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Nijenhuis operators and mock-Lie bialgebras

Tianshui Ma, Sami Mabrouk, Abdenacer Makhlouf, Feiyan Song

TL;DR

The paper develops a comprehensive framework for Nijenhuis operators on mock-Lie algebras, extending to mock-Lie bialgebras and establishing a parallel theory of deformations, abelian extensions, and cohomology. It introduces Nijenhuis mock-Lie algebras and connects them to bialgebra structures through symplectic, dual-quasitriangular, and Manin-triple constructions, culminating in a coboundary theory via the mock-Lie Yang–Baxter equation. Central contributions include the characterization of Nijenhuis mock-Lie bialgebras via matched pairs and Manin triples, the coboundary theory with antisymmetric $r$-matrices, and the role of $ O$-operators as a unifying tool. The results provide a cohesive, cohomology-driven approach to deformations, extensions, and integrable structures in the mock-Lie setting, with potential applications to related non-associative and Jordan-like algebras. The formalism paves the way for further exploration of Loday-type cohomology, Rota–Baxter-like operators, and Yang–Baxter-type equations in nonassociative contexts.

Abstract

A Nijenhuis mock-Lie algebra is a mock-Lie algebra equipped with a Nijenhuis operator. The purpose of this paper is to extend the well-known results about Nijenhuis mock-Lie algebras to the realm of mock-Lie bialgebras. It aims to characterize Nijenhuis mock-Lie bialgebras by generalizing the concepts of matched pairs and Manin triples of mock-Lie algebras to the context of Nijenhuis mock-Lie algebras. Moreover, we discuss formal deformation theory and explore infinitesimal formal deformations of Nijenhuis mock-Lie algebras, demonstrating that the associated cohomology corresponds to a deformation cohomology. Moreover, we define abelian extensions of Nijenhuis mock-Lie algebras and show that equivalence classes of such extensions are linked to cohomology groups. The coboundary case leads to the introduction of an admissible mock-Lie-Yang-Baxter equation (mLYBe) in Nijenhuis mock-Lie algebras, for which the antisymmetric solutions give rise to Nijenhuis mock-Lie bialgebras. Furthermore, the notion of $\mathcal O$-operator on Nijenhuis mock-Lie algebras is introduced and connected to mock-Lie-Yang-Baxter equation.

Nijenhuis operators and mock-Lie bialgebras

TL;DR

The paper develops a comprehensive framework for Nijenhuis operators on mock-Lie algebras, extending to mock-Lie bialgebras and establishing a parallel theory of deformations, abelian extensions, and cohomology. It introduces Nijenhuis mock-Lie algebras and connects them to bialgebra structures through symplectic, dual-quasitriangular, and Manin-triple constructions, culminating in a coboundary theory via the mock-Lie Yang–Baxter equation. Central contributions include the characterization of Nijenhuis mock-Lie bialgebras via matched pairs and Manin triples, the coboundary theory with antisymmetric -matrices, and the role of -operators as a unifying tool. The results provide a cohesive, cohomology-driven approach to deformations, extensions, and integrable structures in the mock-Lie setting, with potential applications to related non-associative and Jordan-like algebras. The formalism paves the way for further exploration of Loday-type cohomology, Rota–Baxter-like operators, and Yang–Baxter-type equations in nonassociative contexts.

Abstract

A Nijenhuis mock-Lie algebra is a mock-Lie algebra equipped with a Nijenhuis operator. The purpose of this paper is to extend the well-known results about Nijenhuis mock-Lie algebras to the realm of mock-Lie bialgebras. It aims to characterize Nijenhuis mock-Lie bialgebras by generalizing the concepts of matched pairs and Manin triples of mock-Lie algebras to the context of Nijenhuis mock-Lie algebras. Moreover, we discuss formal deformation theory and explore infinitesimal formal deformations of Nijenhuis mock-Lie algebras, demonstrating that the associated cohomology corresponds to a deformation cohomology. Moreover, we define abelian extensions of Nijenhuis mock-Lie algebras and show that equivalence classes of such extensions are linked to cohomology groups. The coboundary case leads to the introduction of an admissible mock-Lie-Yang-Baxter equation (mLYBe) in Nijenhuis mock-Lie algebras, for which the antisymmetric solutions give rise to Nijenhuis mock-Lie bialgebras. Furthermore, the notion of -operator on Nijenhuis mock-Lie algebras is introduced and connected to mock-Lie-Yang-Baxter equation.
Paper Structure (19 sections, 36 theorems, 104 equations)

This paper contains 19 sections, 36 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Mock-Lie algebras
  3. Deformations, Nijenhuis operators and abelian extensions
  4. Bialgebras
  5. Layout of the paper
  6. From mock-Lie bialgebras to Nijenhuis mock-Lie algebras
  7. Nijenhuis mock-Lie algebras
  8. Mock-Lie bialgebras revisited
  9. Symplectic mock-Lie algebras from dual quasitriangular mock-Lie bialgebras
  10. Nijenhuis operators from symplectic mock-Lie algebras
  11. Deformations and extensions of Nijenhuis mock-Lie algebras
  12. Deformations of Nijenhuis mock-Lie algebras
  13. Abelian extensions of Nijenhuis mock-Lie algebras
  14. Nijenhuis mock-Lie bialgebras
  15. Matched pairs of Nijenhuis mock-Lie algebras
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $\mu:\mathcal{A}\otimes \mathcal{A}\longrightarrow \mathcal{A}$, $\rho: \mathcal{A}\longrightarrow End(V)$ and $m:\mathcal{A}\otimes \mathcal{A}\longrightarrow \mathcal{A}$, $\varphi: \mathcal{A}\longrightarrow End(V)$ be linear maps (we write $\mu(x\otimes y)=[x,y]$, $m(x\otimes y)=[x,y]_\cdot$ Then

Theorems & Definitions (94)

  • Definition 2.1: BCHM
  • Remark 2.2: BCHM
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Proposition 2.8
  • ...and 84 more