scholarly journals Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

2017 ◽  
Vol 24 (2) ◽  
pp. 149-170
Author(s):  
Jafar Abedi-Fardad ◽  
Adel Rezaei-Aghdam ◽  
Ghorbanali Haghighatdoost
Keyword(s):  
Lie Groups ◽  
Hamiltonian Systems ◽  
Poisson Structures ◽  
Compatible Poisson Structures

scholarly journals DYNAMICAL ASPECTS OF LIE-POISSON STRUCTURES

1993 ◽  
Vol 08 (31) ◽  
pp. 2973-2987 ◽  
Author(s):  
F. LIZZI ◽  
G. MARMO ◽  
G. SPARANO ◽  
P. VITALE
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Poisson Bracket ◽  
Lie Groups ◽  
Hamiltonian Systems ◽  
Quantum Groups ◽  
Equations Of Motion ◽  
Poisson Structures ◽  
Dimensional Phase Space ◽  
Deformation Procedure

Quantum groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU(2) and SU(1, 1), as submanifolds of a four-dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some Hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.

scholarly journals Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

2021 ◽  
Author(s):  
Alexey V. Bolsinov ◽  
Andrey Yu. Konyaev ◽  
Vladimir S. Matveev
Keyword(s):  
Singular Points ◽  
Recursion Operator ◽  
Poisson Structures ◽  
Degeneracy Condition ◽  
Compatible Poisson Structures

AbstractWe study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

1992 ◽  
Vol 07 (05) ◽  
pp. 853-876 ◽  
Author(s):  
V. A. FATEEV ◽  
S. L. LUKYANOV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Group ◽  
Lie Groups ◽  
Group Structure ◽  
Two Dimensional ◽  
Poisson Structures ◽  
Additional Symmetries ◽  
Conformal Field ◽  
Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

Generating Function Methods for Coefficient-Varying Generalized Hamiltonian Systems

2014 ◽  
Vol 6 (01) ◽  
pp. 87-106
Author(s):  
Xueyang Li ◽  
Aiguo Xiao ◽  
Dongling Wang
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Generating Function ◽  
Poisson Structure ◽  
Poisson Structures ◽  
Volterra Systems

AbstractThe generating function methods have been applied successfully to generalized Hamiltonian systems with constant or invertible Poisson-structure matrices. In this paper, we extend these results and present the generating function methods preserving the Poisson structures for generalized Hamiltonian systems with general variable Poisson-structure matrices. In particular, some obtained Poisson schemes are applied efficiently to some dynamical systems which can be written into generalized Hamiltonian systems (such as generalized Lotka-Volterra systems, Robbins equations and so on).

scholarly journals Poisson structures on double Lie groups

1998 ◽  
Vol 26 (3-4) ◽  
pp. 340-379 ◽  
Author(s):  
D. Alekseevsky ◽  
J. Grabowski ◽  
G. Marmo ◽  
P.W. Michor
Keyword(s):  
Lie Groups ◽  
Poisson Structures

scholarly journals COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY

2005 ◽  
Vol 20 (07) ◽  
pp. 1367-1388 ◽  
Author(s):  
HENRIK ARATYN ◽  
KLAUS BERING
Keyword(s):  
Poisson Structure ◽  
Integrable Hierarchy ◽  
Difference Operators ◽  
Baxter Equation ◽  
Poisson Structures ◽  
Algebra Isomorphism ◽  
R Matrix ◽  
Compatible Poisson Structures ◽  
Special Value

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.

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