Sutherland model Ian Marshall March 2015 Ian Marshall Ruijsenaars - PowerPoint PPT Presentation

Ruijsenaars type deformation of hyperbolic BC n Sutherland model Ian Marshall March 2015 Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

History 1. Olshanetsky and Perelomov discovered the hyperbolic BC n Sutherland model by a reduction/projection procedure, but it had only 2 independent parameters 2. Inozemtsev and Meshcheryakov proved integrability of the three parameter version 3. Feher-Pusztai obtained it with all three parameters by reduction: Start with M = T ∗ K and reduce by canonical left and right actions of subgroups K + ≺ K , with K = SU ( n, n ) and K + the compact subgroup satisfying gg † = I : J r is a character, and J l is an appropriate analog of the so-called Kazhdan-Kostant-Sternberg element. n n n � � � p 2 sinh − 2 (2 q i ) + b sinh − 2 ( q i ) 2 H = i + a i =1 i =1 i =1 � sinh − 2 ( q i + q j ) + sinh − 2 ( q i − q j ) � � + c . i,j i � = j Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Relativistic versions of Calogero type models were introduced by Ruijsenaars. It was proposed in several papers of Gorsky and others that Poisson Lie group reduction should be the appropriate setting for these systems. Feher and Klimcik worked on this project and found PLG reduction interpretations for several known Ruijsenaars type systems. My result is a PLG reduction construction of the same kind. It imitates the result of Feher and Pusztai, essentially amounting to the replacement of T ∗ K by the Heisenberg double of K . The product of this procedure is a new integrable system. This is roughly speaking the whole story. The rest is a lot of tricky computations. Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Reduction Suppose we have a group G acting on a differentiable manifold M . The conerstone of reduction is the decision to restrict from the ring C ∞ ( M ) of all smooth functions on M to the subring of invariant functions C ∞ ( M ) G . This means that we shall be interested only in invariant flows. That is, for v ∈ V ect ( M ) , for any ϕ ∈ C ∞ ( M ) we require v ( ϕ ◦ g )( x ) = v ( ϕ )( g · x ) , where ( ϕ ◦ g )( x ) = ϕ ( g · x ) . (1) With ξ ( ϕ )( x ) := d � � e tξ · x � ˆ � ϕ , � dt � t =0 then infinitesimally, (1) amounts to L ˆ ξv = 0 ∀ ξ ∈ g . Interpret v as a vector field on M/G , by restricting it to act only on C ∞ ( M ) G . Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Poisson case : M , G Suppose now that M is a Poisson space, and that G is a Lie group acting on M in such a way that the Poisson structure is preserved. That is, F, K ∈ C ∞ ( M ) G ⇒ { F, K } ∈ C ∞ ( M ) G . (2) In this case, H ∈ C ∞ ( M ) G is a sufficient condition for ensuring that the corresponding Hamiltonian vector field X H ∈ V ect ( M ) is invariant. Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Poisson reduction I : F and F gen ( ν ) Suppose that F ⊂ C ∞ ( M ) is a family of functions, all of whose Hamiltonian vector fields are tangent to the vector fields in ˆ g , i.e., ϕ ∈ F , F ∈ C ∞ ( M ) G ⇒ 0 = X ϕ ( F ) = − X F ( ϕ ) . Restricting to G -invariant functions, all the functions in F may be viewed as constants : (i) Introduce F gen := { ϕ α | α ∈ A} , for which F = �F gen � . (ii) For any set ν := { ν α ∈ R | α ∈ A} , define F gen ( ν ) := { ϕ α − ν α | α ∈ A} (iii) Restrict to the submanifold � ψ − 1 (0) = � N ν := { x ∈ M | ϕ α ( x ) = ν α } . α ∈A ψ ∈F gen ( ν ) Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Poisson reduction II : factor by G ν Suppose that G ν ⊂ G is the maximal subgroup which acts on N ν . � G ν C ∞ ( M ) | N ν ⊃ C ∞ ( M ) G | N ν = � C ∞ ( M ) | N ν We arrive at a Poisson algebra consisting of the functions C ∞ ( N ν ) G ν , or, in other words, to the reduced space N ν /G ν =: M red ( ν ) say . If F := { F i | i = 1 , 2 , . . . } is a collection of invariant functions with the property { F i , F j } = 0 , then their restrictions to N ν will still have zero Poisson bracket, and they define commuting functions on the reduced space M red ( ν ) . Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

M : Heisenberg double, and K : our symmetry group � I n � 0 Let I nn = . G denotes SL (2 n, C ) . − I n 0 K denotes SU ( n, n ) = { g ∈ SL (2 n, C ) | g † I nn g = I nn } . B denotes the set of all upper triangular matrices in SL (2 n, C ) with real, positive diagonal entries, and B n denotes the same set in GL ( n, C ) . �� p �� 0 K ⊃ K + = , p, q, ∈ U ( n ) , K = K + × K + acts 0 q on G by ordinary left and right multiplication. The diagonal subgroup in U ( n ) will be denoted by T . For any k ∈ K we may write � ρ � � Γ � � p � Σ 0 0 k = , with ρ, m, p, q ∈ SU ( n ) , 0 m Σ Γ 0 q . Γ = diag (cosh(∆ i )) , Σ = diag (sinh(∆ i )) Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

g = Lie ( G ) can be decomposed as the sum g = k + b of the two subalgebras k = Lie ( K ) and b = Lie ( B ) , with respect to which the projections P k : g → k and P b : g → b are well-defined. Let � , � : g × g → R denote the non-degenerate, invariant inner product defined by � X, Y � = Im tr XY. Then R := P k − P b defines a classical r-matrix on g , skew-symmetric with respect to � , � . For any function F ∈ C ∞ ( G ) , the left- and right-derivatives, D l,r F : G → g ∼ g ∗ , of F are defined by d � F ( e tX ge tY ) = � D l F ( g ) , X � + � D r F ( g ) , Y � � ∀ X, Y ∈ g . � dt � t =0 The Poisson structure on G , viewed as the Heisenberg double based on the bi-algebra g = k + b , is defined by { F, H } = � D l F, R ( D l H ) � + � D r F, R ( D r H ) � . Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Let c 1 , c 2 ∈ G and define the subspace M ( c 1 , c 2 ) ⊂ G by M ( c 1 , c 2 ) = { bc 1 k | b ∈ B, k ∈ K } ∩ { kc 2 b | b ∈ B, k ∈ K } . Introduce “coordinates” ( b L , k L , b R , k R ) on M ( c 1 , c 2 ) (not independent) by M ( c 1 , c 2 ) ∋ g = b L c 1 k R = k L c 2 b R . Proposition (Alekseev and Malkin – CMP1994) M ( c 1 , c 2 ) is a symplectic leaf, and all symplectic leaves are of this form. The symplectic structure on M ( c 1 , c 2 ) can be written [ Symp ]( g ) = � db L b − 1 , dk L k − 1 L � + � b − 1 , k − 1 ∧ R db R ∧ R dk R � . L Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Constraints Proposition: The functions Φ k ∈ C ∞ ( G ) defined by Φ k ( g ) = − 1 � k gI n,n g † I n,n � 2 k tr are in involution with one another. Constraints. Fixing σ ∈ B n and x, y ∈ R + , constraints are as follows: suppose that when written in the form � x I � ω G ∋ g = k L b R , b R = x − 1 I 0 and that, when written in the form � y − 1 σ y − 1 ν � g = b L k R , b L = , y I 0 with det( σ ) = 1 , and with both ω and ν undetermined in gl ( n ) . σ ∈ B n is the PLG analog of the KKS element. Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Aside : F gen ( ν ) For ξ, η ∈ k + , define � � 0 � 0 gI nn g † + y − 2 gI nn g † gI nn g † F ξ,η ( g ) =Im tr ξ 0 I � σσ † �� 0 − y − 2 0 0 � � I � 0 g † I nn g − x − 2 g † I nn g g † I nn g + Im tr η 0 0 � 0 �� 0 + x − 2 , 0 I and then define the constraint submanifold by N x,y,σ := { g ∈ M ( I , I ) | F ξ,η ( g ) = 0 ∀ ξ, η ∈ k + } . Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

- back to the constraints Factoring on the right by K + and on the left by the subgroup �� I �� 0 ⊂ K + , let’s assume that g is in the gauge 0 p � ρ Γ � � x I � ρ Σ ω g = with ω ∈ gl ( n, C ) , ρ ∈ SU ( n ) x − 1 I Σ Γ 0 Compare the two versions of gI nn g † and get constraint condition, for T ∈ U ( n ) T † Σ 2 T = Σ ρ † σσ † ρ Σ Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

Explanation of where we’re up to: Let us make the substitution Ω = Σ ω + x − 1 Γ , so that ρ Σ − 1 (ΓΩ − x − 1 I ) � xρ Γ � g = . x Σ Ω Now, from gI nn g † = b L I nn b † L we get ΩΩ † = y 2 I + x 2 Σ 2 =: Λ 2 ⇒ Ω = Λ T, T ∈ U ( n ) ν = ρ Σ − 1 ( y 2 Γ − x − 1 Ω † ) and T † Σ 2 T = Σ ρ † σσ † ρ Σ Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

The PLG KKS element is σ ∈ B n , satisfying σσ † = α 2 I + ˆ v † , v ˆ v ∈ C n s.t. det σ = 1 , so the constraint reads for α ∈ R and ˆ T † Σ 2 T = α 2 Σ 2 + vv T v := Σ ρ † ˆ v and can be supposed real with all entries in R ≥ 0 . The constraint condition can be solved(!) After some work (quite a lot) [ Symp ] = � ρ † dρ ∧ , Σ − 1 T † Σ d (Σ − 1 T Σ) � + � T † dT + dTT † ∧ , Σ − 1 d Σ � . Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo

After using a few more tricks, we arrive at n � dp i ∧ Σ − 1 [ Symp ] = i d Σ i i =1 where, for P := e ip ∈ T , the general solution of the constraint condition was T = P ˜ T and ˜ T is a (complicated!) explicit, real matrix function of Σ . Ian Marshall Ruijsenaars type deformation of hyperbolic BC n Sutherland mo