structure of the Calogero-Moser system

arXiv:0902.0953v2 [math-ph] 3 Jun 2009 On the geometric origin of the bi-Hamiltonian structure of the Calogero-Moser system C. Bartocci1 , G. Falqui2 , I. Mencattini3 , G. Ortenzi2 , M. Pedroni4 1 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova, Italy 2 Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Roberto Cozzi 53, I-20125 Milano, Italy 3 SISSA, via Beirut 2/4, I–34014 Trieste, Italy 4 Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, I-24044 Dalmine (BG), Italy E-mail addresses: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero-Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of gl(n, R). The relation with the Lax formalism is also discussed. 1 Introduction In 1971 Francesco Calogero [4] solved the quantum system consisting of n unit-mass particles moving on the line and pairwise interacting via a (repulsive) potential that is proportional to the inverse of the squared distance. (The case n = 3 was treated earlier [3] by Calogero himself). The integrability of the classical counterpart was conjectured in [4] and proved by Moser in [18]. Later, this system was showed to be superintegrable [23]. It is also worthwhile to mention that the classical 3-particle case appeared in the works of Jacobi [13]. More information on the (quantum and classical) Calogero-Moser system can be found in [5]. Recently, this system gained an important role in pure mathematics too. We just cite its relations with 1 quiver varieties [10] and double affine Hecke algebras [7], referring to [6] for a more complete list. Although a lot of papers were devoted to the many facets of the CalogeroMoser system, only a few results concerning its bi-Hamiltonian formulation were found. In [17] (see also [16]) a bi-Hamiltonian structure was constructed with the help of the Lax representation of the system. A (2n−1)-dimensional family of compatible Poisson tensors—apparently unrelated with the above mentioned Poisson pair—was found in [11], in the context of superintegrable systems. In this paper we explain where the bi-Hamiltonian structure of [17] comes from. The spirit is very close to that of the fundamental paper [14], where the Calogero-Moser system is shown to be the Marsden-Weinstein reduction of a trivial system on the cotangent bundle of su(n). In the same vein, we show that the bi-Hamiltonian structure can be obtained—by means of two projections—from a Poisson pair belonging to a wide class of bi-Hamiltonian structures on cotangent bundles. Such class is recalled in Section 2, while in Section 3 the particular example related to the Poisson brackets of the (attractive) Calogero-Moser system is considered. In Section 4 a first reduction is performed, corresponding to the action given by the simultaneous conjugation. A second projection, leading to the phase space of the Calogero-Moser system, is described in Section 5. Finally, Section 6 is devoted to the example of the 2-particle system (trivial from the physical point of view, but not from the mathematical one) and Section 7 to some final remarks. Acknowledgments. The authors are grateful to Franco Magri, Andriy Panasyuk, and Volodya Rubtsov for fruitful discussions, and to Victor Ginzburg for some remarks on the archive version of this paper. Moreover, we would like to thank the anonymous referee for useful suggestions, in particular for Remark 3. This work has been partially supported by the European Community through the FP6 Marie Curie RTN ENIGMA (Contract number MRTNCT-2004-5652) and by the ESF through the research programme MISGAM. M.P. would like to thank for the hospitality the Department Matematica e Applicazioni of the Milano-Bicocca University, the Department of Mathematics of the Genova University, and the Department of Mathematics of the University of North Carolina at Chapell Hill, where part of this work was done. 2 2 Bi-Hamiltonian structures on cotangent bundles In this section we recall from [21] (see also [12]) that a torsionless (1,1) tensor field on a smooth manifold Q gives rise to a (second) Poisson structure on the cotangent space T ∗ Q, compatible with the canonical one. More information on the geometry of bi-Hamiltonian manifolds can be found, e.g., in [16]. Let L : T Q → T Q be a type (1, 1) tensor field on Q, whose Nijenhuis torsion vanishes. This means that T (L)(X, Y ) := [LX, LY ] − L([LX, Y ] + [X, LY ] − L[X, Y ]) = 0 (1) for all pairs of vector fields X, Y on Q. Let θ be the Liouville 1-form on T ∗ Q and ω = dθ the standard symplectic 2-form on T ∗ Q, whose associated Poisson tensor will be denoted with P0 . One can deform the Liouville 1-form to a 1-form θL : hθL , Ziα = hα, L(π∗ Z)iπ(α) , for any vector field Z on T ∗ Q and for any 1-form α on Q, where π : T ∗ Q → Q is the canonical projection. If we choose local coordinates (x1 , . . . , xn ) on Q and consider the corresponding symplectic coordinates (x1 , . . . , xn , y1 , . . . , yn ) on T ∗ Q, we get the local expression θL = Lij yi dxj . Now, it is well-known that the canonical Poisson bracket is defined by F, G ∈ C ∞ (T ∗ Q) , {F, G}0 = ω(XF , XG ) where XF , XG are the Hamiltonian vector fields associated to F , G with respect to the symplectic form ω. A second composition law on C ∞ (T ∗ Q) is given by {F, G}1 = ωL (XF , XG ) , (2) where ωL := dθL . It is easily seen that i j {x , x }1 = 0 , j {yi , x }1 = Lji , {yi , yj }1 = ∂Lkj ∂Lki − ∂xi ∂xj ! yk . Moreover, the vanishing of the torsion of L entails that (2) is a Poisson bracket too, and that it is compatible with {·, ·}0. Thus we have a biHamiltonian structure on T ∗ Q. 3 Remark 1 Since P0 is invertible, one can introduce the so-called recursion operator N := P1 P0−1, whose Nijenhius torsion also vanishes. It turns out to be the complete lift of L (see, e.g., [24]), and it is uniquely determined by the condition dθL (X, Y ) = ω(NX, Y ) for all vector fields X, Y on T ∗ Q. An easy computation shows that     l ∂ ∂ ∂Li ∂Llk i ∂ N = Lk − yl − ∂x ∂xi ∂xk ∂xi ∂yi  k ∂ ∂ N = Lki . ∂yk ∂yi As pointed out in [12] (see also [2, 8] and the references cited therein), the geometry of such bi-Hamiltonian manifolds—often called ωN-manifolds— can be successfully exploited to characterize the Hamiltonian system that are separable in canonical coordinates in which N is diagonal. We conclude this section by recalling that the functions Hk := 1 1 trLk = trN k k 2k (3) form a bi-Hamiltonian hierarchy on T ∗ Q, that is, P1 dHk = P0 dHk+1 for all k ≥ 1. This follows from N ∗ dHk = dHk+1, where N ∗ is the transpose of N, and is well-known to imply the involutivity (with respect to both Poisson brackets) of the Hk . 3 A Bi-Hamiltonian structure on T ∗gl(n) In this section we consider a particular case of the general construction described in the previous section. The manifold Q is the set gl(n) of real n × n matrices, and the (1, 1) torsionless tensor field is defined as LA : V 7→ AV , (4) where A ∈ gl(n) and V ∈ TA gl(n) ≃ gl(n). It is known that the torsion of L vanishes (and one can easily check it by writing (1) for constant vector fields). The cotangent bundle T ∗ gl(n) ≃ gl(n) × gl(n)∗ can be identified with gl(n) × gl(n) by means of the pairing given by the trace of the product. 4 Thus on gl(n)×gl(n) we have a bi-Hamiltonian structure, whose first Poisson bracket is associated with the canonical symplectic form ω0 = tr (dB ∧ dA), where (A, B) ∈ gl(n) × gl(n). In order to determine the second Poisson tensor, we have to consider the 1-form θL = tr (BA dA) and to compute ωL = dθL = tr (dB ∧ A dA + B dA ∧ dA) . Let F1 , F2 be two real functions on gl(n) × gl(n), and let (ξ1 , η1 ) and (ξ2 , η2 ) be their differentials. Since the canonical Poisson tensor P0 acts on a covector (ξ, η) as        η ξ 0 I ξ   =   , P0 :   7→  −ξ η −I 0 η (5) the corresponding Hamiltonian vector fields (with respect to ω) are XFi = (ηi , −ξi ), for i = 1, 2. Therefore {F1 , F2 }1 = ωL (XF1 , XF2 ) = tr (A(η1 ξ2 − η2 ξ1 ) + B[η1 , η2 ]) , so that the second Poisson tensor is        ξ 0 A· ξ Aη   =   , P1 :   7→  η − · A [B, ·] η −ξA + [B, η] (6) (7) and the recursion operator N = P1 P0 −1 and its transpose are given by     ·A [·, B] A· 0  ,  . N = N∗ =  [B, ·] ·A 0 A· From the expression of N ∗ it is evident that the functions Hk = k1 tr Ak , for k ≥ 1, form a bi-Hamiltonian hierarchy. It can be easily checked that this hierarchy coincide with the one mentioned at the end of Section 2, namely, that tr Ak = tr Lk . The corresponding vector fields Xk := −P0 dHk are given by (Xk )(A,B) = (0, Ak−1). Remark 2 One can check that if T ∗ GL(n, R) is seen as an open subset of gl(n) × gl(n) by means of left translations, then the canonical symplectic structure of T ∗ GL(n, R) takes the form (7). 5 Remark 3 Another interesting interpretation of P1 is as follows. Let us consider the Cartesian product gl(n) × gl(n) with the Lie bracket [(A1 , B1 ), (A2 , B2 )] = (B1 A2 − B2 A1 , [B1 , B2 ]) , (8) i.e., the semidirect product of gl(n) with the abelian structure and gl(n) with its usual Lie bracket. Then P1 is the Lie-Poisson structure associated with (8) on the dual space (gl(n) × gl(n))∗ ≃ gl(n) × gl(n), where the identification is given by (A, B) 7→ tr (A·) + tr (B·) . Moreover, in the terminology of, e.g., [1], P0 is the “frozen Lie-Poisson structure” at the point (I, 0). 4 The first projection The main aim of this paper is to show that the bi-Hamiltonian structure of the Calogero-Moser system is a reduction of the one presented in the previous section. The first step is to notice that the Poisson pair (P0 , P1 ) is invariant with respect to the action of G = GL(n, R) on gl(n) × gl(n) given by the simultaneous conjugation:
(g, (A, B)) 7→ gAg −1, gBg −1 . This is obvious for the canonical Poisson tensor P0 , since this action is the lifting to the cotangent bundle of the action (g, A) 7→ gAg −1 on gl(n). Since the latter leaves invariant the tensor field L, the Poisson tensor P1 is invariant too. We would like to obtain a nice quotient, so we consider G acting on the open subset M ⊂ gl(n) × gl(n) formed by the pairs (A, B) such that: • A and B have real distinct eigenvalues; • if {vi }i=1,...,n is an eigenvector basis of B, then Avi ∈ / hv1 , . . . , vˆj , . . . , vn i for all j 6= i. As usual, h. . . i denotes the linear span and vˆj means that vj is not included in the list; • the same condition as before with A and B exchanged. It is clear that M is invariant under the action of G. Moreover, the description of the quotient M/G is made easy by the existence of a subset P ⊂ M 6 S intersecting every orbit in one point. It is given by P = ǫi ∈{+,−} P(ǫ1 ,...,ǫn−1 ) , where P(ǫ1 ,...,ǫn−1 ) is the set of pairs (A, B) ∈ M such that B is diagonal with Bii < Bjj if i < j, and Ai+1,i > 0, Ai,i+1 = ǫi Ai+1,i for all i = 1, . . . , n − 1. For example, if n = 2 we have P = P+ ∪ P− , where the elements of P+ are those in M of the form     A A21 B 0  11  ,  11  , A21 A22 0 B22 with A21 > 0 and B11 < B22 , while the elements of P− are those in M of the form     A −A21 B 0  11  ,  11  , A21 A22 0 B22 again with A21 > 0 and B11 < B22 . Proposition 4 Every orbit of G in M intersects P in just one point. Moreover, for all (A, B) ∈ P the tangent space T(A,B) M is the direct sum of T(A,B) P and the tangent space to the orbit. Proof. Given an orbit of G in M, it is clearly possible to find a point (A, B) on such orbit with B diagonal and Bii < Bjj for all i < j. (9) Then, again by the definition of M, one has that Aij 6= 0 if i 6= j. Because of (9) we can still act on (A, B) only by an invertible diagonal matrix g = diag(d1 , . . . , dn ). We have to show that one can choose the di in such a way that (gAg −1, gBg −1) = (gAg −1, B) ∈ P(ǫ1 ,...,ǫn−1 ) for some ǫi = ±1. Since (gAg −1)ij = di Aij dj −1 , this means that di+1 Ai+1,i di −1 > 0 , di Ai,i+1 di+1 −1 = ǫi di+1 Ai+1,i di −1 . Therefore ǫi is determined by the sign of Ai,i+1 /Ai+1,i , and s di Ai+1,i = ± ǫi , di+1 Ai,i+1 where the ± has to be chosen in such a way that di+1 Ai+1,i di −1 > 0. In this way we have found the matrix g up to a multiple, and the first part of the claim is proved. 7 Let us fix now a point (A, B) ∈ P(ǫ1 ,...,ǫn−1 ) and a tangent vector (V, W ) ∈ T(A,B) M. We have to show that (V, W ) can be uniquely decomposed as (V, W ) = (Ȧ, Ḃ) + ([A, ξ], [B, ξ]) , (10) where (Ȧ, Ḃ) ∈ T(A,B) P(ǫ1 ,...,ǫn−1 ) and ξ ∈ gl(n, R). Since Ḃ is diagonal, we immediately have that the off-diagonal entries of ξ are given by ξij = Wij /(Bii − Bjj ). Then we have to impose the conditions Ȧi,i+1 = ǫi Ȧi+1,i , getting the following equations, n X (ξij Aj,i+1 − Aij ξj,i+1) = ǫi j=1 n X (ξi+1,j Aji − Ai+1,j ξji ) − Vi,i+1 + ǫi Vi+1,i , j=1 for all i = 1, . . . , n − 1. Since Ai,i+1 = ǫi Ai+1,i , we obtain X 2ǫi Ai+1,i (ξii − ξi+1,i+1 ) = − (ξij Aj,i+1 + ǫi ξji Ai+1,j ) j6=i + X (ξj,i+1Aij + ǫi ξi+1,j Aji ) − Vi,i+1 + ǫi Vi+1,i , j6=i+1 for all i = 1, . . . , n−1, namely, (n−1) equations for the variables ξ11 , . . . , ξnn . Thanks to the fact that Ai+1,i > 0, they can be solved, and the solution is unique up to a (common) additive constant. This shows the uniqueness of the vector ([A, ξ], [B, ξ]), tangent to the orbit, and of the decomposition (10). QED Remark 5 It is clear from the previous proof that one can also identify M/G with the submanifold P ′ ⊂ M, whose definition is the same of P, but with the matrices A and B exchanged. Remark 6 The quotient space defined by simultaneous conjugation on ktuples of matrices has been the subject of important investigations by Artin, Procesi, Razmyslov, and others (see, e.g., [15]). For our purposes, it is convenient to restrict to the open subset M, and in this case an explicit description of the quotient is possible in terms of the transversal submanifold P. Next we consider the vector fields on
Xk of the bi-Hamiltonian hierarchy 1 k−1 M, that is, (Xk )(A,B) = 0, A . Since their Hamiltonians Hk = k tr Ak 8 and the bi-Hamiltonian structure are invariant with respect to the action of G, the Xk can be projected on M/G. Their projections are the vector fields associated with the Hamiltonians Hk (seen as functions on the quotient) and the reduced bi-Hamiltonian structure. We can exploit the identification between M/G and the submanifold P ⊂ M in order to explicitly write the projected vector fields. Indeed, we have just seen that we can uniquely find (∂k A, ∂k B) ∈ T(A,B) P and ([A, ξk ], [B, ξk ]), tangent to the orbit passing through (A, B) ∈ P, such that (0, Ak−1) = (∂k A, ∂k B) + ([A, ξk ], [B, ξk ]) . This shows that ∂k A = [ξk , A] , ∂k B = [ξk , B] + Ak−1 , (11) i.e., the projected flows possess a Lax representation. In the next section we will perform a second reduction and we will show that the flows (11) give rise to the (attractive) Calogero-Moser flows. Remark 7 The deduction of the Lax equations (11) is well-known (see, e.g., [19]). Notice however that such equations describe flows on the (n2 + 1)dimensional manifold P ≃ M/G and so they are an extension of the usual Lax representation of the Calogero-Moser system. We close this section with an interesting description of the bi-Hamiltonian structure on the quotient M/G (see [6], where only the first Poisson structure is considered). Let F1 = tr (a1 · · · ar ) and F2 = tr (b1 · · · bs ), where ai and bj are either A or B. Then   X X dF1 =  ai+1 · · · ar a1 · · · ai−1 , aj+1 · · · ar a1 · · · aj−1  j:aj =B i:ai =A and therefore we have the so-called necklace bracket formula X {F1 , F2 }0 = tr (ai+1 · · · ar a1 · · · ai−1 bj+1 · · · bs b1 · · · bj−1 ) (i,j):ai =B,bj =A − X tr (bj+1 · · · bs b1 · · · bj−1 ai+1 · · · ar a1 · · · ai−1 ) . (i,j):ai =A,bj =B 9 (12) As far as the second Poisson bracket is concerned, we have from (6) that X {F1 , F2 }1 = tr (Aai+1 · · · ar a1 · · · ai−1 bj+1 · · · bs b1 · · · bj−1 ) (i,j):ai =B,bj =A − X tr (Abj+1 · · · bs b1 · · · bj−1 ai+1 · · · ar a1 · · · ai−1 ) (i,j):ai =A,bj =B + X tr (B[ai+1 · · · ar a1 · · · ai−1 , bj+1 · · · bs b1 · · · bj−1 ]) (i,j):ai =B,bj =B = X tr (ai ai+1 · · · ar a1 · · · ai−1 bj+1 · · · bs b1 · · · bj−1 ) (i,j):ai =B,bj =A − X tr (bj+1 · · · bs b1 · · · bj−1 ai+1 · · · ar a1 · · · ai−1 ai ) (i,j):ai =A,bj =B + X tr (ai ai+1 · · · ar a1 · · · ai−1 bj+1 · · · bs b1 · · · bj−1 ) (i,j):ai =B,bj =B − X tr (bj bj+1 · · · bs b1 · · · bj−1 ai+1 · · · ar a1 · · · ai−1 ) . (i,j):ai =B,bj =B (13) Now let us pass from the (n2 + 1)-dimensional quotient M/G ≃ P to the phase space of the Calogero-Moser system. 5 The second projection In this section we will perform a second reduction of the bi-Hamiltonian structure on M/G ≃ P. The starting point is the observation that the invariant functions Ik (A, B) = 1 tr Ak = Hk (A, B) , k Jk (A, B) = tr (Ak−1 B) , for k = 1, . . . , n, form a Poisson subalgebra with respect to both Poisson brackets. Indeed, by direct computation or using the necklace bracket formulas (12-13), one finds that {Ik , Il }0 = 0 , {Jl , Ik }0 = (k + l − 2)Ik+l−2 , {Jk , Jl }0 = (l − k)Jk+l−2 , {Ik , Il }1 = 0 , {Jl , Ik }1 = (k + l − 1)Ik+l−1 , {Jk , Jl }1 = (l − k)Jk+l−1 , (14) 10 with the exception that {J1 , I1 }0 = n. In any case, the Cayley-Hamilton theorem implies that all the right-hand sides of (14) can be written in terms of I1 , . . . , In , J1 , . . . , Jn . Thus both Poisson brackets can be further projected on the quotient space defined by the map π : M/G ≃ P → R2n whose components are the functions I1 , . . . , In , J1 , . . . , Jn . Proposition 8 The map π is a submersion, i.e., its differential is surjective at every point of M/G. Proof. It is convenient to identify M/G with P ′ and to consider, among the coordinates on P ′ , the diagonal entries (λ1 , . . . , λn ) of the (diagonal) matrix A and the diagonal entries (µ1 , . . . , µn ) of B. Then n 1X k Ik = λl , k l=1 Jk = n X µl λl k−1 , l=1 which implies that det  ∂I  ∂λ ∂I ∂µ  ∂J ∂λ  ∂J ∂µ 6= 0 since the λi are distinct. This shows that the differential of π is surjective. QED The previous proposition entails that the image of π is an open subset U ⊂ R2n , which is in 1-1 correspondence with the second quotient space (by its very definition). Our final step is to prove that the projection on U gives rise to the phase space of the attractive Calogero-Moser system, with its biHamiltonian flows. To do this, we recall once more that the (first) quotient M/G can be identified with the submanifold P ⊂ M and we restrict to its connected component P(−,...,−) , that is, the set of pairs (A, B) ∈ M such that B is diagonal with Bii < Bjj if i < j, and Ai+1,i > 0, Ai,i+1 = −Ai+1,i for all i = 1, . . . , n − 1. Then we introduce a submanifold Q ⊂ P(−,...,−) which will be shown to be in 1-1 correspondence with an open subset of the second quotient space. The elements of Q are the pairs (L, diag(x1 , . . . , xn )) ∈ P 1 such that xi < xj if i < j, and Lij = xi −x if i 6= j. If we put Lii = yi , we j 11 obtain the Lax matrix of the attractive Calogero-Moser system:   1 1 y · · · x1 −xn x1 −x2  1  ..   1 .. .  x2 −x1 y2 .   . L=  .. . .. .. ..  . .  .    1 1 ··· yn xn −x1 xn −x2 Let us also introduce the submanifold Q′ ⊂ P ′ whose elements are the pairs 1 if i 6= j. (diag(λ1 , . . . , λn ), L′ ) ∈ P ′ such that λi < λj if i < j, and L′ij = λj −λ i In order to identify Q with a subset of the second quotient space, we need the following result. It is a restatement of Proposition 2.6 in [6], but we give its proof for the reader’s sake. Proposition 9 If ρ : M → M/G is the canonical projection, then ρ(Q) coincides with ρ(Q′ ), and is formed by the orbits of the pairs (A, B) such that the rank of [B, A] + I is 1. Proof. We notice that Q = {(A, B) ∈ P | [B, A] = µ} and Q′ = {(A, B) ∈ P ′ | [B, A] = µ} , where µij = 1−δij . Thus, the elements (A, B) in the orbits passing through Q and Q′ satisfy the condition rank([B, A] + I) = 1. Conversely, let us suppose that (A, B) ∈ M and the above condition holds. We can also suppose that B has already been diagonalized. Since the rank of K := [B, A] + I is 1, there exist ai , bi ∈ R, i = 1, . . . , n, such that Kij = ai bj . From [B, A]ij = (Bii − Bjj )Aij we have that Kii = 1 and therefore bi = ai −1 . By acting with diag(a1 , . . . , an ), the entries of the matrix K all become 1 and so (A, B) is mapped into Q. This shows that the orbits in ρ(Q) are precisely those of the pairs (A, B) such that rank([B, A] + I) = 1. Diagonalizing A instead of B, one proves that the same is true for ρ(Q′ ). QED Corollary 10 The restriction to Q of the map π = (I1 , . . . , In , J1 , . . . , Jn ) is injective. 12 Proof. Since π is an invariant map, we can exploit the identification between Q and Q′ given by the previous proposition and show that π is injective on Q′ . If (D ′ , L′ ) ∈ Q′ , with   1 1 · · · λn −λ1 µ λ2 −λ1  1  ..   1 .. .   . µ 2 λ1 −λ2  D ′ = diag(λ1 , . . . , λn ) , L′ =  ..  ,  .. .. .. . .  . .    1 1 ··· µn λ1 −λn λ2 −λn then n 1X k λl , Ik (D , L ) = k l=1 ′ ′ ′ ′ Jk (D , L ) = n X µl λl k−1 . l=1 To conclude that π is injective, we simply have to recall that λi < λj if i < j. QED Remark 11 From the proof of Proposition 9 it is also clear that ρ(Q) = ρ(Q′ ) = {orbits of the pairs (A, B) such that [B, A] = µ}. We have thus shown that an open subset of the quotient defined by the map π can be identified with the submanifold Q, that is, the phase space of the Calogero-Moser system. The bi-Hamiltonian structure on Q is given by the Poisson brackets (14). Remark 12 Formulas (14) appeared in [17] (see also [16]), in the context of the repulsive Calogero-Moser system. In that paper the construction of the bi-Hamiltonian structure on the Calogero-Moser phase space starts from the Lax representation of the system and uses a special class of coordinates defined on regular bi-Hamiltonian manifolds (the so-called Darboux-Nijenhuis coordinates, see [17, 16, 8]). On the contrary, here we recover both the Poisson brackets (14) and the Lax representation from the Poisson pair (5-7) on gl(n) × gl(n). Now we consider the (projected) bi-Hamiltonian hierarchy on P(−,...,−) ⊂ M/G. Since the Hamiltonians are just the functions Hk = Ik , this hierarchy further projects on the second quotient space. In particular, it gives rise to 13 bi-Hamiltonian vector fields on Q, which we will soon see to be those of the attractive Calogero-Moser system. In principle, to write these vector fields we should project the flows (11), as we did after Proposition 4. But they are already tangent to Q, as shown in Proposition 13 Let (A, B) ∈ Q and let (∂k A, ∂k B) ∈ T(A,B) P be given by (11), that is, ∂k A = [ξk , A] , ∂k B = [ξk , B] + Ak−1 . Then (∂k A, ∂k B) ∈ T(A,B) Q for all k ≥ 1. Proof. We know that Q = {(A, B) ∈ P | [B, A] = µ} , where µij = 1 − δij . This entails that (∂k A, ∂k B) ∈ T(A,B) Q if and only if [∂k A, B] + [A, ∂k B] = 0 at the points of Q. But this is equivalent to the assertion that [ξk , µ] = 0.
Let us prove this fact, introducing a matrix ξ such that ξij = (ξk )ij = Ak−1 ij /(xi − xj ) for i 6= j, and ξii = − 1X (ξil + ξli ) . 2 l6=i (15) Since B is diagonal, we have that ∂k B = [ξk , B] + Ak−1 = [ξ, B] + Ak−1 . (16) At the end, it will turn out that ξ is a possible choice for ξk (recall from the proof of Proposition 4 that ξk is determined up to a multiple of the identity matrix). Now let us show that [ξ, µ] = 0. Indeed, [ξ, µ] = [ξ, [B, A]] = [A, [B, ξ]] + [B, [ξ, A]] = −[A, ∂k B] + [B, [ξ, A]] , so that, putting ∂k B = diag(ẋ1 , . . . , ẋn ), we have [ξ, µ]ij = (ẋi − ẋj )Aij + (xi − xj )[ξ, A]ij . (17) Thus [ξ, µ]ii = 0 for all i = 1, . . . , n. Moreover, (15) implies that, for i 6= j, [ξ, µ]ij = 1X 1X (ξil − ξli ) + (ξjm − ξmj ) = [ξ, µ]ii + [ξ, µ]jj . 2 l 2 m 14 Therefore we have that [ξ, µ] = 0. In order to finish the proof, we have to show that ξ is a possible choice for ξk , i.e., that [ξ, A]i,i+1 = −[ξ, A]i+1,i for all i = 1, . . . , n − 1. But from (17), the vanishing of [ξ, µ], and the fact that Aij = −Aji for i 6= j, we have that 0 = (xi − xj ) ([ξ, A]ij + [ξ, A]ji ) and therefore [ξ, A]ij = −[ξ, A]ji . QED As it is well-known, the Calogero-Moser system is given by the second flow P 2 ∂2 A = [ξ2 , A], where (ξ2 )ij = 1/(xi − xj ) for i 6= j and (ξ2 )ii = − j6=i (ξ2 )ij . In general ξk is not symmetric for k > 2. Remark 14 In the repulsive case, corresponding to the connected component P(+,...,+) ⊂ M/G, one can still introduce the submanifold   1 Q+ = (L, diag(x1 , . . . , xn )) | xi < xj and Lij = Lji = if i < j xi − xj and show that it can be identified with (an open subset of) the second quotient space. However, if n > 2 the flows (11) are not tangent to Q+ and therefore they need to be projected on Q+ , where they assume a more complicated form. Remark 15 It is well-known [14, 6, 22] that the first Poisson structure P0 can be reduced on the phase space of the Calogero-Moser system by means of the Marsden-Weinstein reduction and that it gives rise to the canonical structure in the coordinates (xi , yj ). Our reduction consists in a double projection, and has the same effect on P0 . However, we can reduce also the second Poisson structure P1 , on which the Marsden-Weinstein reduction cannot be performed, since it employs the moment map (A, B) → [A, B] of P0 . Notice that this moment map appears in the proof of Proposition 9. Remark 16 We have used two projections to obtain the bi-Hamiltonian structure of the Calogero-Moser system from the one on T ∗ gl(n). Of course, such projections can be composed and we could have found directly the Poisson pair on Q. We decided to study also the first quotient M/G because it is more natural and we think that it might be of interest on its own. 15 6 Example: n = 2 In this section we consider the 2-particle Calogero-Moser system in order to exemplify our construction. The starting point is the set M whose elements are pairs (A, B) of matrices in gl(2, R) such that A and B have real distinct eigenvalues and no common eigenvector. The first quotient space M/G can be identified with the 5-dimensional manifold P = P+ ∪ P− (see Section 4) or with P ′ = P+′ ∪ P−′ , where the elements of P+′ are those in M of the form     A 0 B B12  11  ,  11  , 0 A22 B12 B22 with A11 < A22 and B12 > 0, while the elements of P−′ are those in M of the form     A11 0 B11 B12   , , 0 A22 −B12 B22 again with A11 < A22 and B12 > 0. There are only two independent vector fields X1 and X2 in the bi-Hamiltonian hierarchy on M, corresponding to the invariant functions H1 = tr A and H2 = 21 tr A2 . The equations on M/G ≃ P are given by (11), with ξ1 = I and     A21 A21 0 0 − B22 −B11  B22 −B11  ξ2 =  on P− , ξ2 =  on P+ . A21 A21 0 0 B22 −B11 B22 −B11 The second projection π : P → R4 is given by π = (I1 , I2 , J1 , J2 ), where I1 = H1 = tr A, 1 I2 = H2 = tr A2 , 2 J1 = tr B, J2 = tr (AB) . One can check that the image of π is the set   1 2 4 U = (I1 , I2 , J1 , J2 ) ∈ R s.t. 4I2 − I1 > 0, J2 − I1 J1 6= 0 . 2 The restriction of π to      1   y1 x 0 x1 −x2   1  s.t. x1 < x2 , |y1 − y2 |(x2 − x1 ) > 2 Q =  , 1   y2 0 x2 x2 −x1 16 is a bijection onto   1 2 4 V = (I1 , I2 , J1 , J2 ) ∈ R s.t. 4I2 − I1 > 0, |J2 − I1 J1 | > 1 . 2 The Poisson brackets on Q are given by {I1 , I2 }0 = 0, {J1 , I1 }0 = 2, {J2 , I2 }0 = 2I2 , {I1 , I2 }1 = 0, {J1 , I2 }0 = {J2 , I1 }0 = I1 , {J1 , J2 }0 = J1 , {J1 , I1 }1 = I1 , (18) {J1 , I2 }1 = {J2 , I1 }1 = 2I2 , {J2 , I2 }1 = 3I3 = 3I1 I2 − 12 I1 3 , {J1 , J2 }1 = J2 . In terms of the physical coordinates (x1 , x2 , y1 , y2 ), the first bracket is the canonical one, while {x1 , x2 }1 = 2x12 , ∆ {x1 , y1 }1 = y1 + {y2 , x1 }1 = {x2 , y1 }1 = (y1 −y2 )x12 2 , ∆ (y1 −y2 )x12 2 , ∆ {x2 , y2}1 = y2 − (y1 −y2 )x12 2 , ∆ {y1 , y2 }1 = −x12 3 . (19) where x12 = 1/(x1 − x2 ) and ∆ = 4x12 − (y1 − y2 ) . Notice that the term ∆ appearing in the denominators of (19) is the discriminant of the characteristic polynomial of   2 y1  1 x2 −x1 2 1 x1 −x2  . y2 This means that one cannot reduce the second Poisson bracket on the whole Calogero-Moser phase space, but only on its open subset Q. 7 Final remarks In this paper we have shown that the Poisson pair of the (rational, attractive) Calogero-Moser system is a reduction of a very natural bi-Hamiltonian structure on T ∗ gl(n, R). A first possible development of this result is to extend this construction to other Calogero-Moser systems, such as the trigonometric one (see also Remark 2) and those associated to (root systems of) simple Lie algebras [19]. Secondly, it would be interesting to investigate, from the point of 17 view of bi-Hamiltonian geometry, the problem of duality between CalogeroMoser systems [20, 9] and, more generally, between integrable systems. In particular, on M there is another bi-Hamiltonian structure, obtained by exchanging A with B. (In other words, one can look at B as the “point” in gl(n, R) and A as the “covector” in TB∗ gl(n, R) ≃ gl(n, R), and consider the (1,1) tensor field V 7→ BV on gl(n, R).) The functions Hk′ = k1 tr B k , for k ≥ 1, form a bi-Hamiltonian hierarchy with respect to the new Poisson pair. Being invariant with respect to the action of G, such pair can be projected on the (first) quotient space M/G, along with its hierarchy. 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