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Local Rigidity of Quasi--Lie Brackets on Quaternionic Banach Modules and Applications to Nonlinear PDEs

Nassim Athmouni

TL;DR

We address the local rigidity of quasi-Lie brackets on quaternionic Banach right modules by constructing an explicit bilinear correction that restores the Jacobi identity while preserving right $\mathbb{H}$-linearity. The method combines a radial homotopy operator, a Neumann-series inversion of $\mathrm{Id}+M$, and a finite-rank adjustment to annihilate obstructions, with precise operator bounds. This yields a fully constructive framework linking quaternionic functional analysis and rigidity theory, and enables concrete nonlinear PDE applications, including local well-posedness and Beale--Kato--Majda continuation criteria with explicit thresholds. The results provide a quantitative, nonlinear rigidity mechanism in the quaternionic setting, extendable to other normed algebras, and offer a bridge between deformation theory and PDE analysis with potential for broader noncommutative applications.

Abstract

We establish a local rigidity theorem for quasi--Lie brackets on quaternionic Banach right modules. Under quantitative control of antisymmetry and Jacobi defects, we construct an explicit bilinear correction that preserves right $\mathbb{H}$--linearity and restores the exact Lie property. The approach combines a radial homotopy operator, a controlled Neumann-series inversion, and a finite-rank adjustment, all with explicit operator estimates. This constructive framework bridges quaternionic functional analysis with rigidity theory and yields concrete applications to nonlinear PDEs, including local well-posedness and Beale--Kato--Majda continuation criteria with explicit thresholds.

Local Rigidity of Quasi--Lie Brackets on Quaternionic Banach Modules and Applications to Nonlinear PDEs

TL;DR

We address the local rigidity of quasi-Lie brackets on quaternionic Banach right modules by constructing an explicit bilinear correction that restores the Jacobi identity while preserving right -linearity. The method combines a radial homotopy operator, a Neumann-series inversion of , and a finite-rank adjustment to annihilate obstructions, with precise operator bounds. This yields a fully constructive framework linking quaternionic functional analysis and rigidity theory, and enables concrete nonlinear PDE applications, including local well-posedness and Beale--Kato--Majda continuation criteria with explicit thresholds. The results provide a quantitative, nonlinear rigidity mechanism in the quaternionic setting, extendable to other normed algebras, and offer a bridge between deformation theory and PDE analysis with potential for broader noncommutative applications.

Abstract

We establish a local rigidity theorem for quasi--Lie brackets on quaternionic Banach right modules. Under quantitative control of antisymmetry and Jacobi defects, we construct an explicit bilinear correction that preserves right --linearity and restores the exact Lie property. The approach combines a radial homotopy operator, a controlled Neumann-series inversion, and a finite-rank adjustment, all with explicit operator estimates. This constructive framework bridges quaternionic functional analysis with rigidity theory and yields concrete applications to nonlinear PDEs, including local well-posedness and Beale--Kato--Majda continuation criteria with explicit thresholds.

Paper Structure

This paper contains 27 sections, 26 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notations and local assumptions
  3. Cochain complexes and the Chevalley--Eilenberg differential
  4. Cochain spaces and the operator $d$
  5. Cochain norms and quantitative bounds for the defects
  6. Cochain norms.
  7. Quantitative assumptions on the quasi--Lie defects.
  8. Radial homotopy operator and homotopy formula
  9. Quadratic remainder.
  10. Verification of the Jacobi identity for the corrected bracket
  11. Application: Local quaternionic PDE
  12. Functional--analytic toolkit and dimensional thresholds
  13. Local existence and uniqueness
  14. Extensions and comparisons
  15. Beyond quaternions
  16. ...and 12 more sections

Key Result

Lemma 2.8

For every continuous cochain $\Theta\in\mathcal{C}^3_\varepsilon$ and every $\eta>0$, there exists a homogeneous cochain $\Theta_\eta$ such that $\|\Theta-\Theta_\eta\|_\varepsilon\le \eta$.

Theorems & Definitions (77)

  • Definition 2.1
  • Remark 2.2: Compatibility of $d$ with right $\mathbb{H}$--linearity
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8: Homogeneous approximation
  • proof
  • Lemma 3.1
  • ...and 67 more