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Subalgebras of integrals, commutants, and superintegrable deformations of Lotka-Volterra systems

Ian Marquette, Peter H. van der Kamp, G. R. W. Quispel

Abstract

We consider the Lie-algebraic notion of commutant in the setting of Poisson algebra. This provides a framework for deforming Hamiltonian differential equations. By taking a subalgebra of the algebra of integrals, and considering the set of functions that Poisson commute with that subalgebra, the Hamiltonian can be deformed, while retaining integrability. We deform Liouville integrable and superintegrable Lotka-Volterra systems studied in [19]. We present different explicit constructions considering Abelian and non-Abelian subalgebras of integrals. We obtain superintegrable systems for specific dimensions, and in arbitrary dimension. Polynomial systems are deformed to rational systems, some of which have non-rational integrals. Superintegrability seems to be preserved in this approach.

Subalgebras of integrals, commutants, and superintegrable deformations of Lotka-Volterra systems

Abstract

We consider the Lie-algebraic notion of commutant in the setting of Poisson algebra. This provides a framework for deforming Hamiltonian differential equations. By taking a subalgebra of the algebra of integrals, and considering the set of functions that Poisson commute with that subalgebra, the Hamiltonian can be deformed, while retaining integrability. We deform Liouville integrable and superintegrable Lotka-Volterra systems studied in [19]. We present different explicit constructions considering Abelian and non-Abelian subalgebras of integrals. We obtain superintegrable systems for specific dimensions, and in arbitrary dimension. Polynomial systems are deformed to rational systems, some of which have non-rational integrals. Superintegrability seems to be preserved in this approach.
Paper Structure (20 sections, 13 theorems, 124 equations)

This paper contains 20 sections, 13 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Poisson algebras, commutants and superintegrable systems
  3. A family of superintegrable Lotka-Volterra systems
  4. Hamiltonian deformation using a subalgebra of integrals
  5. Deformation of the odd $n=2m-1$ dimensional LV system \ref{['LV']}
  6. Deformation of the even $n=2m$ dimensional system \ref{['LV']}
  7. Integrable deformations of Lotka-Volterra system \ref{['LV']}, using an Abelian subalgebra of integrals
  8. Deformation of the odd $n=2m-1$ dimensional system \ref{['LV']}
  9. Deformation of the even $n=2m$ dimensional system \ref{['LV']}
  10. Integrable deformations of Lotka-Volterra system \ref{['LV']}, using a multi-parameter one-dimensional subalgebra of integrals
  11. Odd case (linear combination of commuting functions)
  12. Even case (linear combination of commuting functions)
  13. Odd Case (linear combination of noncommuting functions)
  14. Even case (linear combination of noncommuting functions)
  15. Iterative application of the method
  16. ...and 5 more sections

Key Result

Lemma 1

The matrix $f_{i,j}$ defines a bracket on ${\cal F}$, $\{\cdot,\cdot\}_{\cal F}:{\cal F}\times{\cal F}\rightarrow{\cal F}$,

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Example 2
  • Definition 3
  • Example 4
  • Example 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • Proposition 11
  • ...and 17 more