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NL bialgebras

Zohreh Ravanpak

TL;DR

NL bialgebras fuse Poisson-Nijenhuis ideas with Lie bialgebra theory by equipping a Lie bialgebra with a Nijenhuis operator on $\mathfrak{g}$ and studying compatibility to generate a hierarchy of deformed bialgebras. The work develops three deformation avenues (deformed brackets, deformed $1$-cochains, and double deformations) and identifies exact criteria under which each yields a valid Lie bialgebra, including coboundary and $r$-matrix perspectives. An explicit Euler-top example demonstrates a weak NL bialgebra structure underlying $\mathfrak{so}(3)$ dynamics and its associated linear Poisson brackets. The results tie integrable dynamics to an algebraic framework linking NL bialgebras with Poisson-Lie groups and quantum groups, suggesting new avenues for constructing integrable models and their quantizations.

Abstract

In this paper, we introduce the concept of (weak) NL bialgebras. These structures consist of a Lie bialgebra $(\g,[\cdot,\cdot],δ)$ equipped with a Nijenhuis structure on the Lie algebra $(\g,[\cdot,\cdot])$, satisfying specific compatibility conditions. This construction is analogous to Poisson-Nijenhuis structures studied in the context of integrable systems. We further investigate NL bialgebras that generate a compatible hierarchy of bialgebras, both on the original Lie algebra and its deformed versions, through the Nijenhuis structure of any order. Additionally, we demonstrate that the underlying algebraic structure of a particular case of the Euler-top system is a weak NL bialgebra.

NL bialgebras

TL;DR

NL bialgebras fuse Poisson-Nijenhuis ideas with Lie bialgebra theory by equipping a Lie bialgebra with a Nijenhuis operator on and studying compatibility to generate a hierarchy of deformed bialgebras. The work develops three deformation avenues (deformed brackets, deformed -cochains, and double deformations) and identifies exact criteria under which each yields a valid Lie bialgebra, including coboundary and -matrix perspectives. An explicit Euler-top example demonstrates a weak NL bialgebra structure underlying dynamics and its associated linear Poisson brackets. The results tie integrable dynamics to an algebraic framework linking NL bialgebras with Poisson-Lie groups and quantum groups, suggesting new avenues for constructing integrable models and their quantizations.

Abstract

In this paper, we introduce the concept of (weak) NL bialgebras. These structures consist of a Lie bialgebra equipped with a Nijenhuis structure on the Lie algebra , satisfying specific compatibility conditions. This construction is analogous to Poisson-Nijenhuis structures studied in the context of integrable systems. We further investigate NL bialgebras that generate a compatible hierarchy of bialgebras, both on the original Lie algebra and its deformed versions, through the Nijenhuis structure of any order. Additionally, we demonstrate that the underlying algebraic structure of a particular case of the Euler-top system is a weak NL bialgebra.
Paper Structure (18 sections, 29 theorems, 140 equations, 2 figures)

This paper contains 18 sections, 29 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Poisson-Nijenhuis structures and (coboundary) Lie bialgebras
  3. Poisson-Nijenhuis structures on Manifolds
  4. Lie bialgebras
  5. Coboundary Lie bialgebras
  6. Deformation of Lie algebras and Lie bialgebras
  7. One-cocycle on a deformed Lie algebra
  8. Deformation of a one-cocycle on a Lie algebra: almost NL bialgebras
  9. Double Deformation: (weak) NL bialgebras
  10. Solutions of the classical Yang-Baxter equation: coboundary NL bialgebras
  11. The hierarchy of structures on an NL bialgebra
  12. The hierarchy of deformed brackets and adjoint representations on a Lie algebra
  13. The hierarchy of deformed one-cochains on a Lie algebra
  14. The hierarchy of Lie bialgebras on an NL bialgebra
  15. Weak NL bialgebra on Euler-top system
  16. ...and 3 more sections

Key Result

Proposition 2.5

Let $({\mathfrak g},[\cdot,\cdot])$ be a Lie algebra and $r$ an element of $\wedge^2 {\mathfrak g}$. Then, the following are equivalent:

Figures (2)

  • Figure 1: Pair of Lie algebras defined by the corresponding $1$-cocycles of Lie bialgebras.
  • Figure 2: The diagram illustrates all the deformations under consideration. The left-hand column displays the pairs of Lie algebras corresponding to the iterated 1-cocycles $\delta^{n^k}$ in the deformed cohomology defined on the Lie algebras $(\mathfrak{g},[\cdot,\cdot]_{n^k})$. The right-hand column shows the pairs of Lie algebras corresponding to 1-cocycles deformed by means of $(^tn)^k$. The central column represents the double deformations.

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 59 more