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Reynolds Leibniz bialgebras of any weight

Tianshui Ma, Yuguang Ming, Chan Zhao

Abstract

This paper studies bialgebraic structures associated with a Reynolds Leibniz algebra of weight $λ$, that is, a Leibniz algebra equipped with a Reynolds operator of weight $λ$. We first present equivalent characterizations of Reynolds Leibniz bialgebras of weight $λ$, using matched pairs and Manin triples. Next, we examine compatibility conditions between solutions of the classical Leibniz Yang-Baxter equation and Reynolds operators of weight $λ$, framed in terms of triangular Reynolds Leibniz bialgebras. Finally, building on results of Ayupov {\em et al.}, we classify two-dimensional triangular Reynolds Leibniz bialgebras of weight $λ$.

Reynolds Leibniz bialgebras of any weight

Abstract

This paper studies bialgebraic structures associated with a Reynolds Leibniz algebra of weight , that is, a Leibniz algebra equipped with a Reynolds operator of weight . We first present equivalent characterizations of Reynolds Leibniz bialgebras of weight , using matched pairs and Manin triples. Next, we examine compatibility conditions between solutions of the classical Leibniz Yang-Baxter equation and Reynolds operators of weight , framed in terms of triangular Reynolds Leibniz bialgebras. Finally, building on results of Ayupov {\em et al.}, we classify two-dimensional triangular Reynolds Leibniz bialgebras of weight .
Paper Structure (16 sections, 30 theorems, 47 equations)

This paper contains 16 sections, 30 theorems, 47 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Reynolds Leibniz algebras
  3. Leibniz (bi)algebras
  4. Main results and layout of the paper
  5. The (co)representation of Reynolds Leibniz (co)algebra
  6. Representation of Reynolds Leibniz algebras
  7. Dual representation
  8. Reynolds Leibniz bialgebras
  9. Matched pairs of Reynolds Leibniz algebras
  10. Manin triple of Reynolds Leibniz algebra
  11. Reynolds Leibniz bialgebras
  12. Triangular Reynolds Leibniz bialgebras, admissible cLYBes and $\mathcal{O}$-operators
  13. Triangular Reynolds Leibniz bialgebras
  14. Admissible cLYBe in a Reynolds Leibniz algebra
  15. $\mathcal{O}$-operator on a Reynolds Leibniz algebra
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $((\mathfrak{g},[,]),\mathfrak{R})$ be a Reynolds Leibniz algebra. Then $\mathfrak{R}$ induce a new Leibniz algebra structure on $\mathfrak{g}$ given by The Leibniz algebra $(\mathfrak{g},[,]_{\mathfrak{R}})$ is called the induced Leibniz algebra. Moreover, $((\mathfrak{g},[,]_{\mathfrak{R}}),\mathfrak{R})$ is also a Reynolds Leibniz algebra and $\mathfrak{R}$ is a homomorphism from $((\mathf

Theorems & Definitions (65)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 55 more