable Zinbiel bialgebras.

Zinbiel bialgebras, relative Rota-Baxter operators and the related Yang-Baxter Equation

TL;DR

The paper develops a comprehensive bialgebra framework for Zinbiel algebras, introducing matched pairs, Manin triples, and coboundary constructions that lead to a Zinbiel Yang-Baxter equation . It then defines quasi-triangular and factorizable Zinbiel bialgebras, showing how quasi-triangular structures yield relative Rota-Baxter operators of weight on the coregular representation and how factorizability enables a canonical factorization of the underlying Zinbiel algebra. A central contribution is the Rota-Baxter characterization of factorizable Zinbiel bialgebras, proving a one-to-one correspondence with quadratic Rota-Baxter Zinbiel algebras of nonzero weight (and with Rota-Baxter commutative algebras equipped with Connes cocycles). The results unify bialgebra, Yang-Baxter, and Rota-Baxter perspectives in the Zinbiel setting, with the Zinbiel double providing a natural and factorizable extension, and establish explicit equivalences and constructions that may impact integrable systems and operad theory. The work thus extends classical Lie bialgebra concepts to Zinbiel structures, enriching the algebraic toolbox for dual Leibniz-type theories.

Abstract

In this paper, we first introduce the notion of a Zinbiel bialgebra and show that Zinbiel bialgebras, matched pairs of Zinbiel algebras and Manin triples of Zinbiel algebras are equivalent. Then we study the coboundary Zinbiel bialgebras, which leads to an analogue of the classical Yang-Baxter equation. Moreover, we introduce the notions of quasi-triangular and factorizable Zinbiel bialgebras as special cases. A quasi-triangular Zinbiel bialgebra can give rise to a relative Rota-Baxter operator of weight . A factorizable Zinbiel bialgebra can give a factorization of the underlying Zinbiel algebra. As an example, we define the Zinbiel double of a Zinbiel bialgebra, which enjoys a natural factorizable Zinbiel bialgebra structure. Finally, we introduce the notion of quadratic Rota-Baxter Zinbiel algebras, as the Rota-Baxter characterization of factorizable Zinbiel bialgebras. We show that there is a one-to-one correspondence between quadratic Rota-Baxter Zinbiel algebras and factorizable Zinbiel bialgebras.

Paper Structure

This paper contains 16 sections, 29 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Zinbiel algebras
  3. Zinbiel bialgebras
  4. Rota-Baxter characterization of factorizable Zinbiel bialgebras
  5. Outline of the paper
  6. Quadratic Zinbiel algebras and Zinbiel bialgebras
  7. Preliminaries and basic results
  8. Matched pairs, Manin triples of Zinbiel algebras and Zinbiel bialgebras
  9. Coboundary Zinbiel bialgebras and the classical Zinbiel Yang-Baxter equation
  10. Quasi-triangular and factorizable Zinbiel bialgebras
  11. Quasi-triangular Zinbiel bialgebras and relative Rota-Baxter operators
  12. Factorizable Zinbiel bialgebras
  13. Zinbiel double
  14. Rota-Baxter characterization of factorizable Zinbiel bialgebras
  15. Quadractic Rota-Baxter Zinbiel algebras
  16. ...and 1 more sections

Key Result

Proposition 2.2

Every Zinbiel algebra $(A,\circ_A)$ satisfies the equation:

Theorems & Definitions (82)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 72 more