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

Non-Abelian Poisson Brackets . Two Poisson brackets {·,·}1 and {·,·}2 are
compatible if. {·,·}λ = {·,·}1 + λ{· Such brackets possess the following two
properties:.

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