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Reynolds Lie bialgebras

Shuai Hou, Maxim Goncharov

TL;DR

This work develops a comprehensive Reynolds Lie bialgebra theory by integrating Reynolds operators, matched pairs, and Manin triples, and by establishing representations and dualities via invariant forms. It shows that Reynolds operators on quadratic Rota-Baxter Lie algebras naturally induce Reynolds Lie bialgebras, with the Drinfeld double framework extended to this setting. The paper then connects these structures to the classical Yang-Baxter equation through relative Rota-Baxter operators and Reynolds pre-Lie algebras, providing methods to construct CYBE solutions in Reynolds contexts. Overall, the results unify Reynolds operator techniques with Lie bialgebra theory, offering a coherent pathway to quantum-type structures within Reynolds-Lie theory and CYBE-inspired constructions.

Abstract

In this paper, we establish a bialgebra theory for Reynolds Lie algebras. First we introduce the notion of a quadratic Reynolds Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint representation. Then we introduce the notion of matched pairs, Manin triples and bialgebras for Reynolds Lie algebras, and show that Manin triples, bialgebras and certain matched pairs of Reynolds Lie algebras are equivalent. In particular, we introduce the notion of a Reynolds operator on a quadratic Rota-Baxter Lie algebra which can induce a Reynolds Lie bialgebra naturally. Finally, we introduce the notion of the classical Yang-Baxter equation in a Reynolds Lie algebra whose solutions give rise to Reynolds Lie bialgebras. We also introduce the notion of relative Rota-Baxter operators on a Reynolds Lie algebra and Reynolds pre-Lie algebras, and construct solutions of the classical Yang-Baxter equation in terms of relative Rota-Baxter operators and Reynolds pre-Lie algebras.

Reynolds Lie bialgebras

TL;DR

This work develops a comprehensive Reynolds Lie bialgebra theory by integrating Reynolds operators, matched pairs, and Manin triples, and by establishing representations and dualities via invariant forms. It shows that Reynolds operators on quadratic Rota-Baxter Lie algebras naturally induce Reynolds Lie bialgebras, with the Drinfeld double framework extended to this setting. The paper then connects these structures to the classical Yang-Baxter equation through relative Rota-Baxter operators and Reynolds pre-Lie algebras, providing methods to construct CYBE solutions in Reynolds contexts. Overall, the results unify Reynolds operator techniques with Lie bialgebra theory, offering a coherent pathway to quantum-type structures within Reynolds-Lie theory and CYBE-inspired constructions.

Abstract

In this paper, we establish a bialgebra theory for Reynolds Lie algebras. First we introduce the notion of a quadratic Reynolds Lie algebra and show that it induces an isomorphism from the adjoint representation to the coadjoint representation. Then we introduce the notion of matched pairs, Manin triples and bialgebras for Reynolds Lie algebras, and show that Manin triples, bialgebras and certain matched pairs of Reynolds Lie algebras are equivalent. In particular, we introduce the notion of a Reynolds operator on a quadratic Rota-Baxter Lie algebra which can induce a Reynolds Lie bialgebra naturally. Finally, we introduce the notion of the classical Yang-Baxter equation in a Reynolds Lie algebra whose solutions give rise to Reynolds Lie bialgebras. We also introduce the notion of relative Rota-Baxter operators on a Reynolds Lie algebra and Reynolds pre-Lie algebras, and construct solutions of the classical Yang-Baxter equation in terms of relative Rota-Baxter operators and Reynolds pre-Lie algebras.

Paper Structure

This paper contains 16 sections, 39 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Reynolds operators
  3. Lie bialgebras
  4. The classical Yang-Baxter equation and Rota-Baxter operators
  5. Main results and outline of the paper
  6. Reynolds Lie algebras
  7. Matched pairs and Manin triples of Reynolds Lie algebras
  8. Matched pairs of Reynolds Lie algebras
  9. Manin triples of Reynolds Lie algebras
  10. Reynolds Lie bialgebras and Reynolds operators on quadratic Rota-Baxter Lie algebras
  11. Reynolds operators on Lie bialgebras
  12. Reynolds operators on quadratic Rota-Baxter Lie algebras
  13. The classical Yang-Baxter equation, relative Rota-Baxter operators and Reynolds pre-Lie algebras
  14. The classical Yang-Baxter equation in a Reynolds Lie algebra
  15. Relative Rota-Baxter operators on Reynolds Lie algebras
  16. ...and 1 more sections

Key Result

Proposition 2.3

(FoxMil) Let $(\mathfrak g,[\cdot,\cdot]_{\mathfrak g})$ be a Lie algebra and $d\in\mathrm{Der}(\mathfrak g)$ is a derivation of $\mathfrak g$ such that the map $T={\rm{Id}}-d$ is invertible. Then $R=T^{-1}=({\rm{Id}}-d)^{-1}$ is a Reynolds operator on $\mathfrak g$.

Theorems & Definitions (99)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • proof
  • Example 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 89 more