Non-abelian quadratic Poisson brackets - From noncommutative

63 Pages·2011·0.3 MB·English

Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Non-abelian quadratic Poisson brackets
From noncommutative ODE to noncommutative Algebraic
Geometry and back
A. Odesskii1 V. Rubtsov2 V. Sokolov3
1DepartmentofMathematics,UniversityofBrock,Canada
2DepartmentofMathematics,UniversityofAngersandTheoryDivisionofITEP
3LandauInstituteofTheoreticalPhysics,RAS,Moscow
14.06 2011, Figueira da Foz, " Poisson Geometry and
Applications"
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Outline
1 Introduction
Motivations and origins
2 Integrable Systems
3 Non-Abelian Poisson Brackets
"Tensor" interpretations
4 Algebraic characterization
5 Summary
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets Motivationsandorigins
Algebraiccharacterization
Summary
Outline
1 Introduction
Motivations and origins
2 Integrable Systems
3 Non-Abelian Poisson Brackets
"Tensor" interpretations
4 Algebraic characterization
5 Summary
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets Motivationsandorigins
Algebraiccharacterization
Summary
Main sources of inspiration
Integrable Systems
MatrixIntegrablesystemsgeneralizedManakov’s
BihamiltonianpropertyandMagri-Lenardschemes
Noncommutative algebraic Poisson geometry
KontsevichNCsymplecticgeometry
LeBryunNC@geomerty
"Double"PoissonstrucruresandAYBE
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Generalized Manakov’s system
We consider ODE systems of the form
dx
α
= F (x), x = (x ,...,x ), (1)
α 1 N
dt
Herex arem×m -matricesandF are(non-commutative)
i α
polynomials. There exist systems (1) integrable for any m.
The system
u = u2v −vu2, v = 0 (2)
t t
is integrable by the Inverse Scattering Method for any size
m of matrices u and v.
If u - such that uT = −u, and v - a constant diagonal
matrix, then (2) is equivalent to the m-dimensional Euler
top. This is the famous S.V. Manakov model (1976).
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Generalized Manakov’s system
We consider ODE systems of the form
dx
α
= F (x), x = (x ,...,x ), (1)
α 1 N
dt
Herex arem×m -matricesandF are(non-commutative)
i α
polynomials. There exist systems (1) integrable for any m.
The system
u = u2v −vu2, v = 0 (2)
t t
is integrable by the Inverse Scattering Method for any size
m of matrices u and v.
If u - such that uT = −u, and v - a constant diagonal
matrix, then (2) is equivalent to the m-dimensional Euler
top. This is the famous S.V. Manakov model (1976).
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Generalized Manakov’s system
We consider ODE systems of the form
dx
α
= F (x), x = (x ,...,x ), (1)
α 1 N
dt
Herex arem×m -matricesandF are(non-commutative)
i α
polynomials. There exist systems (1) integrable for any m.
The system
u = u2v −vu2, v = 0 (2)
t t
is integrable by the Inverse Scattering Method for any size
m of matrices u and v.
If u - such that uT = −u, and v - a constant diagonal
matrix, then (2) is equivalent to the m-dimensional Euler
top. This is the famous S.V. Manakov model (1976).
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Generalized Manakov’s system
We consider ODE systems of the form
dx
α
= F (x), x = (x ,...,x ), (1)
α 1 N
dt
Herex arem×m -matricesandF are(non-commutative)
i α
polynomials. There exist systems (1) integrable for any m.
The system
u = u2v −vu2, v = 0 (2)
t t
is integrable by the Inverse Scattering Method for any size
m of matrices u and v.
If u - such that uT = −u, and v - a constant diagonal
matrix, then (2) is equivalent to the m-dimensional Euler
top. This is the famous S.V. Manakov model (1976).
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Bihamiltonian structures
Aim - to construct an integrable generalization of (2) to the
case of arbitrary N using the bi-Hamiltonian approach
Definition
Two Poisson brackets {·,·} and {·,·} are compatible if
1 2
{·,·} = {·,·} +λ{·,·} (3)
λ 1 2
is a Poisson bracket for any λ.
If the bracket (3) is degenerate then a hierarchy of
integrable Hamiltonian ODE systems can be constructed
via the following result:
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets
Introduction
IntegrableSystems
Non-AbelianPoissonBrackets
Algebraiccharacterization
Summary
Bihamiltonian structures
Aim - to construct an integrable generalization of (2) to the
case of arbitrary N using the bi-Hamiltonian approach
Definition
Two Poisson brackets {·,·} and {·,·} are compatible if
1 2
{·,·} = {·,·} +λ{·,·} (3)
λ 1 2
is a Poisson bracket for any λ.
If the bracket (3) is degenerate then a hierarchy of
integrable Hamiltonian ODE systems can be constructed
via the following result:
A.Odesskii,V.Rubtsov,V.Sokolov Non-abelianquadraticPoissonbrackets