Marsden- Weinstein Ruijsenaars-Schneider system from reduction Quasi- quasi-Hamiltonian reduction Hamiltonian reduction Classification of compact integrable Timo Kluck systems Larger coupling Mathematisch Instituut, Universiteit Utrecht parameter February 8, 2013 1
Marsden- Weinstein reduction Quasi- Main point Hamiltonian reduction We will explain the integrability of the compact, trigonometric Classification Ruijsenaars-Schneider system as a consequence of the symmetry of compact integrable of a much simpler dynamical system on SU ( N ) × SU ( N ) . systems Larger coupling parameter 2
The trigonometric Ruijsenaars-Schneider system The compact trigonometric Ruijsenaars-Schneider model is the Marsden- following: Weinstein reduction ◮ N particles on a circle, with coordinates q i ∈ [ 0 , π ) for Quasi- i = 1 , · · · , N . Hamiltonian reduction ◮ Hamiltonian with y ∈ ( 0 , π ) a real coupling parameter: Classification of compact integrable sin 2 y 1 / 2 N systems � � H = cos p i 1 − Larger sin 2 ( q i − q j ) coupling i = 1 j � i parameter 3
The trigonometric Ruijsenaars-Schneider system The compact trigonometric Ruijsenaars-Schneider model is the Marsden- following: Weinstein reduction ◮ N particles on a circle, with coordinates q i ∈ [ 0 , π ) for Quasi- i = 1 , · · · , N . Hamiltonian reduction ◮ Hamiltonian with y ∈ ( 0 , π ) a real coupling parameter: Classification of compact integrable sin 2 y 1 / 2 N systems � � H = cos p i 1 − Larger sin 2 ( q i − q j ) coupling i = 1 j � i parameter � � ◮ The requirement � � q i − q j � � ≥ y guarantees that all square roots are real, which only has nonempty solutions if y ≤ π N . 3
The trigonometric Ruijsenaars-Schneider system The compact trigonometric Ruijsenaars-Schneider model is the Marsden- following: Weinstein reduction ◮ N particles on a circle, with coordinates q i ∈ [ 0 , π ) for Quasi- i = 1 , · · · , N . Hamiltonian reduction ◮ Hamiltonian with y ∈ ( 0 , π ) a real coupling parameter: Classification of compact integrable sin 2 y 1 / 2 N systems � � H = cos p i 1 − Larger sin 2 ( q i − q j ) coupling i = 1 j � i parameter � � ◮ The requirement � � q i − q j � � ≥ y guarantees that all square roots are real, which only has nonempty solutions if y ≤ π N . ◮ Why is this integrable? Where do the symmetries come from? 3
Outline Marsden- Weinstein reduction 1. Simple example: Marsden-Weinstein reduction “explains” Quasi- integrability in the rational Calogero-Moser system Hamiltonian reduction 2. Similarly, quasi-Hamiltonian reduction explains integrability Classification in the current case of compact integrable systems 3. Classification of compact integrable systems allow us to Larger describe the topology of the reduced space (it is just coupling parameter C P N − 1 ) 4. New work: extension to coupling parameter values y > π N 4
Outline Marsden- Weinstein Marsden-Weinstein reduction reduction Quasi- Hamiltonian reduction Quasi-Hamiltonian reduction Classification of compact integrable systems Larger Classification of compact integrable systems coupling parameter Larger coupling parameter 5
Rational Calogero-Moser system Example M ⊆ T ∨ R N , interpreted as positions and momenta of N Marsden- particles in 1 dimension with center-of-mass set to 0. Weinstein reduction Quasi- N ϵ 2 H = 1 � � Hamiltonian p 2 i − reduction 2 ( q i − q j ) 2 Classification i = 1 i < j of compact integrable systems Larger coupling parameter 6
Rational Calogero-Moser system Example M ⊆ T ∨ R N , interpreted as positions and momenta of N Marsden- particles in 1 dimension with center-of-mass set to 0. Weinstein reduction Quasi- N ϵ 2 H = 1 � � Hamiltonian p 2 i − reduction 2 ( q i − q j ) 2 Classification i = 1 i < j of compact integrable systems Larger Theorem (Calogero) coupling parameter This is an 2 ( N − 1 ) -dimensional integrable system, with Hamiltonians given by H k = Tr L k , where L is the traceless matrix ϵ p 1 q i − q j ... L = ϵ p N q i − q j 6
Rational Calogero-Moser system (ctd) Marsden- So some of these Hamiltonians are: Weinstein reduction N Quasi- � Hamiltonian H 1 p i = 0 = reduction i = 1 Classification of compact N ϵ 2 integrable � � p 2 systems H 2 ( = 2 H ) i − = ( q i − q j ) 2 Larger i = 1 i � j coupling parameter N ϵ 2 ϵ 3 � � � p 3 H 3 p i ( q i − q j ) 2 + i − = ( q i − q j )( q j − q k )( q k − q i ) i = 1 i � j i , j , k distinct 7
Rational Calogero-Moser system (ctd) Marsden- Weinstein reduction Quasi- Question Hamiltonian reduction Where do all these symmetries / conserved quantities come Classification from? of compact integrable systems Larger coupling parameter 8
Rational Calogero-Moser system (ctd) Marsden- Weinstein reduction Quasi- Question Hamiltonian reduction Where do all these symmetries / conserved quantities come Classification from? of compact integrable systems “Answer” Larger They exist because the motion is very simple (linear) in the coupling parameter matrix space. 8
Linear motion in a matrix space Marsden- Weinstein reduction ◮ G = SL ( N ) , g = sl ( N ) Quasi- Hamiltonian ◮ Phase space M = T ∨ g = g × g using Killing pairing reduction ◮ Hamiltonian H ( P , Q ) = 1 2 � P , P � Classification of compact integrable systems Larger coupling parameter 9
Linear motion in a matrix space Marsden- Weinstein reduction ◮ G = SL ( N ) , g = sl ( N ) Quasi- Hamiltonian ◮ Phase space M = T ∨ g = g × g using Killing pairing reduction ◮ Hamiltonian H ( P , Q ) = 1 2 � P , P � Classification of compact integrable ◮ Solution for given initial value ( P 0 , Q 0 ) : systems Larger coupling P ( t ) P 0 = parameter Q ( t ) Q 0 + tP 0 = 9
Linear motion in a matrix space (ctd) Marsden- ◮ Symmetric under adjoint action of G on g : Weinstein reduction • H and ω invariant under conjugation Quasi- Hamiltonian P , Q �→ gPg − 1 , gQg − 1 reduction • Time evolution commutes with G -action Classification of compact ◮ Conserved quantities in involution: integrable systems Larger H k = Tr P k coupling parameter ◮ Also invariant under conjugation � � n 2 − 1 ◮ But too few: dim M = 2 > 2 ( n − 1 ) 10
Marsden-Weinstein reduction Marsden- Weinstein reduction Idea Quasi- ◮ Since everything is G -invariant, we can quotient out by it. Hamiltonian reduction ◮ Hopefully, this reduces the dimension sufficiently to end up Classification of compact with an integrable system. integrable systems Larger coupling parameter 11
Marsden-Weinstein reduction Marsden- Weinstein reduction Idea Quasi- ◮ Since everything is G -invariant, we can quotient out by it. Hamiltonian reduction ◮ Hopefully, this reduces the dimension sufficiently to end up Classification of compact with an integrable system. integrable systems ◮ But we also need to keep a non-degenerate symplectic form: Larger coupling • If we quotient out a tangent vector ξ ∈ TM , then we parameter should also quotient out its image ω ( ξ ) ∈ T ∨ M . Dually, that means restricting to a submanifold. 11
Marsden-Weinstein reduction (ctd) Definition A group action of G on M is Hamiltonian if its infinitesimal Marsden- Weinstein vector fields v ξ for ξ ∈ g are of the form reduction Quasi- v ξ = { f ξ , ·} Hamiltonian reduction Classification of compact integrable systems Larger coupling parameter 12
Marsden-Weinstein reduction (ctd) Definition A group action of G on M is Hamiltonian if its infinitesimal Marsden- Weinstein vector fields v ξ for ξ ∈ g are of the form reduction Quasi- v ξ = { f ξ , ·} Hamiltonian reduction Classification of compact Definition integrable systems A Hamiltonian group action is generated by a moment map Larger µ : M → g ∨ if coupling parameter f ξ = � µ , ξ � and if µ is G -equivariant. 12
Marsden-Weinstein reduction (ctd) Definition A group action of G on M is Hamiltonian if its infinitesimal Marsden- Weinstein vector fields v ξ for ξ ∈ g are of the form reduction Quasi- v ξ = { f ξ , ·} Hamiltonian reduction Classification of compact Definition integrable systems A Hamiltonian group action is generated by a moment map Larger µ : M → g ∨ if coupling parameter f ξ = � µ , ξ � and if µ is G -equivariant. Important example A commuting set of Hamiltonians h = ( h 1 , · · · , h n ) forms a moment map for an R n -action, or for a T n -action. 12
Marsden-Weinstein reduction (ctd) Marsden- Weinstein reduction Quasi- Theorem (Marsden, Weinstein) Hamiltonian reduction If µ 0 ∈ g ∨ is a regular value of µ , then the space µ − 1 ( µ 0 ) / G µ 0 is Classification a symplectic manifold. of compact integrable systems Larger coupling parameter 13
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