Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of GL(n,C), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard GL(n,C) Poisson--Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.


Introduction
The theory of integrable systems is an interesting field of mathematics motivated by influential examples of exactly solvable models of theoretical physics. For reviews, see e.g. [4,5,22,25]. There exist several approaches to integrability. One of the most popular ones in connection with classical integrable systems is the bi-Hamiltonian method, which originates from the work of Magri [18] on the KdV equation, and plays an important role in generalizations of this infinite dimensional bi-Hamiltonian system [8]. As can be seen in the reviews, among finite dimensional integrable systems the central position is occupied by Toda models and the models that carry the names of Calogero, Moser, Sutherland, Ruijsenaars and Schneider. The Toda models have a relatively well developed bi-Hamiltonian description [25]. The Calogero-Moser type models and their generalizations are much less explored from this point of view, except for the rational Calogero-Moser model [2,7,11]. In our recent work [12,13] we made a step towards improving this situation by providing a bi-Hamiltonian interpretation for a family of spin extended hyperbolic and trigonometric Sutherland models. In these references we investigated real-analytic Hamiltonian systems, and here wish to extend the pertinent results to the corresponding complex holomorphic case.
Specifically, the aim of this paper is to derive a bi-Hamiltonian description for the hierarchy of holomorphic evolution equations of the forṁ where Q is an invertible complex diagonal matrix of size n×n, L is an arbitrary n×n complex matrix, and the subscript 0 means diagonal part. The eigenvalues Q j of Q are required to be distinct, ensuring that the formula 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 basic dynamical r-matrix [6,9]. Like in the real case [12], it follows from the classical dynamical Yang-Baxter equation satisfied by R(Q) that the evolutional derivations (1.1) pairwise commute if they act on such 'observables' f (Q, L) that are invariant with respect to conjugations of L by invertible diagonal matrices. The system (1.1) has a well known interpretation as a holomorphic Hamiltonian system [17]. This arises from the parametrization 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 diagonal part of φ represents a constraint on the linear Poisson space gl(n, C), and this is responsible for the gauge transformations acting on L as conjugations by diagonal matrices. The k = 1 member of the hierarchy (1. ( 1.4) For this reason, we may refer to (1.1) as the holomorphic spin Sutherland hierarchy. It is also known (see e.g. [21]) that the holomorphic spin Sutherland hierarchy is a reduction of a natural integrable system on the cotangent bundle M := T * GL(n, C) equipped with its canonical symplectic form. Before reduction, the elements of M can be represented by pairs (g, L), where g belongs to the configuration space and (g, L) → L is the moment map for left-translations. The Hamiltonians tr(L k ) generate an integrable system on M, which reduces to the spin Sutherland system by keeping only the observables that are invariant under simultaneous conjugations of g and L by arbitrary elements of GL(n, C). This procedure is called 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 spin Sutherland 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 reg 0 , the subset of M consisting of the pairs (Q, L) with diagonal and regular Q ∈ GL(n, C). Moreover, the restricted function inherits invariance with respect to the normalizer of the diagonal subgroup G 0 < GL(n, C), which includes G 0 . This explains the gauge symmetry of the hierarchy (1.1), and lends justification to the restriction on the eigenvalues of Q.
The bi-Hamiltonian structure on M involves in addition to the canonical Poisson bracket associated with the universal cotangent bundle symplectic form another one that we construct from Semenov-Tian-Shansky's Poisson bracket of the Heisenberg double of GL(n, C) endowed with its standard Poisson-Lie group structure [23]. Surprisingly, we could not find it in the literature that the canonical symplectic structure of the cotangent bundle M can be complemented to a bi-Hamiltonian structure in this manner. So this appears to be a novel result, which is given by Theorems 2.1, 2.2 and Proposition 2.4 in Section 2. The actual derivation of the second Poisson bracket (2.13) is relegated to an appendix. The heart of the paper is Section 3, where we derive the bi-Hamiltonian structure of the system (1.1) by Poisson reduction. The main results are encapsulated by Theorem 3.5 and Proposition 3.7. The first reduced Poisson bracket (3.34) is associated with the spin Sutherland interpretation by means of the parametrization (1.3). The formula of the second reduced Poisson bracket is given by equation (3.35). After deriving the holomorphic bi-Hamiltonian structure in Section 3, we shall explain in Section 4 that it allows us to recover the bi-Hamiltonian structures of the hyperbolic and trigonometric real forms derived earlier by different means [12,13]. In the final section, we summarize the main results once more, and highlight a few open problems.
2 Bi-Hamiltonian hierarchy on the cotangent bundle Let us denote G := GL(n, C) and equip its Lie algebra G := gl(n, C) with the trace form This is a non-degenerate, symmetric bilinear form that enjoys the invariance property Any X ∈ G admits the unique decomposition into strictly upper triangular part X > , diagonal part X 0 , and strictly lower triangular part X < . Thus G is the vector space direct sum of the corresponding subalgebras We shall use the standard solution of the modified classical Yang-Baxter equation on G, r ∈ End(G) given by and define also Our aim is to present two holomorphic Poisson structures on the complex manifold Denote Hol(M) the commutative algebra of holomorphic functions on M. For anyF ∈ Hol(M), introduce the G-valued derivatives ∇ 1 F , ∇ ′ 1 F and d 2 F by the defining relations and where z is a complex variable and X ∈ G is arbitrary. In addition, it will be convenient to define the G-valued functions ∇ 2 F and ∇ ′ 2 F by and a similar relations holds between ∇ 2 F and ∇ ′ 2 F whenever L is invertible. Theorem 2.1. For holomorphic functions F, H ∈ Hol(M), the following formulae define two Poisson brackets:

12)
and 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 canonical symplectic 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 identity of the second bracket can be verified by direct calculation. More conceptually, they follow from the fact that locally, in a neighbourhood of (1 n , 1 n ) ∈ G × G, the second bracket can be transformed into Semenov-Tian-Shansky's [23] Poisson bracket on the Heisenberg double of the standard Poisson-Lie group G. This is explained in the appendix.
Let us display the explicit formula of the Poisson brackets of the evaluation functions given by the matrix elements g ij and the linear functions L a := T a , L associated with an arbitrary basis T a of G, whose dual basis is T a , T b , T a = δ b a . One may use the standard basis of elementary matrices, e ij defined by (e ij ) kl = δ ik δ jl , but we find it convenient to keep a general basis. We obtain directly from the definitions (gT a ) ij T a = e ji g, dL a = T a . (2.14) These give the first Poisson bracket immediately Then elementary calculations lead to the following formulae of the second Poisson bracket, where sgn is the usual sign function, and By using the standard basis and evaluating the matrix multiplications, one may also spell out the last two equations as where δ (i>k) := 1 if i > k and is zero otherwise. Let us recall that two Poisson brackets on the same manifold are called compatible if their arbitrary linear combination is also a Poisson bracket [18]. Compatible Poisson brackets often arise by taking the Lie derivative of a given Poisson bracket along a suitable vector field. If W is a vector field and { , } is Poisson bracket, then the Lie derivative bracket is given by where 1 n is the unit matrix. Consequently, the two Poisson brackets are compatible.
Proof. By the general result quoted above [10,24] (2.24) Comparison with (2.15) implies the claim of the theorem.
The derivatives of the matrix elements of (g, L) ∈ M give

28)
and the flow of V 2 Proof. We obtain the derivatives As a result of (2.10), and thus, by (2.7), The substitution of these relations into the formulae of Proposition 2.1 gives By the very meaning of the Hamiltonian vector field associated with a function, these Poisson brackets imply (2.28), and then (2.29) follows, too.
Like in the compact case [13], we call the H m 'free Hamiltonians' and conclude from Proposition 2.4 that they generate a bi-Hamiltonian hierarchy on the holomorphic cotangent bundle M.

The reduced bi-Hamiltonian hierarchy
The essence of Hamiltonian symmetry reduction is that one keeps only the 'observables' that are invariant with respect to the pertinent group action. Here, we apply this principle to the adjoint action of G on M, for which η ∈ G acts by the holomorphic diffeomorphism A η , Thus we keep only the G invariant holomorphic functions on M, whose set is denoted For invariant functions, the formula of the second Poisson brackets simplifies drastically.
Lemma 3.1. For F, H ∈ Hol(M) G , the formula (2.13) can be rewritten as follows: We start by noting that for a G invariant function H the relation , taking the derivative of both sides of (3.4) at z = 0 gives Since X is arbitrary, (3.5) follows. Formally, (3.3) is obtained from (2.13) by setting r to zero, i.e., r cancels from all terms. The verification of this cancellation relies on the identity (3.5) and is completely straightforward. We express ∇ ′ 1 H through the other derivatives with the help of (3.5), apply the same to ∇ ′ 1 F , and then collect terms in (2.13). To cancel all terms containing r we use also that rX, Y = − X, rY . After cancelling those terms, the equality (3.3) is obtained by utilizing the identity which is verified by means of the definitions (2.1) and (2.10). Proof. Let us observe that the derivatives of the G invariant functions are equivariant, and similar for ∇ ′ i H. In order to see this, notice that holds for any X ∈ G and η ∈ G if H is an invariant function. By taking derivative, we obtain This leads to the i = 1 case of (3.8). The property follows in a similar manner, and it implies the i = 2 case of (3.8).
By combining the formulas (2.12) and (3.3) with the equivariance property of the derivatives of F and H, we may conclude from (2.2) that if F, H are invariant, then so is {F, H} i for i = 1, 2.
We wish to characterize the Poisson algebras of the G invariant functions. To start, we consider the diagonal subgroup G 0 < G, (3.12) and its regular part The normalizer contains G 0 as a normal subgroup, and the corresponding quotient is the permutation group, (3.14) We also let G reg ⊂ G denote the dense open subset consisting of the conjugacy classes having representatives in G reg 0 . Next, we define and These are complex manifolds, equipped with their own holomorphic functions. Now we introduce the chain of commutative algebras The last two sets contain the respective invariant elements of Hol(M reg 0 ), and Hol(M) red contains the restrictions of the elements of Hol(M) G to M reg 0 . To put this in a more formal manner, let ι : be the tautological embedding. Then pull-back by ι provides an isomorphism between Hol(M) G and Hol(M) red . We here used that any holomorphic (or even continuous) function on M is uniquely determined by its restriction to M reg . Similar, we obtain the map which is also injective and surjective. It may be worth elucidating why the pull-back (3.19) is an isomorphism. To this end, consider any map η : yields a well-defined, G invariant function on M reg , which restricts to f . The function F is holomorphic, since locally, on an open set around any fixed g 0 ∈ G reg , one can choose η(g) to depend holomorphically on g. Regarding this classical result of perturbation theory, see, e.g., Theorem 2.1 in [1]. The main goal of this paper is to derive formulae for the reduced Poisson brackets (3.21). To do so, we now note that any f ∈ Hol(M reg 0 ) has the G 0 -valued derivative ∇ 1 f and the G-valued derivative d 2 f , defined by (3.22) which are required for all X 0 ∈ G 0 (2.4), X ∈ G. For any Q ∈ G 0 , the linear operator Ad Q : G → G acts as Ad Q (X) = QXQ −1 . Set where G < (resp. G > ) is the strictly lower (resp. upper) triangular subalgebra of G introduced in (2.4). Notice that for Q ∈ G reg 0 the operator (Ad Q − id) maps G ⊥ to G ⊥ in an invertible manner. Building on (2.3), we have the decomposition (3.24) Using this, for any Q ∈ G reg 0 , the 'dynamical r-matrix' R(Q) ∈ End(G) is given by and we remark its antisymmetry property This can be seen by writing Q = e q with q ∈ G 0 , whereby we obtain Here ad q (X ⊥ ) = [q, X ⊥ ], which gives an anti-symmetric, invertible linear operator on G ⊥ . (The invertibility holds since Q ∈ G reg 0 , and is needed for coth 1 2 ad q to be well defined on G ⊥ .) Below, we shall also employ the shorthand Proof. The equalities (3.29) hold since f is the restriction of F . In particular, it satisfies (3.31) Concerning (3.30), the equality of the G 0 parts, (∇ 1 F (Q, L)) 0 = (∇ 1 f (Q, L)) 0 , is obvious. Then take any T ∈ G ⊥ , for which we have which implies (3.30).
Theorem 3.5. For f, h ∈ Hol(M) reg , the reduced Poisson brackets defined by (3.21) can be described explicitly as follows:

34)
and {f, h} red 2 (Q, L) = where all derivatives are taken at (Q, L) ∈ M reg 0 , and the notation (2.10) is in force. These formulae give two compatible Poisson brackets on Hol(M) red .
Proof. Let us begin with the first bracket, and note that at (Q, L) ∈ M reg 0 we have This holds at (Q, L), since f, h are the restrictions of F, H ∈ Hol(M) G . We then combine (3.37) with the second term in (3.3). Collecting terms and using the antisymmetry (3.26), we obtain Moreover, we have which cancels the contribution of the last two terms of (3.3). In conclusion, we see that the first and second lines in (3.37) and their counterparts ensuring antisymmetry give the claimed formula (3.35). We know from Theorem 2.2 that the original Poisson brackets on Hol(M) are compatible, which means that their arbitrary linear combination { , } := x{ , } 1 +y{ , } 2 satisfies the Jacobi identity. In particular, the Jacobi identity holds for elements of Hol(M) G as well. It is thus plain from Definition 3.3 that the arbitrary linear combination { , } red = x{ , } red 1 + y{ , } red 2 also satisfies the Jacobi identity. In this way, the compatibility of the two reduced Poisson brackets is inherited from the compatibility of the original Poisson brackets.
Remark 3.6. It can be shown that the formulae of Theorem 3.5 give Poisson brackets on Hol(M reg 0 ) N and on Hol(M reg 0 ) G 0 as well. Indeed, we can repeat the reduction starting from Hol(M reg ) G using the map (3.19), and this leads to the reduced Poisson brackets on Hol(M reg 0 ) N . Then the closure on Hol(M reg 0 ) G 0 follows from (3.14) and the local nature of the Poisson brackets. Because of (3.14), the quotient by N can be taken in two steps, Since the action of S n is free, the Poisson structure on M reg 0 /N , which carries the functions Hol(M reg 0 ) N , lifts to a Poisson structure on M reg 0 /G 0 , whose ring of functions is Hol(M reg 0 ) G 0 . Now we turn to the reduction of the Hamiltonian vector fields (2.28) to vector fields on M reg 0 . There are two ways to proceed. One may either directly associate vector fields to the reduced Hamiltonians using the reduced Poisson brackets, or can suitably 'project' the original Hamiltonian 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 , which are given by (3.45) To obtain this, note that We conclude from Proposition 3.7 that the evolutional vector fields on M reg 0 that underlie the equations (1.1) induce commuting bi-Hamiltonian derivations of the commutative algebra 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 if we replace Hol(M) red by either of the two spaces of functions in the chain (3.17). According to (3.19), Hol(M reg 0 ) 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 functions on the quotient M/G that inherits complete flows from the bi-Hamiltonian hierarchy on M.
According to general principles [20], the flows on the singular Poisson space M/G are just the projections of the unreduced flows displayed explicitly in (2.29).

Recovering the real forms
It is interesting to see how the bi-Hamiltonian structures of the real forms of the system (1.1), described in [12,13], can be recovered from the complex holomorphic case. First, let us consider 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 we replace M reg 0 by the 'real slice' and consider the real functions belonging to C ∞ (ℜM reg 0 ) T n , where T n is the unitary subgroup of G 0 . For such a function 1 , say f , we can take ∇ 1 f to be a real diagonal matrix and d 2 f to be a Hermitian matrix. In fact, in [12] we applied X, Y R := ℜ X, Y (4.2) and defined the derivatives by where t ∈ R, δq is an arbitrary real-diagonal matrix and δL is an arbitrary Hermitian matrix. Notice that the definitions entail If we consider f, h ∈ C ∞ (ℜM reg 0 ) T n with (4.1) and insert their derivatives as defined above into the right-hand sides of the formulae of Theorem 3.5, then we obtain the following real Poisson brackets: and which reproduce the real bi-Hamiltonian structure given in Theorem 1 of [12]. 1 We could also consider real-analytic functions.
Proof. The proof relies on the identity This can be seen, for example, from the formula (3.27), since Consequently, we obtain the formula (4.6) from (3.34) Turning to the second bracket, the equality ∇ ′ 2 h = (∇ 2 h) † (4.5) implies simply because X, Y * = X † , Y † holds for all X, Y ∈ G. Thus the first line of (3.35) correctly gives the first two terms of (4.7). Moreover, on account of (4.5) and (4.8), we obtain Therefore, (3.35) gives (4.7).
Comparison with Theorem 1 in [12] shows that the formulae (4.6) and (4.7) reproduce the real bi-Hamiltonian structure derived in that paper. We remark that our d 2 f (4.3) was denoted ∇ 2 f , and our variable q corresponds to 2q in [12]. Taking this into account, the Poisson brackets of Proposition 4.1, multiplied by an overall factor 2, give precisely the Poisson brackets of [12].
The real form treated above yields the hyperbolic spin Sutherland model, and now we deal with the trigonometric case. For this purpose, we introduce the alternative real slice and consider the real functions belonging to C ∞ (ℜ ′ M reg 0 ) T n . A bi-Hamiltonian structure on this space of functions was derived in [13], where we used the pairing X, Y I := ℑ X, Y (4.14) and defined the derivatives D 1 f , which is a real diagonal matrix, and D 2 f , which is an anti-Hermitian matrix, by the requirement where t ∈ R, δq is an arbitrary real-diagonal matrix and δL is an arbitrary Hermitian matrix. It is readily seen that iδq, and comparison with (2.8) motivates the definitions This implies that ∇ 2 f := Ld 2 f and ∇ ′ 2 f := (d 2 f )L satisfy (4.5) in this case as well. An important difference is that instead of (4.8) in the present case we have because in (3.27) q gets replaced by iq with a real q, and then instead of (4.9) we have ad iq X † = (ad iq X) † .

Proposition 4.2.
If we consider f, h ∈ C ∞ (ℜ ′ M reg 0 ) T n with (4.13) and insert their derivatives as defined in (4.17) into the right-hand sides of the formulae of Theorem 3.5, then we obtain the following purely imaginary Poisson brackets: and Then i{f, h} I 1 and i{f, h} I 2 reproduce the real bi-Hamiltonian structure given in Theorem 4.5 of [13].
Proof. We detail only the first bracket, for which the first term of (3.34) gives Collecting terms, the formula (4.19) is obtained. The proof of (4.20) is analogous to the calculation presented in the proof of (4.7). The difference arises from the fact that now we have (4.18) instead of (4.8). The last statement of the proposition is a matter of obvious comparison with the formulae of Theorem 4.5 of [13] (but one should note that what we here call D 2 f was denoted d 2 f in that paper, and , I was denoted , ).

Conclusion
In this paper we developed a bi-Hamiltonian interpretation for the system of holomorphic evolution equations (1.1). The bi-Hamiltonian structure was found by interpreting this hierarchy as the Poisson reduction of a bi-Hamiltonian hierarchy on the holomorphic cotangent bundle T * GL(n, C), described by Theorems 2.1, 2.2 and Proposition 2.4. Our main result is given by Theorem 3.5 together with Proposition 3.7, which characterize the reduced bi-Hamiltonian hierarchy. Then we reproduced our previous results on real forms of the system [12,13] by considering real slices of the holomorphic reduced phase space. The first reduced Poisson structure and the associated interpretation as a spin Sutherland model is well known, and it is also known that the restrictions of the system to generic symplectic leaves of T * GL(n, C)/GL(n, C) are integrable in the degenerate sense [21]. Experience with the real forms [13] indicates that the second Poisson structure should be tied in with a relation of the reduced system to spin Ruijsenaars-Schneider models, and degenerate integrability should also hold on the corresponding symplectic leaves. We plan to come back to this issue 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 union of symplectic leaves. This follows from general results on singular Hamiltonian reduction [20].
We finish by highlighting a few open problems for future work. First, it could be interesting to explore degenerate integrability directly on the Poisson space T * GL(n, C)/GL(n, C), suitably adapting the formalism of the paper [15]. Second, we wish to gain a better conceptual understanding of the process whereby one goes from holomorphic Poisson spaces and integrable systems to their real forms, and apply it to our case. The results of the recent study [3] should be relevant in this respect. Finally, it is a challenge to generalize our construction from the hyperbolic/trigonometric case to elliptic systems. The existence of a bi-Hamiltonian structure for the elliptic spin Calogero-Moser system appears to follow from the existence of such a structure for an integrable elliptic top on GL(n, C) [14] via the symplectic Hecke correspondence [16,19].
Acknowledgements. I wish to thank Maxime Fairon for several useful remarks on the manuscript. I am also grateful to Mikhail Olshanetsky for drawing my attention to relevant references. This work was supported by the NKFIH research grant K134946, and was also supported partially by the Project GINOP-2.3.2-15-2016-00036 co-financed by the European Regional Development Fund and the budget of Hungary.
A The origin of the second Poisson bracket on G × G In this appendix we outline how the Poisson bracket { , } 2 (2.13) arises from the standard Poisson bracket [23] on the Heisenberg double of the GL(n, C) Poisson-Lie group.
We start with the complex Lie group G × G and denote its elements as pairs (g 1 , g 2 ). We equip the corresponding Lie algebra G ⊕ G with the nondegenerate bilinear form , 2 , given by for all (X 1 , X 2 ) and (Y 1 , Y 2 ) from G ⊕ G. Then we have the isotropic subalgebras, and Recall that r ± are defined in (2.6), and note that G ⊕ G is the vector space direct sum of the disjunct subspaces G δ and G * ; G δ is isomorphic to G, and G * can be regarded as its linear dual space. We also introduce the corresponding subgroups of G × G, G δ := {g δ | g δ := (g, g), g ∈ G}, (A.4) and G * = g * | g * := g > g 0 , (g 0 g < ) −1 , g > ∈ G > , g 0 ∈ G 0 , g < ∈ G < , (A. 5) where G > , G < and G 0 are the connected subgroups of G associated with the Lie subalgebras in the decomposition (2.4). That is, G 0 contains the diagonal, invertible complex matrices, and G > (resp. G < ) consists of the upper triangular (resp. lower triangular) complex matrices whose diagonal entries are all equal to 1. In order to describe the pertinent Poisson structures, we need the Lie algebra valued derivatives of holomorphic functions. For F ∈ Hol(G × G), we denote its G ⊕ G-valued leftand right-derivatives, respectively, by DF and D ′ F . For example, we have (X 1 , X 2 ), DF (g 1 , g 2 ) 2 := d dz z=0 F (e zX 1 g 1 , e zX 2 g 2 ), (A. 6) where z ∈ C and (X 1 , X 2 ) runs over G ⊕ G. Defined using , 2 , a holomorphic function φ on G δ has the G * -valued left-and right-derivatives, Dφ and D ′ φ. Analogously, the left-and right-derivatives Dχ and D ′ χ of χ ∈ Hol(G * ) are G δ -valued. Now we recall [23] that G × G carries two natural Poisson brackets, which are given by where R := 1 2 (P G δ − P G * ) with the projections P G δ onto G δ and P G * onto G * defined via the vector space direct sum G ⊕ G = G δ + G * . The minus bracket is called the Drinfeld double bracket, and the plus one the Heisenberg double bracket. The former makes G × G into a Poisson-Lie group, having the Poisson submanifolds G δ and G * , and the latter is symplectic in a neighbourhood of the identity.
Let us consider an open neighbourhood of the identity in G × G whose elements can be factorized as with g δL , g δR ∈ G δ and g * L , g * R ∈ G * . Restricting (g 1 , g 2 ) as well as all constituents in the factorizations to be near enough to the respective identity elements, the map (g 1 , g 2 ) → (g δR , g * R ) (A.9) yields a local, biholomorphic diffeomorphism. As the first step towards deriving the bracket in (2.13), we use this diffeomorphism to transfer the plus Poisson bracket to a neighbourhood of the identity of G δ × G * . The resulting Poisson structure then extends holomorphically to the 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: where the derivatives on the right-hand side are taken at (g δ , g * ) ∈ G δ × G * . The subscript 1 and 2 refer to derivatives with respect to the first and second arguments; they are G * and G δ valued, respectively. For example, we have F ((e zX , e zX )g δ , g * ) (A.11) and D 2 F (g δ , g * ), (r + X, r − X) 2 = d dz z=0 F (g δ , (e zr + X , e zr − X )g * ). (A.12) It is worth noting that The derivatives D ′ 1 and D ′ 2 are defined analogously, cf. (2.8). The derivation of the formula (A.10) from { , } + in (A.7) can follow closely the proof of Proposition 2.1 in [13], where another Heisenberg double was treated. The formula (A.10) itself has the same structure as formula (2.18) in [13], and thus we here omit its derivation.