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Admissible Yang-Baxter equation for Nijenhuis perm algebras

Tianshui Ma, Feiyan Song

TL;DR

This work develops a framework that generalizes Yang-Baxter theory to perm algebras in the presence of Nijenhuis operators. It introduces S-admissible perm Yang-Baxter equations, defines Nijenhuis perm bialgebras, and establishes multiple equivalent characterizations via matched pairs and Manin triples, including the coboundary and O-operator formalisms. The authors classify 2‑dimensional quasitriangular noncommutative perm bialgebras and provide construction methods for Nijenhuis perm algebras from symplectic and dual-quasitriangular data, tying deformation theory to integrable structures in the perm setting. The results offer a cohesive toolkit for generating and studying Nijenhuis perm algebras and their bialgebraic analogues with potential applications in deformation theory and integrable systems. Overall, the paper extends perm-Yang-Baxter theory into a robust Nijenhuis framework with explicit constructions and classifications.

Abstract

In this paper, on one hand, based on the classical perm Yang-Baxter equation, we investigate under what conditions a perm algebra must be a Nijenhuis perm algebra. On the other hand, we derive the compatible conditions between classical perm Yang-Baxter equation and Nijenhuis operator by a class of Nijenhuis perm bialgebras.

Admissible Yang-Baxter equation for Nijenhuis perm algebras

TL;DR

This work develops a framework that generalizes Yang-Baxter theory to perm algebras in the presence of Nijenhuis operators. It introduces S-admissible perm Yang-Baxter equations, defines Nijenhuis perm bialgebras, and establishes multiple equivalent characterizations via matched pairs and Manin triples, including the coboundary and O-operator formalisms. The authors classify 2‑dimensional quasitriangular noncommutative perm bialgebras and provide construction methods for Nijenhuis perm algebras from symplectic and dual-quasitriangular data, tying deformation theory to integrable structures in the perm setting. The results offer a cohesive toolkit for generating and studying Nijenhuis perm algebras and their bialgebraic analogues with potential applications in deformation theory and integrable systems. Overall, the paper extends perm-Yang-Baxter theory into a robust Nijenhuis framework with explicit constructions and classifications.

Abstract

In this paper, on one hand, based on the classical perm Yang-Baxter equation, we investigate under what conditions a perm algebra must be a Nijenhuis perm algebra. On the other hand, we derive the compatible conditions between classical perm Yang-Baxter equation and Nijenhuis operator by a class of Nijenhuis perm bialgebras.

Paper Structure

This paper contains 13 sections, 35 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. $S$-admissible perm Yang-Baxter equations
  3. Nijenhuis perm algebras and representations
  4. Nijenhuis perm bialgebras and equivalent characterizations
  5. Matched pairs of Nijenhuis perm algebras
  6. Manin triple of a Nijenhuis perm algebra
  7. Nijenhuis perm bialgebras
  8. Coboundary Nijenhuis perm bialgebra
  9. $\mathcal{O}$-operators on Nijenhuis perm algebra
  10. Classification of 2-dimensional quasitriangular noncommutative perm bialgebras
  11. A method to obtain a Nijenhuis perm algebra
  12. Symplectic perm algebras from dual quasitriangular perm bialgebras
  13. Nijenhuis operators from symplectic perm algebras

Key Result

Lemma 2.2

Let $\mu, m:\mathfrak{g}\otimes \mathfrak{g}\longrightarrow \mathfrak{g}$ (we write $\mu(x\otimes y)=x\cdot y$, $m(x\otimes y)=x\bullet y$), $\eta^{\ell}, \eta^{r}, \tau^{\ell}, \tau^{r}: \mathfrak{g}\longrightarrow End(V)$ be linear maps and $s, t$ be parameters. Define the following new multiplica where $x, y\in \mathfrak{g}$. Then

Theorems & Definitions (86)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Proposition 2.7
  • proof
  • ...and 76 more