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On the Lie noncommutative integrability

A. V. Tsiganov

Abstract

The Lie theory of non-commutative integrability is used to reconstruct some integrable systems of ordinary differential equations in three dimensional Eucledian space. The Darboux-Brioschi-Halphen system is an example of the Lie integrable system associated with the simple Lie algebra sl(2,R). Other examples are related with solvable three dimensional real Lie algebras of Bianchi B class.

On the Lie noncommutative integrability

Abstract

The Lie theory of non-commutative integrability is used to reconstruct some integrable systems of ordinary differential equations in three dimensional Eucledian space. The Darboux-Brioschi-Halphen system is an example of the Lie integrable system associated with the simple Lie algebra sl(2,R). Other examples are related with solvable three dimensional real Lie algebras of Bianchi B class.
Paper Structure (14 sections, 75 equations)

This paper contains 14 sections, 75 equations.

Table of Contents

  1. Introduction
  2. On the Lie theory of function groups
  3. On the Jacobi theory of invariant measure
  4. The Lie prolongation theory
  5. Real 3D algebras and their invariant systems
  6. The Bianchi class A
  7. Deformation of solvable algebra $e(2)$
  8. Simple $sl(2,\mathbb R)$ algebra
  9. Darboux-Brioschi-Halphen system
  10. The Bianchi class B algebras
  11. Case $\mathrm d=-(\kappa+1)=0$
  12. First possible realization at $\Delta\neq 0$
  13. Second possible realization at $\Delta\neq 0$
  14. Conclusion