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Classification of four dimensional real Lie bialgebras of symplectic type and their Poisson-Lie groups

J. Abedi-Fardad, A. Rezaei-Aghdam, Gh. Haghighatdoost

Abstract

In this paper we classify all four dimensional real Lie bialgebras of symplectic type. The classical r- matrices for these Lie bialgebras and Poisson structures on all of the related four dimensional Poisson-Lie groups are also obtained. Some new integrable models for which the Poisson-Lie group plays the role as a phase space and its dual Lie group plays the role of a symmetry group of the system, are obtained.

Classification of four dimensional real Lie bialgebras of symplectic type and their Poisson-Lie groups

Abstract

In this paper we classify all four dimensional real Lie bialgebras of symplectic type. The classical r- matrices for these Lie bialgebras and Poisson structures on all of the related four dimensional Poisson-Lie groups are also obtained. Some new integrable models for which the Poisson-Lie group plays the role as a phase space and its dual Lie group plays the role of a symmetry group of the system, are obtained.

Paper Structure

This paper contains 13 sections, 61 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Classification of four dimensional real Lie bialgebras of symplectic type.
  4. Four dimensional real Lie algebras of symplectic type
  5. Review of the classification method for low dimensional real Lie bialgebras
  6. Classification of four dimensional real Lie bialgebra of symplectic type
  7. Four dimensional coboundary real Lie bialgebras of symplectic type
  8. Calculation of Poisson and symplectic structures on four dimensional real Poisson-Lie groups
  9. Calculation of Poisson and symplectic structures by Sklyanin bracket
  10. Calculation of Poisson and symplectic structures on four dimensional real Poisson-Lie group by Manin triple and map $\pi (g)$ between the Lie subalgebras $\bf g$ and $\tilde{\bf g}$.
  11. Compatible symplectic structure on four dimensional real Poisson-Lie groups
  12. Physical application; example of integrable systems by use of symplectic Lie bialgebra
  13. Conclusion