Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case
aa r X i v : . [ m a t h - ph ] J a n Bi-Hamiltonian structure of spin Sutherland models:the holomorphic case
L. Feh´er a,ba
Department of Theoretical Physics, University of SzegedTisza Lajos krt 84-86, H-6720 Szeged, Hungarye-mail: [email protected] b Department of Theoretical Physics, WIGNER RCP, RMKIH-1525 Budapest, P.O.B. 49, Hungary
Abstract
We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hier-archy based on collective spin variables. The construction relies on Poisson reductionof a bi-Hamiltonian structure on the holomorphic cotangent bundle of GL( n, C ), whichitself arises from the canonical symplectic structure and the Poisson structure of theHeisenberg double of the standard GL( n, C ) Poisson–Lie group. The previously obtainedbi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recoveredon real slices of the holomorphic spin Sutherland model. Contents G × G Introduction
The theory of integrable systems is an interesting field of mathematics motivated by influentialexamples of exactly solvable models of theoretical physics. For reviews, see e.g. [3, 4, 18, 21].There exist several approaches to integrability. One of the most popular ones in connectionwith classical integrable systems is the bi-Hamiltonian method, which originates from thework of Magri [15] on the KdV equation, and plays an important role in generalizations of thisinfinite dimensional bi-Hamiltonian system [7]. As can be seen in the reviews, among finitedimensional integrable systems the central position is occupied by Toda models and the modelsthat carry the names of Calogero, Moser, Sutherland, Ruijsenaars and Schneider. The Todamodels have a relatively well developed bi-Hamiltonian description [21]. The Calogero–Mosertype models and their generalizations are much less explored from this point of view, exceptfor the rational Calogero–Moser model [1, 6, 10]. In our recent work [11, 12] we made a steptowards improving this situation by providing a bi-Hamiltonian interpretation for a familyof spin extended hyperbolic and trigonometric Sutherland models. In these references weinvestigated real-analytic Hamiltonian systems, and here wish to extend the pertinent resultsto the corresponding complex holomorphic case.Specifically, the aim of this paper is to derive a bi-Hamiltonian description for the hierarchyof holomorphic evolution equations of the form˙ Q = ( L k ) Q, ˙ L = [ R ( Q )( L k ) , L ] , ∀ k ∈ N , (1.1)where Q is an invertible complex diagonal matrix of size n × n , L is an arbitrary n × n complexmatrix, and the subscript 0 means diagonal part. The eigenvalues Q j of Q are required to bedistinct, ensuring that the formula R ( Q ) := 12 (Ad Q + id)(Ad Q − id) − , with Ad Q ( X ) := QXQ − , (1.2)gives a well-defined linear operator on the off-diagonal subspace of gl( n, C ). By definition, R ( Q ) ∈ End(gl( n, C )) vanishes on the diagonal matrices, and one can recognize it as the basicdynamical r -matrix [5, 8]. Like in the real case [11], it follows from the classical dynamicalYang–Baxter equation satisfied by R ( Q ) that the evolutional derivations (1.1) pairwise com-mute if they act on such ‘observables’ f ( Q, L ) that are invariant with respect to conjugationsof L by invertible diagonal matrices.The system (1.1) has a well known interpretation as a holomorphic Hamiltonian system[14]. This arises from the parametrization L = p + ( R ( Q ) + 12 id)( φ ) , (1.3)where p is an arbitrary diagonal and φ is an arbitrary off-diagonal matrix. The diagonal entries p j of p and q j in Q j = e q j form canonically conjugate pairs. The vanishing of the diagonalpart of φ represents a constraint on the linear Poisson space gl( n, C ), and this is responsiblefor the gauge transformations acting on L as conjugations by diagonal matrices. The k = 1member of the hierarchy (1.1) is governed by the standard spin Sutherland Hamiltonian H Suth ( Q, p, φ ) = 12 tr (cid:0) L ( Q, p, φ ) (cid:1) = 12 n X i =1 p i + 18 X k = l φ kl φ lk sinh q k − q l . (1.4)For this reason, we may refer to (1.1) as the holomorphic spin Sutherland hierarchy.It is also known (see e.g. [17]) that the holomorphic spin Sutherland hierarchy is a reductionof a natural integrable system on the cotangent bundle M := T ∗ GL( n, C ) equipped with2ts canonical symplectic form. Before reduction, the elements of M can be represented bypairs ( g, L ), where g belongs to the configuration space and ( g, L ) L is the moment mapfor left-translations. The Hamiltonians tr( L k ) generate an integrable system on M , whichreduces to the spin Sutherland system by keeping only the observables that are invariant undersimultaneous conjugations of g and L by arbitrary elements of GL( n, C ). This procedure iscalled Poisson reduction. We shall demonstrate that the unreduced integrable system on M possesses a bi-Hamiltonian structure that descends to a bi-Hamiltonian structure of the spinSutherland hierarchy via the Poisson reduction .A holomorphic (or even a continuous) function on M that is invariant under the GL( n, C )action (3.1) can be recovered from its restriction to M reg0 , the subset of M consisting of the pairs( Q, L ) with diagonal and regular Q ∈ GL( n, C ). Moreover, the restricted function inheritsinvariance with respect to the normalizer of the diagonal subgroup G < GL( n, C ), whichincludes G . This explains the gauge symmetry of the hierarchy (1.1), and lends justificationto the restriction on the eigenvalues of Q .The bi-Hamiltonian structure on M involves in addition to the canonical Poisson bracketassociated with the universal cotangent bundle symplectic form another one that we constructfrom Semenov-Tian-Shansky’s Poisson bracket of the Heisenberg double of GL( n, C ) endowedwith its standard Poisson–Lie group structure [19]. Surprisingly, we could not find it inthe literature that the canonical symplectic structure of the cotangent bundle M can becomplemented to a bi-Hamiltonian structure in this manner. So this appears to be a novelresult, which is given by Theorem 2.1 and Proposition 2.3 in Section 2. The actual derivation ofthe second Poisson bracket (2.10) is relegated to an appendix. The heart of the paper is Section3, where we derive the bi-Hamiltonian structure of the system (1.1) by Poisson reduction.The main results are encapsulated by Theorem 3.4 and Proposition 3.6. The first reducedPoisson bracket (3.26) is associated with the spin Sutherland interpretation by means of theparametrization (1.3). The formula of the second reduced Poisson bracket (3.27) contains thestandard constant classical r -matrix, given in (2.3), which stems from the Poisson–Lie groupstructure on GL( n, C ). After deriving the holomorphic bi-Hamiltonian structure in Section3, we shall explain in Section 4 that it allows us to recover the bi-Hamiltonian structures ofthe hyperbolic and trigonometric real forms derived earlier by different means [11, 12]. In thefinal section, we summarize the main results once more, and highlight a few open problems. Let us denote G := GL( n, C ) and equip its Lie algebra G := gl( n, C ) with the trace form h X, Y i := tr( XY ) , ∀ X, Y ∈ G . (2.1)Any X ∈ G admits the unique decomposition X = X > + X + X < (2.2)into strictly upper triangular part X > , diagonal part X , and strictly lower triangular part X < . We shall use the standard solution of the modified classical Yang–Baxter equation on G , r ∈ End( G ) given by r ( X ) := 12 ( X > − X < ) , (2.3)and define also r ± := r ±
12 id . (2.4)3ur aim is to present two holomorphic Poisson structures on the complex manifold M := G × G = { ( g, L ) | g ∈ G, L ∈ G} . (2.5)Denote Hol( M ) the commutative algebra of holomorphic functions on M . For any F ∈ Hol( M ),introduce the G -valued derivatives ∇ F and d F by the defining relations h∇ F ( g, L ) , X i = ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 F ( e zX g, L ) , h d F ( g, L ) , X i = ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 F ( g, L + zX ) , (2.6)where z is a complex variable and X ∈ G is arbitrary. We also have ∇ ′ F ( g, L ) = g − ( ∇ F ( g, L )) g. (2.7)In addition, it will be convenient to define the G -valued functions ∇ F and ∇ ′ F by( ∇ F )( g, L ) := Ld F ( g, L ) , ( ∇ ′ F )( g, L ) := ( d F ( g, L )) L. (2.8) Theorem 2.1.
For holomorphic functions
F, H ∈ Hol( M ) , the following formulae define twocompatible Poisson brackets: { F, H } ( g, L ) = h∇ F, d H i − h∇ H, d F i + h L, [ d F, d H ] i , (2.9) and { F, H } ( g, L ) = h r ∇ F, ∇ H i − h r ∇ ′ F, ∇ ′ H i (2.10)+ h∇ F − ∇ ′ F, r + ∇ ′ H − r − ∇ H i + h∇ F, r + ∇ ′ H − r − ∇ H i − h∇ H, r + ∇ ′ F − r − ∇ F i , where the derivatives are evaluated at ( g, L ) , and we put rX for r ( X ) .Proof. The first bracket is easily seen to be the Poisson bracket associated with the canonicalsymplectic form of the holomorphic cotangent bundle of G , which is identified with G × G using right-translations and the trace form on G . The antisymmetry and the Jacobi identityof the second bracket can be verified by direct calculation. More conceptually, they followfrom the fact that locally, in a neighbourhood of ( n , n ) ∈ G × G , the second bracket canbe transformed into Semenov-Tian-Shansky’s [19] Poisson bracket on the Heisenberg doubleof the standard Poisson–Lie group G . This is explained in the appendix. One can check thatthe first bracket is the Lie derivative of the second bracket along the holomorphic vector field, W , on M whose integral curve through the initial value ( g, L ) is φ z ( g, L ) = ( g, L + z n ) , (2.11)where n is the unit matrix. This means that the relation { F, H } = W [ { F, H } ] − { W [ F ] , H } − { F, W [ H ] } (2.12)holds. It is enough to check this for the basic functions provided by the matrix elements of g and L for ( g, L ) ∈ M , which is easy. It is well known [9, 20] that the Lie derivative relationimplies the compatibility of the two Poisson brackets. Remark 2.2.
The first line in (2.10) represents the standard multiplicative Poisson structureon the group G . The second line of { , } can be recognized as the holomophic extensionof the well known Semenov-Tian-Shanksy bracket from G to G , where G is regarded as anopen submanifold of G . We recall that the Semenov-Tian-Shansky bracket originates from thePoisson–Lie group dual to G [3, 19]. 4enote by V iH ( i = 1 ,
2) the Hamiltonian vector field associated with the holomorphicfunction H through the respective Poisson bracket { , } i . For any holomorphic function, wehave the derivatives V iH [ F ] = { F, H } i . (2.13)We are interested in the Hamiltonians H m ( g, L ) := 1 m tr( L m ) , ∀ m ∈ N . (2.14) Proposition 2.3.
The vector fields associated with the functions H m are bi-Hamiltonian, sincewe have { F, H m } = { F, H m +1 } , ∀ m ∈ N , ∀ F ∈ Hol( M ) . (2.15) The derivatives of the matrix elements of ( g, L ) ∈ M give V H m [ g ] = V H m +1 [ g ] = L m g, V H m [ L ] = V H m +1 [ L ] = 0 , ∀ m ∈ N , (2.16) and the flow of V H m = V H m +1 through the initial value ( g (0) , L (0)) is ( g ( z ) , L ( z )) = (exp( zL (0) m ) g (0) , L (0)) . (2.17) Proof.
One has ∇ H m ( g, L ) = 0 , d H m ( g, L ) = L m − , ∀ m = 1 , , . . . , (2.18)and substitution into the formulae of Theorem 2.1 implies the claimed results.Like in the compact case [12], we call the H m ‘free Hamiltonians’ and conclude fromProposition 2.3 that they generate a bi-Hamiltonian hierarchy on the holomorphic cotangentbundle M . The essence of Hamiltonian symmetry reduction is that one keeps only the ‘observables’ thatare invariant with respect to the pertinent group action. Here, we apply this principle to theadjoint action of G on M , for which η ∈ G acts by the holomorphic diffeomorphism A η , A η : ( g, L ) ( ηgη − , ηLη − ) . (3.1)Thus we keep only the G invariant holomorphic functions on M , whose set is denotedHol( M ) G := { F ∈ Hol( M ) | F ( g, L ) = F ( ηgη − , ηLη − ) , ∀ ( g, L ) ∈ M , η ∈ G } . (3.2) Lemma 3.1.
Hol( M ) G is closed with respect to both Poisson brackets of Theorem 2.1.Proof. The closure is obvious for the canonical Poisson bracket { , } . To deal with the otherPoisson bracket, we note that for a G invariant function H the relation H ( ge zX , L ) = H ( e zX g, e zX Le − zX ) , ∀ z ∈ C , ∀ X ∈ G , (3.3)implies the identity ∇ ′ H = ∇ H + ∇ H − ∇ ′ H. (3.4)5y using this, if both F and H are invariant functions, then we can straightforwardly rewritethe formula (2.10) as2 { F, H } = h∇ F, ∇ H + ∇ ′ H i − h∇ H, ∇ F + ∇ ′ F i + h∇ F, ∇ ′ H i − h∇ H, ∇ ′ F i . (3.5)Formally, this is obtained from (2.10) by setting r to zero, i.e., r cancels from all terms byvirtue of (3.4). The derivatives of the G invariant functions are equivariant, ∇ i H ( ηgη − , ηLη − ) = η ( ∇ i H ( g, L ))) η − , i = 1 , , (3.6)and similar for ∇ ′ i H . Hence, (3.5) shows that if F, H are invariant, then so is { F, H } .We wish to characterize the Poisson algebras of the G invariant functions. To start, weconsider the diagonal subgroup G < G , G := { Q | Q = diag( Q , . . . , Q n ) , Q i ∈ C ∗ } , (3.7)and its regular part G reg0 , where Q i = Q j for all i = j . We let N < G denote the normalizer of G in G . The normalizer contains G as a normal subgroup, and the corresponding quotientis the permutation group, N /G = S n . (3.8)We also let G reg ⊂ G denote the dense open subset consisting of the conjugacy classes havingrepresentatives in G reg0 . Next, we define M reg := { ( g, L ) ∈ M | g ∈ G reg } (3.9)and M reg0 := { ( Q, L ) ∈ M | Q ∈ G reg0 } . (3.10)These are complex manifolds, equipped with their own holomorphic functions. Now we intro-duce the chain of commutative algebrasHol( M ) red ⊂ Hol( M reg0 ) N ⊂ Hol( M reg0 ) G . (3.11)The last two sets contain the respective invariant elements of Hol( M reg0 ), and Hol( M ) red con-tains the restrictions of the elements of Hol( M ) G to M reg0 . To put this in a more formalmanner, let ι : M reg0 → M (3.12)be the tautological embedding. Then pull-back by ι provides an isomorphism between Hol( M ) G and Hol( M ) red . Similar, we obtain the map ι ∗ : Hol( M reg ) G → Hol( M reg0 ) N , (3.13)which is injective. We believe that this map is surjective as well, but shall not use that. Definition 3.2.
Let f, h ∈ Hol( M ) red be related to F, H ∈ Hol( M ) G by f = F ◦ ι and h = H ◦ ι . In consequence of Lemma 3.1, we can define { f, h } red i ∈ Hol( M ) red by the relation { f, h } red i := { F, H } i ◦ ι, i = 1 , . (3.14)This gives rise to the reduced Poisson algebras (Hol( M ) red , { , } red i ).6he main goal of this paper is to derive formulae for the reduced Poisson brackets (3.14).To do so, we now note that any f ∈ Hol( M reg0 ) has the G -valued derivative ∇ f and the G -valued derivative d f , defined by h∇ f ( Q, L ) , X i = ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 f ( e zX Q, L ) , h d f ( Q, L ) , X i = ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 f ( Q, L + zX ) , (3.15)which are required for all X ∈ G , X ∈ G . For any Q ∈ G , the linear operator Ad Q : G → G acts as Ad Q ( X ) = QXQ − . Set G ⊥ := G < + G > , (3.16)where G < (resp. G > ) denotes the strictly lower (resp. upper) triangular subalgebra of G . Noticethat for Q ∈ G reg0 the operator (Ad Q − id) maps G ⊥ to G ⊥ in an invertible manner. Buildingon (2.2), we have the decomposition X = X + X ⊥ with X ⊥ = X < + X > , ∀ X ∈ G . (3.17)Using this, for any Q ∈ G reg0 , the ‘dynamical r -matrix’ R ( Q ) ∈ End( G ) is given by R ( Q ) X = 12 (Ad Q + id) ◦ (Ad Q − id) − |G ⊥ X ⊥ , ∀ X ∈ G , (3.18)and we remark its antisymmetry property hR ( Q ) X, Y i = −h X, R ( Q ) Y i , ∀ X, Y ∈ G . (3.19)Below, we shall also employ the shorthand[ X, Y ] R ( Q ) := [ R ( Q ) X, Y ] + [ X, R ( Q ) Y ] , ∀ X, Y ∈ G . (3.20) Lemma 3.3.
Consider f ∈ Hol( M reg0 ) N given by f = F ◦ ι , where F ∈ Hol( M reg ) G . Then thederivatives of f and F satisfy the following relations at any ( Q, L ) ∈ M reg0 : d F ( Q, L ) = d f ( Q, L ) , [ L, d f ( Q, L )] = 0 , (3.21) ∇ F ( Q, L ) = ∇ f ( Q, L ) − ( R ( Q ) + 12 id)[ L, d f ( Q, L )] . (3.22) Proof.
The equalities (3.21) hold since f is the restriction of F . In particular, it satisfies0 = ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 f ( Q, e zX Le − zX ) = h d f ( Q, L ) , [ X , L ] i = h [ L, d f ( Q, L )] , X i , ∀ X ∈ G . (3.23)Concerning (3.22), the equality of the G parts, ( ∇ F ( Q, L )) = ( ∇ f ( Q, L )) , is obvious.Then take any T ∈ G ⊥ , for which we have0 = ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 F ( e zT Qe − zT , e zT Le − zT ) = h T, (id − Ad Q − ) ∇ F ( Q, L ) + [
L, d F ( Q, L )] i . (3.24)Therefore (Ad Q − − id)( ∇ F ( Q, L )) ⊥ = [ L, d F ( Q, L )] ⊥ , (3.25)which implies (3.22). 7 heorem 3.4. For f, h ∈ Hol( M ) reg , the reduced Poisson brackets defined by (3.14) can bedescribed explicitly as follows: { f, h } red1 ( Q, L ) = h∇ f, d h i − h∇ h, d f i + h L, [ d f, d h ] R ( Q ) i , (3.26) and { f, h } red2 ( Q, L ) = 12 h∇ f, ∇ h + ∇ ′ h i − h∇ h, ∇ f + ∇ ′ f i (3.27)+ hR ( Q )( ∇ h − ∇ ′ h ) , r + ∇ ′ f − r − ∇ f i−hR ( Q )( ∇ f − ∇ ′ f ) , r + ∇ ′ h − r − ∇ h i , where all derivatives are taken at ( Q, L ) ∈ M reg0 , and the notation (2.8) is in force. Theseformulae give two compatible Poisson brackets on Hol( M ) red .Proof. Let us begin with the first bracket, and note that at (
Q, L ) ∈ M reg0 we have h∇ F, d H i = h∇ f, d h i − hR ( Q )[ L, d f ] , d h i − h [ L, d f ] , d h i , (3.28)since this follows from (3.22). Now the third term together with the analogous one coming from −h∇ H, d F i cancel the last term of (2.9). Taking advantage of (3.19), the terms containing R ( Q ) give the expression written in (3.26).Turning to the second bracket, let us begin by noting that ∇ F ( Q, L ) = Ad Q ∇ ′ F ( Q, L ) and r ◦ Ad Q = Ad Q ◦ r, (3.29)whereby the first line of (2.10) vanishes at ( Q, L ). Using (3.22) with [
L, d f ] = ∇ f − ∇ ′ f ,we can write h∇ F, r + ∇ ′ H − r − ∇ H i = 12 h∇ f, ∇ ′ h + ∇ h i−hR ( Q )( ∇ f − ∇ ′ f ) , r + ∇ ′ h − r − ∇ h i (3.30) − h∇ f − ∇ ′ f, r + ∇ ′ h − r − ∇ h i . This holds at (
Q, L ), since f, h are the restrictions of
F, H ∈ Hol( M ) G . To obtain the firstline on the right-hand side, we also used that ∇ f ( Q, L ) ∈ G . The third line of (3.30) andits counterpart coming from −h∇ H, r + ∇ ′ F − r − ∇ F i cancel the second line of (2.10). Thefirst and second lines in (3.30) and their counterparts ensuring antisymmetry then directlygive the claimed formula (3.27).The compatibility of the two reduced Poisson brackets is inherited from the compatibilityof the original Poisson brackets on Hol( M ) G . Remark 3.5.
It can be shown that the formulae of Theorem 3.4 give Poisson brackets onHol( M reg0 ) N and on Hol( M reg0 ) G as well.Now we turn to the reduction of the Hamiltonian vector fields (2.16) to vector fields on M reg0 . There are two ways to proceed. One may either directly associate vector fields to thereduced Hamiltonians using the reduced Poisson brackets, or can suitably ‘project’ the originalHamiltonian vector fields. Of course, the two methods lead to the same result.We apply the first method to the reduced Hamiltonians h m := H m ◦ ι ∈ Hol( M ) red , whichare given by h m ( Q, L ) = 1 m tr( L m ) . (3.31)8e have to find the vector fields Y im on M reg0 that satisfy Y im [ f ] = { f, h m } red i , ∀ f ∈ Hol( M ) red , i = 1 , . (3.32)These vector fields are of course not unique, since one may add any vector field to Y im that istangent to the orbits of the residual gauge transformations belonging to the group G . Thisambiguity does not effect the derivatives of the elements of Hol( M ) red , and we call any Y im satisfying (3.32) the reduced Hamiltonian vector field associated with h m and the respectivePoisson bracket.Now a vector field Y on M reg0 is characterized by the corresponding derivatives of theevaluation functions that map M reg0 ∋ ( Q, L ) to Q and L , respectively. We denote thesederivatives by Y [ Q ] and Y [ L ]. Then, for any f ∈ Hol( M reg0 ), the chain rule gives Y [ f ] = h∇ f, Q − Y [ Q ] i + h d f, Y [ L ] i . (3.33) Proposition 3.6.
For all m ∈ N , the reduced Hamiltonian vector fields Y im (3.32) can bespecified by the formulae Y m +1 [ Q ] = Y m [ Q ] = ( L m ) Q and Y m +1 [ L ] = Y m [ L ] = [ R ( Q ) L m , L ] . (3.34) Proof.
It is enough to verify that any f ∈ Hol( M ) red and h m (3.31), for m ∈ N , satisfy { f, h m +1 } ( Q, L ) = { f, h m } red2 ( Q, L ) = h∇ f ( Q, L ) , ( L m ) i + h d f ( Q, L ) , [ R ( Q ) L m , L ] i . (3.35)To obtain this, note that d h m +1 ( Q, L ) = ∇ h m ( Q, L ) = ∇ ′ h m ( Q, L ) = L m . (3.36)Putting this into (3.26) immediately gives the claim for { f, h m +1 } . To get { f, h m } red2 , wealso use that r + ∇ ′ h m − r − ∇ h m = L m and ∇ f − ∇ ′ f = [ L, d f ] , (3.37)together with the antisymmetry of R ( Q ) (3.19).We conclude from Proposition 3.6 that the evolutional vector fields on M reg0 that underliethe equations (1.1) induce commuting bi-Hamiltonian derivations of the commutative alge-bra of functions Hol( M ) red . In this sense, the holomorphic spin Sutherland hierarchy (1.1)possesses a bi-Hamiltonian structure. It is worth noting that the same statement holds ifwe replace Hol( M ) red by either of the two spaces of functions in the chain (3.11). Accordingto (3.13), Hol( M reg0 ) N arises by considering the invariants Hol( M reg ) G instead of Hol( M ) G .However, it is the latter space that should be regarded as the proper algebra of functionson the quotient M /G that inherits complete flows from the bi-Hamiltonian hierarchy on M .According to general principles [16], the flows on the singular Poisson space M /G are just theprojections of the unreduced flows displayed explicitly in (2.17). It is interesting to see how the bi-Hamiltonian structures of the real forms of the system(1.1), described in [11, 12], can be recovered from the complex holomorphic case. First, let usconsider the hyperbolic real form which is obtained by taking Q to be a real, positive matrix, Q = e q with a real diagonal matrix q , and L to be a Hermitian matrix. This means that wereplace M reg0 by the ‘real slice’ ℜ M reg0 := { ( Q, L ) ∈ M reg0 | Q i = e q i , q i ∈ R , L † = L } (4.1)9nd consider the real functions belonging to C ∞ ( ℜ M reg0 ) T n , where T n is the unitary subgroupof G . For such a function , say f , we can take ∇ f to be a real diagonal matrix and d f tobe a Hermitian matrix. In fact, in [11] we applied h X, Y i R := ℜh X, Y i (4.2)and defined the derivatives by h δq, ∇ f i R + h δL, d f i R := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( e tδq Q, L + tδL ) , (4.3)where t ∈ R , δq is an arbitrary real-diagonal matrix and δL is an arbitrary Hermitian matrix.Notice that the definitions entail h δq, ∇ f i R + h δL, d f i R = h δq, ∇ f i + h δL, d f i , (4.4)and, with ∇ f ≡ Ld f , ∇ ′ f ≡ ( d f ) L = ( Ld f ) † = ( ∇ f ) † . (4.5) Proposition 4.1.
If we consider f, h ∈ C ∞ ( ℜ M reg0 ) T n with (4.1) and insert their derivativesas defined above into the right-hand sides of the formulae of Theorem 3.4, then we obtain thefollowing real Poisson brackets: { f, h } ℜ ( Q, L ) = h∇ f, d h i R − h∇ h, d f i R + h L, [ d f, d h ] R ( Q ) i R , (4.6) and { f, h } ℜ ( Q, L ) = h∇ f, ∇ h i R − h∇ h, ∇ f i R + 2 h∇ f, R ( Q )( ∇ h ) i R , (4.7) which reproduce the real bi-Hamiltonian structure given in Theorem 1 of [11].Proof. We detail only the more complicated case of the second bracket. First of all, we have12 h∇ f, ∇ h + ∇ ′ h i = 12 h∇ f, ∇ h i + 12 h ( ∇ f ) † , ( ∇ h ) † i = h∇ f, ∇ h i R , (4.8)simply because h X, Y i ∗ = h X † , Y † i holds for all X, Y ∈ G . Thus the first line of (3.27)correctly gives the first two terms of (4.7). To continue, note that R ( Q ) maps G > to G > and G < to G < , and that now R ( Q )( X † ) = − ( R ( Q ) X ) † , ∀ X ∈ G . (4.9)This can be seen, for example, from the formula R ( Q ) X ⊥ = (cid:18)
12 coth 12 a d q (cid:19) X ⊥ , ∀ X ⊥ ∈ ( G > + G < ) , (4.10)with a d q X = [ q, X ]. The second line of (3.27) gives the expression A := hR ( Q ) ∇ h − R ( Q )( ∇ h ) † , ( ∇ f ) < + (( ∇ f ) < ) † i , (4.11)where we used (4.5) and that R ( Q ) vanishes on G . Taking into account (4.9), one easilychecks that A = 2 hR ( Q ) ∇ h, ( ∇ f ) < i R + 2 hR ( Q ) ∇ h, (( ∇ f ) < ) † i R . (4.12) We could also consider real-analytic functions. B = − hR ( Q ) ∇ f, ( ∇ h ) < i R − hR ( Q ) ∇ f, (( ∇ h ) < ) † i R = 2 h∇ f, R ( Q )( ∇ h ) < i R − h∇ f, ( R ( Q )( ∇ h ) < ) † i R , (4.13)where (3.19) and (4.9) were used. Then we obtain2 hR ( Q ) ∇ h, ( ∇ f ) < i R + 2 h∇ f, R ( Q )( ∇ h ) < i R = 2 hR ( Q ) ∇ h, ∇ f i R , (4.14)and 2 hR ( Q ) ∇ h, (( ∇ f ) < ) † i R − h∇ f, ( R ( Q )( ∇ h ) < ) † i R = 2 hR ( Q )( ∇ h ) < , (( ∇ f ) < ) † i R − h ( ∇ f ) † , R ( Q )( ∇ h ) < ) i R = 0 . (4.15)Thus we have shown that (3.27) indeed reduces to (4.7). The case of the first bracket is verysimple, we only note that now [ d f, d h ] R ( Q ) is a Hermitian matrix, and therefore h L, [ d f, d h ] R ( Q ) i = h L, [ d f, d h ] R ( Q ) i R . (4.16)Comparison with Theorem 1 in [11] shows that the formulae (4.6) and (4.7) reproduce the realbi-Hamiltonian structure derived in that paper. We remark that our d f (4.3) was denoted ∇ f , and our variable q corresponds to 2 q in [11]. Taking this into account, the Poissonbrackets of Proposition 4.1, multiplied by an overall factor 2, give precisely the Poisson bracketsof [11].The real form treated above yields the hyperbolic spin Sutherland model, and now we dealwith the trigonometric case. For this purpose, we introduce the alternative real slice ℜ ′ M reg0 := { ( Q, L ) ∈ M reg0 | Q j = e i q j , q j ∈ R , L † = L } (4.17)and consider the real functions belonging to C ∞ ( ℜ ′ M reg0 ) T n . A bi-Hamiltonian structure onthis space of functions was derived in [12], where we used the pairing h X, Y i I := ℑh X, Y i (4.18)and defined the derivatives D f , which is a real diagonal matrix, and D f , which is an anti-Hermitian matrix, by the requirement h i δq, D f i I + h δL, D f i I := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( e t i δq Q, L + tδL ) , (4.19)where t ∈ R , δq is an arbitrary real-diagonal matrix and δL is an arbitrary Hermitian matrix.It is readily seen that h i δq, D f i I + h δL, D f i I = h i δq, − i D f i + h δL, − i D f i , (4.20)and comparison with (2.6) motivates the definitions ∇ f := − i D f, d f := − i D f. (4.21)This implies that ∇ f := Ld f and ∇ ′ f := ( d f ) L satisfy (4.5) in this case as well. Animportant difference is that instead of (4.9) in the present case we have R ( Q ) X † = ( R ( Q ) X ) † , ∀ X ∈ G , (4.22)because in (4.10) q gets replaced by i q . 11 roposition 4.2. If we consider f, h ∈ C ∞ ( ℜ ′ M reg0 ) T n with (4.17) and insert their derivativesas defined in (4.21) into the right-hand sides of the formulae of Theorem 3.4, then we obtainthe following purely imaginary Poisson brackets: { f, h } I ( Q, L ) = − i (cid:0) h D f, D h i I − h D h, D f i I + h L, [ D f, D h ] R ( Q ) i I (cid:1) , (4.23) and { f, h } I ( Q, L ) = − i (cid:0) h D f, LD h i I − h D h, LD f i I + 2 h LD f, R ( Q )( LD h ) i I (cid:1) . (4.24) Then i { f, h } I and i { f, h } I reproduce the real bi-Hamiltonian structure given in Theorem 4.5of [12].Proof. We detail only the first bracket, for which the first term of (3.26) gives h∇ f, d h i = −h D f, D h i = − i h D f, D h i I , (4.25)since h D f, D h i is purely imaginary. The second term of (3.27) is similar, and the third termgives h L, [ d f, d h ] R ( Q ) i = −h L, [ D f, D h ] R ( Q ) i = − i h L, [ D f, D h ] R ( Q ) i I , (4.26)since [ D f, D h ] R ( Q ) is anti-Hermitian. Collecting terms, the formula (4.23) is obtained. Theproof of (4.24) is analogous to the calculation presented in the proof of (4.7). The differencearises from the fact that now we have (4.22) instead of (4.9). The last statement of theproposition is a matter of obvious comparison with the formulae of Theorem 4.5 of [12] (butone should note that what we here call D f was denoted d f in that paper, and h , i I wasdenoted h , i ). In this paper we developed a bi-Hamiltonian interpretation for the system of holomorphicevolution equations (1.1). The bi-Hamiltonian structure was found by interpreting this hier-archy as the Poisson reduction of a bi-Hamiltonian hierarchy on the holomorphic cotangentbundle T ∗ GL( n, C ), described by Theorem 2.1 and Proposition 2.3. Our main result is givenby Theorem 3.4 together with Proposition 3.6, which characterize the reduced bi-Hamiltonianhierarchy. Then we reproduced our previous results on real forms of the system [11, 12] byconsidering real slices of the holomorphic reduced phase space.The first reduced Poisson structure and the associated interpretation as a spin Sutherlandmodel is well known, and it is also known that the restrictions of the system to generic sym-plectic leaves of T ∗ GL( n, C ) / GL( n, C ) are integrable in the degenerate sense [17]. Experiencewith the real forms [12] indicates that the second Poisson structure should be tied in with arelation of the reduced system to spin Ruijsenaars–Schneider models, and degenerate integra-bility should also hold on the corresponding symplectic leaves. We plan to come back to thisissue elsewhere. We remark in passing that although T ∗ GL( n, C ) / GL( n, C ) is not a manifold,this does not cause any serious difficulty, since it still can be decomposed as a disjoint unionof symplectic leaves. This follows from general results on singular Hamiltonian reduction [16].We finish by highlighting a few open problems for future work. First, it could be inter-esting to explore degenerate integrability directly on the Poisson space T ∗ GL( n, C ) / GL( n, C ),suitably adapting the formalism of the paper [13]. Second, we wish to gain a better con-ceptual understanding of the process whereby one goes from holomorphic Poisson spaces andintegrable systems to their real forms, and apply it to our case. The results of the recent study[2] should be relevant in this respect. Finally, it is a challenge to generalize our constructionfrom the hyperbolic/trigonometric case to elliptic systems.12 cknowledgements. I wish to thank Maxime Fairon for several useful remarks on themanuscript. This work was supported in part by the NKFIH research grant K134946.
A The origin of the second Poisson bracket on G × G In this appendix we outline how the Poisson bracket { , } (2.10) arises from the standardPoisson bracket [19] on the Heisenberg double of the GL( n, C ) Poisson–Lie group. We focuson the logic of the derivation, and omit the (rather straightforward) calculational details.We start with the complex Lie group G × G and denote its elements as pairs ( g , g ). Weequip the corresponding Lie algebra G ⊕ G with the nondegenerate bilinear form h , i , givenby h ( X , X ) , ( Y , Y ) i := h X , Y i − h X , Y i (A.1)for all ( X , X ) and ( Y , Y ) from G ⊕ G . Then we have the isotropic subalgebras, G δ := { ( X, X ) | X ∈ G} , (A.2)and G ∗ := { ( r + ( X ) , r − ( X )) | ∀ X ∈ G} . (A.3)Recall that r ± are defined in (2.4), and note that G ⊕ G is the vector space direct sum of thedisjunct subspaces G δ and G ∗ ; G δ is isomorphic to G , and G ∗ can be regarded as its linear dualspace. We also introduce the corresponding subgroups of G × G , G δ := { g δ | g δ := ( g, g ) , g ∈ G } , (A.4)and G ∗ = (cid:8) g ∗ | g ∗ := (cid:0) g > g , ( g g < ) − (cid:1) , g > ∈ G > , g ∈ G , g < ∈ G < (cid:9) , (A.5)where G > , G < and G are the connected subgroups of G associated with the Lie subalgebrasthat underlie the decomposition (2.2). That is, G contains the diagonal, invertible complexmatrices, and G > (resp. G < ) consists of the upper triangular (resp. lower triangular) complexmatrices whose diagonal entries are all equal to 1.In order to describe the pertinent Poisson structures, we need the Lie algebra valuedderivatives of holomorphic functions. For F ∈
Hol( G × G ), we denote its G ⊕ G -valued left-and right-derivatives, respectively, by DF and D ′ F . For example, we have h ( X , X ) , DF ( g , g ) i := ddz (cid:12)(cid:12)(cid:12)(cid:12) z =0 F ( e zX g , e zX g ) , (A.6)where z ∈ C and ( X , X ) runs over G ⊕ G . Defined using h , i , a holomorphic function φ on G δ has the G ∗ -valued left- and right-derivatives, Dφ and D ′ φ . Analogously, the left- andright-derivatives Dχ and D ′ χ of χ ∈ Hol( G ∗ ) are G δ -valued.Now we recall [19] that G × G carries two natural Poisson brackets, which are given by {F , H} ± := hDF , R DHi ± hD ′ F , R D ′ Hi , (A.7)where R := ( P G δ − P G ∗ ) with the projections P G δ onto G δ and P G ∗ onto G ∗ defined via thevector space direct sum G ⊕ G = G δ + G ∗ . The minus bracket is called the Drinfeld doublebracket, and the plus one the Heisenberg double bracket. The former makes G × G into aPoisson–Lie group, having the Poisson submanifolds G δ and G ∗ , and the latter is symplecticin a neighbourhood of the identity. 13et us consider an open neighbourhood of the identity in G × G whose elements can befactorized as ( g , g ) = g δL g − ∗ R = g ∗ L g − δR (A.8)with g δL , g δR ∈ G δ and g ∗ L , g ∗ R ∈ G ∗ . Restricting ( g , g ) as well as all constituents in thefactorizations to be near enough to the respective identity elements, the map( g , g ) ( g δR , g ∗ R ) (A.9)yields a local, biholomorphic diffeomorphism. As the first step towards deriving the bracketin (2.10), we use this diffeomorphism to transfer the plus Poisson bracket to a neighbourhoodof the identity of G δ × G ∗ . The resulting Poisson structure then extends holomorphically tothe full of G δ × G ∗ . For G , H ∈
Hol( G δ × G ∗ ) we denote the resulting Poisson bracket by {F , H} ′ + . One can verify that it takes the following form: {F , H} ′ + ( g δ , g ∗ ) = (cid:10) g ∗ ( D ′ F ) g − ∗ , D H (cid:11) − (cid:10) g δ ( D ′ F ) g − δ , D H (cid:11) + h D F , D Hi − h D H , D F i , (A.10)where the derivatives on the right-hand side are taken at ( g δ , g ∗ ) ∈ G δ × G ∗ . The subscript 1and 2 refer to derivatives with respect to the first and second arguments; they are G ∗ and G δ valued, respectively. The derivation of the formula (A.10) from { , } + in (A.7) can follow thelines of [12], where another Heisenberg double was treated.In the second step towards getting { , } in (2.10), we make use of a biholomorphicdiffeomorphism between open neighbourhoods of the identity element of G δ × G ∗ and theelement ( n , n ) ∈ G × G . For g δ = ( g, g ) and g ∗ = ( g > g , ( g g < ) − ), this is given by the map( g δ , g ∗ ) ( g, L ) with L := g > g g < . (A.11)A (locally defined) function F on G δ × G ∗ then corresponds to a (locally defined) function F on G × G according to F ( g δ , g ∗ ) ≡ F ( g, L ) . (A.12)It is not difficult to check that this implies the following relation between the derivatives D F and D F and the G -valued derivatives ∇ F and ∇ F (defined in (2.6)–(2.8)): D F ( g δ , g ∗ ) = ( r + ∇ F ( g, L ) , r − ∇ F ( g, L )) , (A.13) D ′ F ( g δ , g ∗ ) = ( r + ∇ ′ F ( g, L ) , r − ∇ ′ F ( g, L )) ,D F ( g δ , g ∗ ) = ( r + ∇ ′ F ( g, L ) − r − ∇ F ( g, L ) , r + ∇ ′ F ( g, L ) − r − ∇ F ( g, L )) , (A.14)and P G ∗ (cid:0) g ∗ D ′ F ( g δ , g ∗ ) g − ∗ (cid:1) = P G ∗ (( ∇ F ( g, L ) , ∇ ′ F ( g, L ))) . (A.15)We apply the local diffeomorphism (A.11) to transfer the Poisson bracket { , } ′ + (A.10) to aPoisson bracket of holomorphic functions defined locally on G × G . The transferred Poissonbracket turns out to have the form { , } given in (2.10), and it naturally extends to a globallywell defined Poisson bracket on Hol( M ).Based on the outline of the derivation of the second Poisson bracket on T ∗ G that we gave,the interested reader can reproduce the details.14 eferences [1] I. Aniceto, J. Avan and A. Jevicki, Poisson structures of Calogero–Moser andRuijsenaars–Schneider models , J. Phys. A (2010) 185201; arXiv:0912.3468[hep-th] [2] P. Arathoon and M. Fontaine, Real forms of holomorphic Hamiltonian systems , arXiv:2009.10417 [math.SG] [3] G. Arutyunov, Elements of Classical and Quantum Integrable Systems, Springer, 2019[4] O. Babelon, D. Bernard, and M. Talon, Introduction to Classical Integrable Systems,Cambridge University Press, 2003.[5] J. Balog, L. D¸abrowski and L. Feh´er, Classical r -matrix and exchange algebra in WZNWand Toda theories , Phys. Lett. B (1990) 227-234[6] C. Bartocci, G. Falqui, I. Mencattini, G. Ortenzi and M. Pedroni, On the geometric originof the bi-Hamiltonian structure of the Calogero–Moser system , Int. Math. Res. Not. arXiv:0902.0953 [math-ph] [7] A. De Sole, V.G. Kac and D. Valeri
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