A decoupling property of some Poisson structures on ${\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\times n}(\mathbb{C})$ supporting ${\rm GL}(n,\mathbb{C}) \times {\rm GL}(d,\mathbb{C})$ Poisson-Lie symmetry

We study a holomorphic Poisson structure defined on the linear space $S(n,d):= {\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\times n}(\mathbb{C})$ that is covariant under the natural left actions of the standard ${\rm GL}(n,\mathbb{C})$ and ${\rm GL}(d,\mathbb{C})$ Poisson-Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the $d=1$ special case gives a Poisson structure on $S(n,1)$, and we construct a local Poisson map from the Cartesian product of $d$ independent copies of $S(n,1)$ into $S(n,d)$, which is a holomorphic diffeomorphism in a neighborhood of zero. The Poisson structure on $S(n,d)$ is the complexification of a real Poisson structure on ${\rm Mat}_{n\times d}(\mathbb{C})$ constructed by the authors and Marshall, where a similar decoupling into $d$ independent copies was observed. We also relate our construction to a Poisson structure on $S(n,d)$ defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars-Schneider system by Hamiltonian reduction.

which appeared in recent derivations of trigonometric spin Ruijsenaars-Schneider models [6] by Hamiltonian reduction [2,3].The decoupling means that the Poisson algebra of S(n, d) will be realized using d independent (pairwise Poisson commuting) copies of the Poisson algebra of S(n, 1).The spaces S(n, d) are defined for arbitrary pairs of natural numbers, but the decoupling requires that both n and d are greater than 1.Our result is expected to be useful, for example, for the further studies of the holomorphic spin Ruijsenaars-Schneider systems.
To set the stage, for any natural number ℓ we introduce the Drinfeld-Jimbo classical r-matrix r ℓ by where E jk (ℓ) is the usual elementary matrix of size ℓ × ℓ.We also need E jk (ℓ) ⊗ E kj (ℓ). (1.3) Note that for ℓ = 1 r ℓ = 0 and I ℓ can be viewed as 1 ⊗ 1. Denoting the elements of S(n, d) as pairs (A, B), and employing the standard tensorial notation [1,4], the pertinent Poisson bracket can be written as follows: ) .Here, we use the notations (1.2), (1.3) together with where E n×d iα ∈ Mat n×d (C) is the elementary matrix having a single non-zero entry, equal to 1, at the iα position.One could fix the arbitrary constant κ ∈ C * without loss of generality, but it will be advantageous not to do so.
The Poisson structure (1.4) represents the complexification of a U(n) × U(d) covariant real Poisson structure on Mat n×d (C) ≃ R 2nd considered in [3].By simple changes of variables (see below) it also reproduces the holomorphic Poisson bracket defined on S(n, d) by Arutyunov and Olivucci [2].In the papers mentioned it was natural to assume that n > 1, but here we assume only that either n or d is greater than 1.The d = 1 (or n = 1) cases provide the building blocks from which the general S(n, d) cases will be realized via the decoupling.
The above Poisson brackets have remarkable Poisson-Lie covariance properties.(For background on the theory of Poisson-Lie groups, one may consult, for example, [1,5,9,10].)To describe these, we equip the group GL(ℓ, C) with the standard multiplicative Poisson bracket given in tensorial notation by The subscript G expresses that this Poisson bracket lives on the group G = GL(ℓ, C). (1.9) and using the functions which are well-defined only locally, including a neighborhood of zero.The fundamental property of the map (g + , g − ) (1.10) is the factorization identity (1.17) the identity (1.16) and formulae of Theorem 1.1 imply the further identity (1.18) These properties, which are easily verified, actually motivated our construction.Their meaning will be enlightened in Section 3 (see Remark 3.5) utilizing the theory of Poisson-Lie moment maps.
We can also give an analogous realization of the Poisson bracket (1.4) on S(n, d) in terms of n copies of the Poisson bracket on S (1, d).Such a map can be obtained by combining Theorem 1.1 with the swap map ν from S(n, d) to S(d, n) that operates according to where for any ℓ ∈ N we let In addition, we shall present decoupling results for the 'oscillator Poisson brackets' of Arutyunov-Olivucci [2], who introduced two Poisson structures on S(n, d).Denoting the elements of S(n, d) now as pairs (A, B), one of their Poisson structures, called { , } + κ , is given by ).An alternative decoupling map from (S(n, 1), { , } κ ) ×d to (S(n, d), { , } + κ ) will be presented in Section 5.
Remark 1.3.It is known that the brackets { , } κ and { , } + κ satisfy the Jacobi identity, but the interested reader can also check this by routine calculation.

Basic facts about Poisson-Lie groups
We will recall the embedding of the Poisson-Lie group GL(ℓ, C) and its dual into their Drinfeld double D(ℓ).Then we will present the notion of the Poisson-Lie moment map.We do not give proofs here, since the relevant statements can be found in many reviews [1,5,9,10].
Let us consider the complex Lie group D(ℓ) (1.9) and equip its Lie algebra, with the non-degenerate, invariant bilinear form using a constant κ ∈ C * .Let us also introduce the triangular decomposition where gl(ℓ, C) 0 is the set of diagonal matrices, while gl(ℓ, C) > (resp.gl(ℓ, C) < ) contains the upper (resp.lower) triangular matrices with zero diagonal.Then D(ℓ) can be represented as the vector space direct sum of the isotropic subalgebras and ).We may identify gl(ℓ, C) with the diagonal subalgebra gl(ℓ, C) δ , and identify its linear dual space with the subalgebra gl(ℓ, C) * δ .We also let GL(ℓ, C) δ and GL(ℓ, C) * δ denote the subgroups of D(ℓ) corresponding to the subalgebras in the decomposition The group D(ℓ) carries a natural multiplicative Poisson structure.To describe it, let us take arbitrary bases T a of gl(ℓ, C) δ and T a of gl(ℓ, C) * δ that are in duality with respect to the pairing (2.2).The Poisson bracket of two holomorphic functions F and H on D(ℓ) is given by where for any T ∈ D(ℓ) we have It is well-known that GL(ℓ, C) δ and GL(ℓ, C) * δ are Poisson submanifolds of D(ℓ), and we equip them with the inherited Poisson structures.
The above Poisson structures can be conveniently presented in terms of the functions given by the matrix elements on the respective groups.Denoting the elements of D(ℓ) as pairs (u, v), and employing the tensorial notation of the Faddeev school, one has using the r-matrices (1.2) and (1.3).On the subgroup GL(ℓ, C) δ with elements denoted (g, g), this reduces to the bracket (1.6).The group GL(ℓ, C) * δ consists of the pairs (h + , h − ) ∈ D(ℓ) for which h + (resp.h − ) is upper triangular (resp.lower triangular) and the diagonal part of h + is the inverse of the diagonal part of h − .Restriction from D(ℓ) gives the following Poisson bracket on this dual group: We stress that D(ℓ), GL(ℓ, C) ≡ GL(ℓ, C) δ and GL(ℓ, C) * := GL(ℓ, C) * δ with the above Poisson brackets are Poisson-Lie groups.This means, for example, that the group product Let us briefly explain how (2.9) follows from (2.8).For T = (X, Y ) ∈ D(ℓ), the derivatives of the matrix elements are For any dual bases T a = (X a , X a ) and T a = (Z a , W a ), one can calculate that (2.12) By using these relations, one readily obtains (2.9) from (2.8).
There is an important mapping of GL(ℓ, C) * onto GL(ℓ, C), which is given by This mapping is 2 n to 1, since the image does not change if we replace (h + , h − ) by (h + τ, h − τ ) for any diagonal matrix τ whose entries are taken from the set {+1, −1}.The map χ yields a holomorphic diffeomorphism 1 between respective neighborhoods of the identity elements.Moreover, it is a Poisson map with respect to the so-called Semenov-Tian-Shansky Poisson structure [1,9] on GL(ℓ, C): We now recall [7] what is meant by a moment map for a Poisson action of GL(ℓ, C).Suppose that GL(ℓ, C) acts on a holomorphic Poisson manifold (P, { , } P ) in such a way that the action map, GL(ℓ, C) × P → P, is Poisson, where the product Poisson structure on GL(ℓ, C) × P is built from the bracket { , } κ G (1.6) on GL(ℓ, C) and { , } P on P. For any X ∈ gl(ℓ, C), let X P be the vector field on P given by the flow of exp(tX).We can take the derivative L X P F of any holomorphic function on P. We then say that a holomorphic map (φ + , φ − ) : P → GL(ℓ, C) * δ is the (Poisson-Lie) moment map for the action if it satisfies the following two conditions.First, we must have the equality for all X and F .Second, we also require that (φ + , φ − ) is a Poisson map with respect to the bracket (2.10) on the dual group.This second condition is equivalent to the requirement that the map ) is Poisson with respect to the Semenov-Tian-Shansky bracket (2.14) on GL(ℓ, C).Indeed, the Semenov-Tian-Shansky bracket is just the push-forward of the Poisson bracket (2.10) on the dual group GL(ℓ, C) * .The first condition can also be recast in terms of the map φ, as we shall see in our concrete example in the next section.

Covariance properties of the Poisson structure (1.4)
We now characterize the behavior of the Poisson bracket (1.4) on S(n, d) under the natural left-action of GL(n, C).Throughout this section, d ≥ 1 and n ≥ 2, otherwise they are arbitrary.The statements presented below can also be obtained as consequences of known [2,3] analogous properties of the Arutyunov-Olivucci Poisson bracket (1.21).We sketch the proofs in order to make this paper basically self-contained. 1 The multi-valued inverse of the map χ is closely related to the Gauss decomposition [8, §6.1] of regular matrices.If we Gauss decompose any g ∈ GL(ℓ, C) as g = g > g 0 g < , and have g 0 = h 2 0 for the diagonal constituent, then g = h + h −1 − with h + = g > h 0 and h −1 − = h 0 g < , so that (h + , h − ) ∈ GL(ℓ, C) * .The subtlety is that we have to take the square root of the diagonal matrix g 0 .
Proof.This is very easy and goes as follows.We can calculate {g 1 A 1 , g 2 A 2 } using the product Poisson structure defined by combining (1.4) and (1.6).This gives which agrees with the Poisson bracket on S(n, d).The next line of (1.4) is handled in the same way.Finally, one needs to show that We refrain from spelling this out, but note that the direct verification of this equality relies on the identity g 1 C n×d 12 = C n×d 12 g 2 .Proposition 3.2.Suppose that (φ + , φ − ) : S(n, d) → GL(n, C) * is a (possibly only locally defined) moment map for the Poisson action (3.1).Then the condition (2.15) is equivalent to the equalities where the usual tensorial notation is employed.For φ := φ + φ −1 − , these relations imply Proof.Let T a = (X a , X a ) and T a = (Z a , W a ) be dual bases of gl(n, C) δ and gl(n, C) * δ .Consider an arbitrary matrix element A iα as a function on S(n, d).Its derivative along the vector field induced by X a ∈ gl(n, C) equals (X a A) iα , and (2.15) gives the identity Since T a is a basis of gl(n, C) δ , this implies that This is equivalent to the relations By using the identities (2.12), these two equations are just the componentwise form of the first tensorial formulae in (3.4).The relations involving B are verified in the same way.The equalities in (3.4) are converted into those in (3.5) by a short calculation.Since the matrix elements of A and B form a coordinate system on S(n, d), the proof is complete.
Proposition 3.3.Define the map Γ : S(n, d) → gl(n, C) by the formula As a consequence of the Poisson brackets (1.4), this map satisfies the relation together with In a neighborhood of zero, Γ can be represented in the form Γ = Γ + Γ −1 − so that (Γ + , Γ − ) : S(n, d) → GL(n, C) * serves (locally) as the Poisson-Lie moment map for the action (3.1).
Proof.The equalities (3.10) and (3.11) can be verified by an easy calculation.For this, one needs to use the identities Since Γ(0) = 1 n , it is clear that Γ admits a unique factorization of the form Γ = Γ + Γ −1 − if we restrict (A, B) to be near enough to zero, require the continuity of Γ ± and impose the condition Γ ± (0) = 1 n .The so obtained (Γ + , Γ − ) can be written as holomorphic functions of the matrix entries of Γ. Equation (3.10) entails that (Γ + , Γ − ) gives a Poisson map into the dual group GL(ℓ, C) * carrying the brackets (2.10), and the relations (3.11) are equivalent to the moment map conditions given in (3.4).Here, we used the coincidence of (3.5) with (3.11) and that Γ and (Γ + , Γ − ) are related by a local Poisson diffeomorphism.Remark 3.4.A natural generalization of Proposition 3.3 holds around an arbitrary point (A 0 , B 0 ) ∈ S(n, d) for which (1 n + A 0 B 0 ) is an invertible matrix.See footnote 1 for the construction of (Γ + , Γ − ).
Remark 3.5.We observe from Proposition 3.3 and the first factorization identity (1.16) that (g + , g − ) given by (1.10) is nothing but the (local) GL(n, C) * -valued Poisson-Lie moment map on S(n, 1).For d > 1, the meaning of the second factorization identity (1.18) is that the (local) moment map (Γ + , Γ − ) mentioned in Proposition 3.3 satisfies the equality ) where m is the map of Theorem 1.1 and G ± are defined in (1.17).

Derivation of Theorem 1.1
The Poisson structure is derived in § 4.1, and we prove that m is a diffeomorphism in § 4.
For fixed α ∈ {1, . . ., d}, we use the pair (a α , b α ) to define G α j locally by (1.13), then the upper and lower triangular matrices g +,α , g −1 −,α by (1.11)-(1.12).Using (1.16) in the form g we can combine Propositions 3.2 and 3.3 for each copy S(n, 1), and we obtain ) We also use the formulae (2.10) to write is an anti-Poisson automorphism by Remark A.3 in the appendix.Using this map, we observe the following identities of matrix-valued functions The consistency of the anti-Poisson property with the Poisson brackets collected above is straightforward to check.
To ease computations, we introduce ) It will also be convenient to introduce for 1 and we set h α;α−1 . We note in particular that under the involution ι (4.5) which satisfies (4.6), we can write Here, the value of δ (α≤β) is 1 if the condition α ≤ β holds, and is zero otherwise.
Proof.We have from (4.3a) that {a α 1 , (g if α ≤ β, while it vanishes for β < α.We then get the first equality from (4.3a).The second equality is found in the same way, and the following two are obtained by applying the anti-Poisson automorphism ι.
Lemma 4.3.The following identities hold Proof.For the first identity, since α ≤ β we use the decomposition and note that the Poisson bracket appearing in the sum is given by (4.4a).This directly leads to the claimed result.For the second identity, we write for α ≤ β then we use (4.4c) to get the desired result.The case α ≥ β is obtained in a similar way.
Note that the identities from Lemmae 4.2 and 4.3 can be used with h 0 ± = 1 n as well.

The Poisson brackets {A
This follows by spelling out the action of r d using (1.2).In order to get (4.14),we note that (4.8) yields We can then use (4.1a) and Lemmae 4.2, 4.3 to reduce this expression.If α = β, we directly get (4.17) Upon using the identity we find Thus, we have derived (4.14) for all α ≤ β, hence it holds for all α, β by antisymmetry.
We can check that we obtain the claimed Poisson bracket for {B 1 , B 2 } κ either by a direct computation, or using the anti-Poisson automorphism ι (4.5) under which We have by (4.8) that which can be computed using (4.1b) and Lemmae 4.2, 4.3.If α < β, only the first and third sums in (4.21) do not trivially vanish, and we find that If α > β, we also obtain (4.22) by a similar computation.If α = β, only the first and fourth sums in (4.21) are nonzero, and we obtain We deduce from (4.2) that Thus, we can write {A α 1 , B β 2 } κ for all α, β in the desired form (4.20).

Diffeomorphism property.
Let us consider a point where the map m (1.14) is well-defined, i.e. we can construct g ±,α = g ± (a α , b α ) for α = 1, . . ., d using the formulae (1.11), (1.12) with (1.13).In a sufficiently small neighborhood of this point, the entries of the matrices g ±,α are analytic functions, hence m is holomorphic.From the image of this neighborhood, we can define inductively which is the inverse of the map m.The inverse map is holomorphic since the elements g ±,α are analytic functions in (A β i , B β i ) for β ≤ α.
The fact that the map F is a local diffeomorphism is similar to the argument used in § 4.2.We begin by observing the identities which follow from (5.1a), (5.1b) using g ±,α = g ± (a α , b α ), with g ±,d+1 := 1 n , and applying the analogue of (1.16) for all α.Then, picking Â and B near zero, we define the functions (ĝ +,α , ĝ−,α ) ∈ GL(n, C) * for 1 ≤ α ≤ d, by considering the factorization problems and iteratively This procedure uniquely specifies ĝ±,α for all α if we set ĝ±,α = 1 n for vanishing Â and B, and further require that these matrices depend continuously on Â, B in an open neighborhood of zero.As the final step, we define The definitions guarantee that if on the left-hand sides of (5.5) and (5.6) we use (5.1a) and (5.1b), then we obtain and hence the map that we constructed by (5.7) is indeed the local inverse of F .
It should be noted that although the map F from Theorem 5.1 is only a local diffeomorphism, the formulae (5.3) yield a holomorphic Poisson structure on the full space S(n, d).

.14)
This map enjoys the identity

.15)
The observation that (1 n + κAB) can be realized by applying the mapping (2.13) on the inverse (G + , G − ) −1 of a Poisson map (G + , G − ) into (GL(n, C) * , { , } κ * ) played an important role in the derivation of the trigonometric complex spin Ruijsenaars-Schneider model by Arutyunov and Olivucci [2].(To be precise, they locally realized (G + , G − ) as a moment map generating a Poisson-Lie action of (GL(n, C), { , } κ G ) on (S(n, d), { , } + κ ).)Our result (5.15) provides decoupled variables (the (a α , b α ) for α = 1, . . ., d) that give such a realization explicitly.These new variables (a, b) are expected to be useful for further studies of the reduction treatment of the complex spin Ruijsenaars-Schneider model, similarly as proved to be the case for the real form of this important integrable Hamiltonian system [3].

Conclusion
In this paper we presented a detailed analysis of the GL(n, C)×GL(d, C) covariant Poisson structures (1.4)  the Poisson structure is non-degenerate, and the associated symplectic form can be written as (b i da i − a i db i ) ∧ (b s da s + a s db s ) , (A.12) where we set a n+1 = b n+1 = 0 and G n+1 = 1.
Proof.Without loss of generality, we take κ = 2.We will prove using the 2-form (A.12) that for any 1 ≤ j ≤ n ι Xa j ω = −da j , (A.13) where X a j = {−, a j } denotes the Hamiltonian vector field of a j , which is given by .14)By symmetry between a and b, we will also have that ι X b j ω = −db j .These conditions then imply that ω is non-degenerate and corresponds to the Poisson bracket on S(n, 1), hence it is also closed.
To prove (A.13), let us denote the three terms appearing in the vector field (A.14) as X 1 , X 2 , X 3 .Contracting with the 2-form, we compute (b s da s + a s db s ) (b i da i − a i db i ) . (A.17)

1 . Introduction 2 .
Basic facts about Poisson-Lie groups 3. Covariance properties of the Poisson structure (1.4) 4. Derivation of Theorem 1.1 4.1.The Poisson structure 4.2.Diffeomorphism property 5.A decoupling property of the Arutyunov-Olivucci bracket 6.Conclusion Appendix A. Additional properties of the Poisson bracket on S(n, d) A.1.The Zakrzewski Poisson brackets A.2. Degeneracy of the Poisson bracket A.3.The symplectic form on a dense open subset References 1. Introduction In this paper we prove a remarkable 'decoupling property' of a holomorphic Poisson structure defined on the space S(n, d) := Mat n×d (C) × Mat d×n (C), (1.1)

Theorem 1 . 1 .
For any n and d greater than 1, take d copies of S(n, 1), each equipped with the Poisson bracket (1.4), and denote their elements by (a α , b α ), α = 1, . . ., d.Let (a, b) stand for the collection of the (a α , b α ), A α and B α for the columns and the rows of the matrices (A, B) ∈ S(n, d), respectively.Define the (local) map .15b) Then the map m is a local, holomorphic Poisson diffeomorphism from the d-fold product Poisson space S(n, 1) × • • • × S(n, 1) to S(n, d), where S(n, 1) and S(n, d) are equipped with the relevant Poisson brackets of the form (1.4).

2 . 4 . 1 . 4 . 1 . 1 .
The strategy of the derivation is analogous to[3, Lemma 5.1].The Poisson structure.Preparation and notations.The product Poisson structure on S(n, 1) ×d can be written in tensor notation using the pairs (a α , b α ) as follows

A. 3 .
and (1.21) on the linear space S(n, d) (1.1) for arbitrary natural numbers n and d.Our main results are encapsulated by Theorem 1.1 and Theorem 5.1 with Corollary 5.3 that provide new realizations of the corresponding Poisson algebras in terms of d independent copies of 'elementary spin variables' living in S(n, 1).The subsequent appendix highlights further relevant properties of these Poisson structures, especially by giving the underlying symplectic form on a dense open subset of S(n, 1).These results may contribute, for example, to deepening the understanding of Ruijsenaars-Schneider type integrable many-body models with spin having hidden GL(n, C) Poisson-Lie symmetry.It is also an interesting open question to search for their quantum mechanical analogues in the future.The symplectic form on a dense open subset.The holomorphic Poisson bracket (1.4) is non-degenerate on a dense subset of S(n, d), since it is non-degenerate at the origin.We now present the corresponding symplectic form for d = 1.For d ≥ 2, the symplectic form can be obtained around the origin by combining this result with Theorem 1.1.Proposition A.4.Consider S(n, 1) with the Poisson bracket (1.4), and denote its elements by (a, b).On the open subset where G i := 1 + r≥i a r b r = 0 , ∀ 1 ≤ i ≤ n , (A.11)

ι X 1 =j s>l a j a l b l G l G l+1 (b s da s + a s db s ) + 1 4 l
l b l G l G l+1 (b s da s + a s db s ) l G i G i+1 (b i da i − a i db i ) , =j i<l a j a l b l G i G i+1 (b i da i − a i db i ) , (A.16)and finallyι X 3 ω = − da j + 1 2 s>j a j 1 G j+1

Let ξ A and ξ B be arbitrary constants for which ξ
.14) With this Poisson bracket, GL(ℓ, C) can serve, at least locally, as a model of the dual Poisson-Lie group GL(ℓ, C) *.We note in passing that this quadratic Poisson bracket naturally extends to a Poisson structure on gl(ℓ, C), which is compatible with its linear Lie-Poisson bracket.