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$ω$-Lie bialgebras and $ω$-Yang-Baxter equation

Yining Sun, Zeyu Hao, Ziyi Zhang, Liangyun Chen

TL;DR

This work addresses the lack of a cohesive bialgebra framework for $ω$-Lie algebras by establishing multiplicative $ω$-Lie bialgebras through Manin triples and matched pairs, and by studying the $ω$-Yang-Baxter equation and its associated Yang-Baxter $ω$-Lie bialgebras. The authors develop generalized representations, including dual and adjoint-like structures, and introduce $ω$-$ ext{𝓞}$-operators derived from $ω$-left-symmetric algebras to construct skew-symmetric solutions to the $ω$-Yang-Baxter equation, with a clear path to the classical Lie theory under degeneration. The paper provides a robust set of equivalence theorems linking Manin triples, matched pairs, and bialgebra structures, and presents a precise cobracket framework Δ together with compatibility conditions for Yang-Baxter $ω$-Lie bialgebras. By connecting $ω$-left-symmetric algebras, generalized representations, and operator formalisms, it offers new tools for integrable systems and quantum algebra in the broader $ω$-Lie setting.

Abstract

In this paper, we introduce the definition of multiplicative $ω$-Lie bialgebra, which is equivalent to the Manin triples and matched pairs. We also study the $ω$-Yang-Baxter equation and Yang-Baxter $ω$-Lie bialgebra. The skew-symmetric solutions of the $ω$-Yang-Baxter equation can be used to construct Yang-Baxter $ω$-Lie bialgebra. We further introduce the concept of the $ω$-$\mathcal{O}$-operator, which can be constructed from a left-symmetric algebras, and based on the $ω$-$\mathcal{O}$-operator, we construct skew-symmetric solutions to the $ω$-Yang--Baxter equation.

$ω$-Lie bialgebras and $ω$-Yang-Baxter equation

TL;DR

This work addresses the lack of a cohesive bialgebra framework for -Lie algebras by establishing multiplicative -Lie bialgebras through Manin triples and matched pairs, and by studying the -Yang-Baxter equation and its associated Yang-Baxter -Lie bialgebras. The authors develop generalized representations, including dual and adjoint-like structures, and introduce --operators derived from -left-symmetric algebras to construct skew-symmetric solutions to the -Yang-Baxter equation, with a clear path to the classical Lie theory under degeneration. The paper provides a robust set of equivalence theorems linking Manin triples, matched pairs, and bialgebra structures, and presents a precise cobracket framework Δ together with compatibility conditions for Yang-Baxter -Lie bialgebras. By connecting -left-symmetric algebras, generalized representations, and operator formalisms, it offers new tools for integrable systems and quantum algebra in the broader -Lie setting.

Abstract

In this paper, we introduce the definition of multiplicative -Lie bialgebra, which is equivalent to the Manin triples and matched pairs. We also study the -Yang-Baxter equation and Yang-Baxter -Lie bialgebra. The skew-symmetric solutions of the -Yang-Baxter equation can be used to construct Yang-Baxter -Lie bialgebra. We further introduce the concept of the --operator, which can be constructed from a left-symmetric algebras, and based on the --operator, we construct skew-symmetric solutions to the -Yang--Baxter equation.

Paper Structure

This paper contains 5 sections, 23 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $\omega$-Lie algebras and their representations
  3. Matched pairs, Manin triples and Multiplicative $\omega$-Lie bialgebras
  4. Yang-Baxter $\omega$-Lie Bialgebras and the $\omega$-Yang-Baxter Equation
  5. $(\omega)$-Left-symmetric algebras, $\omega$-$\mathcal{O}$-operators and the $\omega$-Yang-Baxter equation

Key Result

Proposition 2.4

Let $(L, [\cdot, \cdot], r)$ be a multiplicative $\omega$-Lie algebra, and let $(\rho, V)$ be a representation of $L$. Define $\rho^* : L \to \mathrm{gl}(V^*)$ by $\rho^*(x)(\xi)(v) = -\xi(\rho(x)(v)) + 2r(x)\xi(v)$, for all $x \in L$, $\xi \in V^*$, and $v \in V$. Then $(\rho^*, V^*)$ is also a rep

Figures (1)

  • Figure :

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 48 more