Poisson Reductions of Master Integrable Systems on Doubles of Compact Lie Groups

We consider three ‘classical doubles’ of any semisimple, connected and simply connected compact Lie group G: the cotangent bundle, the Heisenberg double and the internally fused quasi-Poisson double. On each double we identify a pair of ‘master integrable systems’ and investigate their Poisson reductions. In the simplest cotangent bundle case, the reduction is defined by taking quotient by the cotangent lift of the conjugation action of G on itself, and this naturally generalizes to the other two doubles. In each case, we derive explicit formulas for the reduced Poisson structure and equations of motion and find that they are associated with well known classical dynamical r-matrices. Our principal result is that we provide a unified treatment of a large family of reduced systems, which contains new models as well as examples of spin Sutherland and Ruijsenaars–Schneider models that were studied previously. We argue that on generic symplectic leaves of the Poisson quotients the reduced systems are integrable in the degenerate sense, although further work is required to prove this rigorously.


Introduction
The variants of the method of Hamiltonian reduction [5,38,43] play a pivotal role in deriving and analyzing integrable Hamiltonian systems. The starting point in the applications is always a manifestly integrable system on a higher dimensional phase space that possesses a large symmetry group, which is used for setting up its reduction. As examples, it is sufficient to mention that key properties of the ubiquitous Calogero-Moser-Sutherland models [7,34,52] and their relativistic [45] and spin generalizations [22,28,30] became transparent from investigations based on this method [6,8,15,21,24,41]. For reviews of the subject, see [5,36,37,40]. Building on our experience gained from previous studies [11,12,13,14,16,19] here we wish to explore a general set of reductions of important families of unreduced 'master systems'.
Let G be a compact, connected and simply connected Lie group whose Lie algebra G is simple. In this paper we study Poisson reductions of three phase spaces associated with G. The first is the cotangent bundle M := T * G ≃ G × G, (1.1) presented by means of right-trivialization and the identification G * ≃ G. Its Poisson-Lie generalization is the Heisenberg double [48] M := G × B, (1.2) which is obtained by combining the standard multiplicative Poisson structures on G and its dual Poisson-Lie group B into a symplectic structure. This is a natural generalization since T * G is the Heisenberg double for G equipped with the zero Poisson structure. The third unreduced phase space is the so-called internally fused quasi-Poisson double [1], denoted that is closely related to the moduli space of flat G-connections on the punctured torus. Each of these spaces carries a pair of degenerate integrable systems, and reductions of those to integrable many-body models and their spin extensions have already received considerable attention (see, e.g., [12,19,41,42] and references therein). The goal of this paper is to describe a very general reduction of these 'master integrable systems' in all three cases. We shall apply the same technique in our study of the distinct cases, and shall highlight the similarities between the resulting reduced systems. The principal case of our interest is the Heisenberg double M. We include the cotangent bundle in our treatment mainly in order to motive the generalizations, although new results will be obtained also in this familiar case. The unified treatment that we present has not yet been developed in the literature, and could be useful for further detailed explorations of the reduced systems descending from the three doubles. The doubles of G are G-manifolds, where M carries the cotangent lift of the conjugation action of G on itself, G acts on D by diagonal conjugations, and there is a similar action on M built from the conjugation action and the dressing action of G on B. The Poisson brackets on M and M and the quasi-Poisson bracket on D share the property that the G-invariant smooth functions form a closed Poisson algebra. By Poisson reduction, we mean the restriction to this Poisson algebra of invariant functions, which is to be thought of as a Poisson structure on the corresponding quotient space defined by the G-action. The first principal goal of our work is to derive an effective description of these 'reduced Poisson algebras'.
Denote C ∞ (G) G , C ∞ (G) G and C ∞ (B) G the respective rings of invariant real functions. The functional dimension of these rings of functions equals the rank ℓ of G. All three doubles are Cartesian products as manifolds, and we let π 1 and π 2 denote the projections onto the first and second factors of those Cartesian products. Then, for each of the three doubles, π * 1 (C ∞ (G) G ) provides an Abelian Poisson subalgebra of the Poisson algebra of the G-invariant functions. We call the elements of π * 1 (C ∞ (G) G ) pullback invariants. Using π 2 in the analogous manner, one also obtains Abelian Poisson algebras of pullback invariants. The Poisson and quasi-Poisson structures allow one to associate a (Hamiltonian or quasi-Hamiltonian) vector field to every function, defining an evolution equation. We shall explain that the evolution equation obtained from any pullback invariant gives rise to a degenerate integrable system [33,35,42], which means 1 that it admits a ring of constants of motion whose functional dimension is equal to 2 dim(G) − ℓ, where 2 dim(G) is the dimension of the phase space. We shall also explicitly describe the integral curves of the pullback invariants and their constants of motion in each case. This yields generalizations of well-known results concerning T * G. Our second principal goal is to characterize the reductions of the degenerate integrable systems induced on the master phase spaces by the pullback invariants.
For clarity, recall that the functional dimension of a ring of smooth functions F on a manifold X is k if there exists an open dense submanifoldX ⊆ X such that the exterior derivatives of the elements of F span a k-dimensional subspace of T * x X for every x ∈X. The fact that the rings of invariants of our concern have functional dimension ℓ = rank(G) follows from basic Lie theoretic results, and the pullback invariants obviously have the same functional dimension as the original invariants. Below, the functional dimension of a Poisson algebra is understood to mean the functional dimension of the underlying ring of functions.
The quotient spaces of the master phase spaces are not smooth manifolds, but stratified Poisson spaces [38,50,51], which still can be decomposed into disjoint unions of smooth symplectic leaves. However, this is quite a complicated structure, and we will be content with describing the Poisson algebras of the invariants, and the reductions of the evolution equations generated by the pullback invariants, in terms of convenient partial gauge fixings. To explain what this means, we next outline the case of the cotangent bundle. We then briefly summarize how the picture generalizes to the other cases.
The motivating example of T * G and its generalizations. Let us fix a maximal torus G 0 < G and let G 0 < G be its Lie algebra. The group G acts on itself by conjugations and on G by the adjoint action. We denote by G reg and G reg the dense open subsets formed by the elements whose isotropy subgroups are maximal tori in G, and let G reg (1.7) Here, dϕ denotes the G-valued gradient of ϕ, the subscript zero refers to the orthogonal decomposition G = G 0 + G ⊥ , and R(Q) ∈ End(G) is a well-known trigonometric solution of the modified classical dynamical Yang-Baxter equation [10]. It vanishes on G 0 and, writing Q = exp(iq) with q ∈ iG reg 0 , it is given by R(Q) = 1 2 coth( i 2 ad q ) on G ⊥ . (Here, iG 0 is a subset of the complexification of G.) Of course, the so-obtained vector fields and evolution equations are unique only up to the addition of arbitrary vector fields that are tangent to the G 0 -orbits in M reg 0 , which generate infinitesimal residual gauge transformations. This ambiguity drops out under the eventual projection to the reduced phase space M/G. Thus, our slight abuse of the term reduced is harmless.
Similarly, the pullback invariants π * 1 (h) associated with the functions h ∈ C ∞ (G) G lead to interesting reduced evolution equations on M ′ reg 0 . We find (Proposition 2.6) that they take the following form: g = [g, r(λ)∇h(g)],λ = −(∇h(g)) 0 . (1.8) Using the Killing form −, − G of G, ∇h(g) ∈ G is defined by the relation X, ∇h(g) G = d dt t=0 h(e tX g) for all X ∈ G, and r(λ) ∈ End(G) is the rational dynamical r-matrix that vanishes on G 0 and operates on G ⊥ as (ad λ ) −1 . These evolution equations matter up to residual gauge transformations like in the case of (1.7).
By parametrizing J in (1.7) according to and taking ϕ(J) = − 1 2 J, J G , the system (1.7) can be recognized as a spin Sutherland system, for which the components of q and p form canonically conjugate pairs and ξ is a so-called collective spin variable [19,30]. (See also equation (2.37).) For G = SU(n), restriction to a small symplectic leaf in the reduced phase space gives the trigonometric (spinless) Sutherland system [24]. On the same symplectic leaf, but using a different parametrization and the Hamiltonian h(g) = ℜtr(g), the system (1.8) yields a specific real form of the rational Ruijsenaars-Schneider system, which enjoys a duality relation with the trigonometric Sutherland system [14,21].
The above sketched results about reductions of the cotangent bundle are known to experts, especially the reduced system described in terms of M reg 0 . In this paper we take the lead from this example and characterize the reductions of the Heisenberg double M and the quasi-Poisson double D in a similar manner. To highlight a key feature of these generalizations, note that the first model M reg 0 was obtained by 'diagonalizing' the first one out of the pairs of elements forming M, and the second model M ′ reg 0 was obtained by diagonalizing the second constituent of those pairs. The pullback invariants built by using π * 2 then led to interesting reduced evolution equations on M red 0 , and those built on π * 1 led to interesting evolution equations on M ′ reg 0 . The situation turns out fully analogous for the reductions of the other two doubles. In particular, we shall derive two presentations of the Poisson algebras of the G-invariant functions, and describe the form of the interesting reduced evolution equations induced by the two rings of pullback invariants. Concerning the Heisenberg double, these results are summarized by Theorem 3.5 together with Proposition 3.8, and Theorem 3.10 with Proposition 3.12, which are tied with two partial gauge fixings akin to what is displayed in (1.4) for T * G. The analogous results pertaining to the quasi-Poisson double are formulated in Theorem 4.3 and Proposition 4.4. These theorems and propositions constitute the main new results of the present paper.
Motivated by the case of T * G [41] and the results of [14,15,16,21], we say that the two kinds of reduced systems that arise from the same double are in duality with each other. In the case of the quasi-Poisson double, duality actually becomes self-duality. The meaning of these dualities will be elaborated in the text.
Degenerate integrability and reduction. First of all, let us specify the precise notion of degenerate integrability used in this paper. Definition 1.1. By definition [35], a degenerate integrable system on a symplectic manifold of dimension N consists of an Abelian Poisson subalgebra of the Poisson algebra of smooth functions such that its functional dimension, δ, is smaller than N/2, and the functional dimension of its centralizer is (N − δ). To put it more plainly, the system is built on 1 ≤ δ < N/2 functionally independent, mutually Poisson commuting Hamiltonians that admit (N − δ) functionally independent joint constants of motion. An additional requirement is that the commuting Hamiltonians should possess complete flows.
Degenerate integrability is a stronger property than Liouville integrability, which corresponds to the limiting case δ = N/2. For the structure of the systems having this property, see [33,35,42,43]. Further variants of the notion of integrability, as well as their extension to Poisson manifolds, and even to Abelian Lie algebras of non-Hamiltonian vector fields, are also discussed in the literature [23,29,53].
The restrictions of our reduced systems are expected to give degenerate integrable systems on generic symplectic leaves of the quotient space of the double in each case. Reshetikhin has argued [41] that this is the case for the complex holomorphic analog of the cotangent bundle T * G, and his arguments can be adapted to the compact real form. His joint paper with Arthamonov [4] leads to the same conclusion regarding the quasi-Poisson double. It may well be that integrability holds on all symplectic leaves (with only Liouville integrability on exceptional leaves), but we cannot prove this at present. Nevertheless, we deem it worthwhile to outline two mechanisms that point toward the heuristic statement that 'degenerate integrability is generically inherited by the reduced systems engendered by Poisson reduction'.
Let V be a G-invariant vector field on a G-manifold X. Equivalently, if x(t) is an integral curve of V , then A η (x(t)) is also an integral curve for each η ∈ G, where A η denotes the diffeomorphism of X associated with η ∈ G. Suppose now that G is compact and denote by d G the probability Haar measure on G. For any real function F ∈ C ∞ (X) define the function F G ∈ C ∞ (X) G by averaging the functions Clearly, if F is a constant of motion for the vector field V , then F G is a G-invariant constant of motion for V . In [53] averaging was used for arguing that, generically, degenerate integrability survives Poisson reduction. In this work it was assumed that the G-action is generated by an equivariant moment map into G * . However, the fine structure of the quotient space of X was treated only rather casually. See also the review [23]. The averaging of the unreduced constants of motion is applicable in all cases that we study. The Hamiltonian vector fields of the pullback invariants are invariant under the 'conjugation action' of G on the unreduced phase space, except for the pullback invariants from π * 1 (C ∞ (G) G ) ⊂ C ∞ (M). In the latter case, G-invariance of the Hamiltonian vector fields holds with respect to an action that has the same orbits as the conjugation action; this is explained in Appendix C. Now we formulate a second mechanism whereby integrability can descend to reduced systems. We extracted this mechanism from the work of Reshetikhin [41]. It will turn out to be applicable to all of our examples of interest. We begin by listing a number of strong assumptions. First, consider two G-manifolds X and Y for which both quotient spaces X/G and Y /G are manifolds such that π X : X → X/G and π Y : Y → Y /G are smooth submersions. Second, suppose that Ψ : X → Y is a smooth, G-equivariant, surjective map. Then, Ψ gives rise to a well-defined smooth, surjective map Ψ red : (1.11) Third, suppose that we have a vector field V on X that is projectable to a vector field V red on X/G. Coming to the crux, if we now assume that Ψ is constant along the integral curves of V , then we obtain that Ψ red is constant along the integral curves of V red . Indeed, this holds since the integral curves of V red result by applying π X to the integral curves of V . In such a situation, Ψ * (C ∞ (Y )) gives constants of motion for V and Ψ * red (C ∞ (Y /G)) gives constants of motion for V red . In particular, the functional dimension of the ring of constants of motion for the projected vector field V red is at most dim(G) less than the dimension of Y . Under favourable circumstances, this mechanism can be used to show the degenerate integrability of the reduced system on X/G that descends from the commuting (Hamiltonian) vector fields of a degenerate integrable system on X. The unreduced commuting Hamiltonians must be G-invariant, and must remain independent after reduction. To put this mechanism into practice, one may have to restrict oneself to dense open submanifolds and to generic symplectic leaves of the quotient Poisson structure. This will become clear in the examples.
Layout and notations. The organization of the rest of the paper is shown by the table of contents. Sections 2, 3 and 4 are devoted to the three doubles, starting from the cotangent bundle. In each case, we first describe the unreduced phase space and its degenerate integrable systems, and then turn to their reductions. We have already delineated the theorems and proposition that contain our main new results.
These results and open problems are briefly discussed in Section 5. Three appendices are also included, which contain auxiliary material. In particular, Appendix A summarizes some Lie theoretic background that the reader may wish to look at before reading Section 3.
Throughout the paper, our notations 'pretend' that we are dealing with matrix Lie groups. For example, ηJη −1 in equation (2.5) denotes the adjoint action of η ∈ G on J ∈ G. As another example, Xg in (2.13) stands for the value at g ∈ G of the right-invariant vector field on G associated with the element X from the Lie algebra G of G. Such matrix notations simplify many formulas considerably, and can be easily converted into more abstract notation if desired. Then one can verify that our results are valid for abstract Lie groups as well. Alternatively, one may employ faithful matrix representations of the underlying Lie groups.
2. The case of the cotangent bundle T * G Let G be a connected and simply connected compact Lie group whose Lie algebra G is simple. In this section we describe two degenerate integrable systems on the cotangent bundle T * G and characterize their Poisson reduction induced by the conjugation action of G. The first system contains the Hamiltonian that generates free geodesic motion on G, and its reduction leads to a trigonometric spin Sutherland model. The reduction of the other system on T * G gives rational spin Ruijsenaars-Schneider type models. Most of the results presented in this section are available in the literature [19,41]. We include their treatment mainly in order to motivate the subsequent generalizations. However, the descriptions of the reduced Poisson brackets and equations of motion as given by Theorem 2.4 and Proposition 2.6 appear to be new.
Let us identify the dual space G * with G using the (negative definite) inner product −, − G , which is a multiple of the Killing form, and then identify M := T * G with G × G using right-translations. The canonical Poisson bracket on the phase space can be written as where the derivatives are taken at (g, J). Here and below, we use the G-valued derivatives of any F ∈ C ∞ (M), defined by and The group G acts by simultaneous conjugations of g and J, i.e., the action of η ∈ G on M is furnished by the map That is, H = π * 2 (ϕ) using the natural projection π 2 : M → G. There are ℓ := rank(G) functionally independent Hamiltonians in this set, since the ring of invariants for the adjoint action of G on G, C ∞ (G) G , is freely generated by ℓ basic invariants (see, e.g., [32], Section 30). The Hamiltonian vector field engendered by H can be written asġ = (dϕ(J))g,J = 0, (2.8) and its integral curve through the initial value (g(0), J(0)) reads (g(t), J(t)) = (exp(tdϕ(J(0)))g(0), J(0)). (2.9) The corresponding constants of motion are given by arbitrary functions of J andJ (2.6). Since ψ(J) = ψ(J) for every function ψ ∈ C ∞ (G) G , and this gives ℓ relations, the functional dimension of the ring of constants of motion is 2 dim(G) − ℓ. Therefore the Hamiltonians (2.7) form a degenerate integrable system. Another degenerate integrable system arises from the Hamiltonians H ∈ C ∞ (M) G of the form In other words, H = π * 1 (h) with the projection π 1 : M → G. These Hamiltonians are in involution and form a ring of functional dimension rank(G), too. The corresponding evolution equations reaḋ g = 0,J = −∇h(g), (2.11) and their flows are given by (g(t), J(t)) = (g(0), J(0) − t∇h(g(0))) . (2.12) The constants of motion are now found as arbitrary functions of the pair (g, Φ), where Φ is the moment map (2.6). To show that the functional dimension of this ring of functions is 2 dim(G) − rank(G), consider the isotropy subalgebra of g, G(g) := {X ∈ G | Xg − gX = 0}, (2.13) whose dimension equals ℓ = rank(G) for generic g. Then notice the identity Φ(g, J), X G = 0 for all X ∈ G(g).
(2.14) 6 On a dense open subset of M, this implies ℓ relations between the components of Φ(g, J), and apart from this Φ varies freely if g is generic. It follows that the functional dimension of the ring of constants of motion is 2 dim(G) − ℓ, proving that the Hamiltonians (2.10) yield a degenerate integrable system. An element of G is regular if its isotropy group with respect to conjugations is a maximal torus in G, and an element of G is regular if its centralizer in G is the Lie algebra of a maximal torus. We fix a maximal torus G 0 < G and let G 0 denote its Lie algebra. Then G reg , G reg 0 and G reg , G reg 0 stand for the corresponding open dense subsets of regular elements. We also introduce the following sets and The submanifolds M reg 0 ⊂ M reg and M ′ reg 0 ⊂ M ′ reg are stable under the action of the normalizer of G 0 in G, which we denote by N: Note that G 0 is a normal subgroup of N, and the factor group N/G 0 is the Weyl group of the pair (G 0 , G). Any continuous function F on M can be recovered from its restriction to M reg 0 , as well as from its restriction to M ′ reg 0 . The restrictions of the G-invariant functions enjoy residual N-invariance. It is also easy to see that the restrictions of functions provide the following isomorphisms: and In preparation, now we introduce the dynamical r-matrices that will feature below. For this purpose, we consider the decomposition where G ⊥ is the orthogonal complement of the fixed maximal Abelian subalgebra G 0 < G with respect to the Killing form. Accordingly, we may write any X ∈ G as Then, for any Q ∈ G reg 0 we introduce R(Q) ∈ End(G) by using that (Ad Q − id) is invertible on G ⊥ . Moreover, for any λ ∈ G reg 0 we define r(λ) ∈ End(G) by using that ad λ is invertible on G ⊥ . These linear operators are well-known solutions of the (modified) classical dynamical Yang-Baxter equation [10]. They vanish identically on G 0 and are antisymmetric With the necessary definitions at hand, we are ready to derive convenient characterizations of the Poisson algebras of the invariant functions. We begin by noting that every G-invariant function on M satisfies the basic identity This is a consequence of the property d dt t=0 F (e tX ge −tX , e tX Je −tX ) = 0, ∀X ∈ G, (2.26) taking into account the equality ∇ ′ F (g, J) = g −1 ∇F (g, J)g. 7 2.1. Spin Sutherland models from reduction. For any the following formula holds:

29)
where R(Q) is given by (2.22) and the derivatives are taken at (Q, J).
Proof. In order to evaluate the right-hand side of (2.28), we have to express the derivatives of F and H in terms of the derivatives of the corresponding restricted functions. Plainly, we have where we use the decomposition (2.20). The invariance with respect to This can be solved: where the inverse is understood to be taken on G ⊥ . By using these equalities as well as the antisymmetry (2.24) and the invariance property of the Killing form, at (Q, J) we obtain   [18], in the G = SU(n) case the ordered eigenvalues of g ∈ G give such functions.) On the other hand, the same formula (2.29) yields a Poisson bracket also on C ∞ (M reg 0 ) G0 , since M reg 0 /G 0 is a covering space of M reg 0 /N, with the fibers labeled by the elements of the Weyl group N/G 0 . To avoid any possible confusion, we note that in (2.29) and similarly for the second term.
The following statement is an immediate consequence of Theorem 2.1 and the identity This gives the derivative of F with respect to an evolution vector field on M reg 0 , and the corresponding 'reduced evolution equation' on M reg 0 can be taken to bė (2.36) The solutions of the evolution equation (2.36) result by applying suitable (point dependent) Gtransformations to the unreduced integral curves (2.9), and they project onto the reduced dynamics on M reg /G ≃ M reg 0 /N. This follows from the general theory of Hamiltonian reduction [38]. Of course, the evolution vector field on M reg 0 is not unique, because the derivative of F ∈ C ∞ (M reg 0 ) N is zero along any vector field that is tangent to the orbits of G 0 in M reg 0 . We fixed this ambiguity by requiring that the derivative of any F ∈ C ∞ (M reg 0 ) should be given by the right-hand side of (2.35).
The reduced system governed by the Poisson bracket (2.29) and equations of motion (2.36) can be interpreted as a spin Sutherland model. Since this is well known [19,30], we only note that for ϕ(J) := − 1 2 J, J G the parametrization (1.9) of J by the new variables q (with Q = e iq ), p and ξ leads to which is a standard spin Sutherland Hamiltonian. Here, the sum is over the positive roots of the complexification G C of G with respect to the Cartan subalgebra G C 0 < G C , and the spin variable ξ ∈ G ⊥ is expanded as ξ = α>0 (ξ α E α − ξ * α E −α ) using root vectors E ±α (normalized according to Appendix A). 2.1.1. Degenerate integrability after reduction. We now discuss how the mechanism outlined around equation (1.11) is applicable to the present case. By inspecting the restriction on M reg 0 , it is easily seen that π * 2 (C ∞ (G) G ) gives rise to ℓ = rank(G) generically independent Hamiltonians on M/G. Let us now define the smooth, G-equivariant map Ψ 1 : M → G × G by where η ∈ G acts on G×G by applying Ad η to both components of (a, b) ∈ G×G. Then, taking any function We next outline a train of thought indicating that these constants of motion guarantee degenerate integrability after reduction.
The isotropy subgroup of generic elements from the image of Ψ 1 is clearly just the center Z G of G. These generic elements form a manifold Y of dimension 2 dim(G) − ℓ, and its pre-image X ⊂ M is a dense, open, G-invariant subset. Thus, taking Ψ := Ψ 1 in (1.11), we obtain dim(G) − ℓ functionally independent constants of motion for the restriction of the reduced system to X/G ⊂ M/G. By using the moment map Φ (2.6), the G-invariant functions of the form φ • Φ, with any φ ∈ C ∞ (G) G , descend to ℓ independent Casimir functions on M/G. Fixing the values of these Casimir functions, generically one obtains a symplectic leaf of dimension dim(G) − ℓ in M/G. Thus, on the intersection of such a generic symplectic leaf with X/G, there remain dim(G) − 2ℓ independent constants of motion. This is sufficient for degenerate integrability since the commuting reduced Hamiltonians remain independent on the generic symplectic leaves.
The above arguments make us confident to expect degenerate integrability on generic symplectic leaves of M/G. These arguments essentially coincide with those presented by Reshetikin [41] for the corresponding complex holomorphic systems. A more complete, rigorous analysis of reduced integrability is beyond the scope of the present paper.  39) and the derivatives with respect to the first variable are given by (2.3).

Defining the reduced Poisson brackets of F and H by
the following formula holds: where r(λ) is given by (2.23) and the derivatives are taken at (g, λ).
Proof. First of all, we remark that and, as a consequence of the invariance under G 0 < G, The subscript 0 refers to the decomposition (2.21). In view of the formula (2.2), we have to express (d 2 F (g, λ)) ⊥ in terms of the derivatives of F with respect to the variable g. By applying (2.25) at (g, λ) and using the above relations, we find with r(λ) (2.23). Then, substitution in the right-hand side of (2.40) leads to This can be simplified by virtue of the classical dynamical Yang-Baxter equation [10], which can be written as [19] [r(λ)X, We now take X : By inserting this into (2.45) and collecting terms, we arrive at the claimed formula (2.41).

The next result follows from Theorem 2.4 by using that
The corresponding reduced evolution equation on M ′ reg 0 can be taken to bė The counterpart of the discussion presented after Proposition 2.3 is applicable in this case as well. We merely note that the solutions of the evolution equations (2.51) can be obtained by applying suitable G-transformations to those unreduced integral curves (2.12), whose initial values belong to M ′ reg 0 . It is known [14] that in the G = SU(n) case the above reduced system contains a real form of the rational Ruijsenaars-Schneider model on a special symplectic leaf. The leaf in question arises by fixing the Casimir functions φ•Φ (φ ∈ C ∞ (G) G ) in such a way that the corresponding joint level surface in G ≃ G * is a minimal (co)adjoint orbit of dimension 2(n − 1). The main Hamiltonian of this model is associated with the function h(g) = ℜtr(g) on G. This lends justification to the terminology 'spin Ruijsenaars-Schneider type models' [41] as a name for the models that stem from the integrable Hamiltonians (2.10) in general. However, in contrast to the spin Sutherland models described in the preceding subsection, it is still an open problem to separate the variables of these models into canonically conjugate pairs complementing the components of λ and additional 'spin' degrees of freedom.

2.2.1.
Degenerate integrability and duality. The degenerate integrability of the reduced systems built on the pullback invariants π * 1 (C ∞ (G) G ) can be analyzed quite similarly to the previous case of π * 2 (C ∞ (G) G ). Now one may use the map which is constant along the flows of any H ∈ π * 1 (C ∞ (G) G ), and is G-equivariant with respect to the same action that operates on M. The arguments presented at the end of Subsection 2.1 go through with little modification, as is discussed in [41] in the holomorphic case. In particular, employing any χ ∈ C ∞ (G × G) G , the function Ψ * 2 (χ) ∈ C ∞ (M) G is a smooth, G-invariant constant of motion. Incidentally, the maps Ψ 1 (2.38) and Ψ 2 (2.52) are Poisson maps with respect to suitable Poisson structures on the target spaces G × G and G × G, which can be easily found by requiring this property to hold. Therefore the just mentioned G-invariant constants of motion Ψ * 2 (χ) form a closed Poisson subalgebra of C ∞ (M) G (and similarly for Ψ 1 ).
Finally, let us comment on the duality between the spin Sutherland and the spin Ruijsenaars-Schneider systems. To this end, we regard the functions of Q in (2.15) and λ in (2.16) as 'position variables' for the respective models. Those functions of Q that descend to well-defined functions on M/G arise from π * 1 (C ∞ (G) G ) and the functions of λ having the same property arise from π * 2 (C ∞ (G) G ). In this way, one of the two sets of pullback invariants plays the role of 'global position variables' in every reduced system, while the other set engenders the commuting Hamiltonians of interest of the same system. The role of the two sets of pullback invariant is interchanged in the two systems. That is, since both systems leave on the same phase space M/G, the global position variables of one system are the interesting Hamiltonians of the other one, and vice versa. This kind of duality was originally discovered by Ruijsenaars for spinless models (see the review [44] and references therein). We call it Ruijsenaars duality or actionposition duality, taking into account that in integrable models the commuting Hamiltonians are in bijective correspondence with the action variables. We prefer this to the term action-angle duality, which is also used in the literature.

Integrable systems from the Heisenberg double
In this section we first describe the Heisenberg double associated with a compact Lie group G, and specify two degenerate integrable systems on this phase space. We then study the Poisson reduction of these systems. For notations, see the remark at the end of Section 1, and also Appendix A. For the underlying theory of Poisson-Lie groups, one may consult the reviews [27,49].
3.1. The basics of the Heisenberg double. We start with a compact simple Lie algebra, G, and pick a maximal Abelian subalgebra, G 0 . These can be regarded as real forms of a complex simple Lie algebra, G C , and its Cartan subalgebra, G C 0 . Choosing a system of positive roots, we obtain the triangular decomposition where G C > is spanned by the eigenvectors associated with the positive roots. Referring to this, we may present any X ∈ G C as with the terms taken from the corresponding subspaces. The real vector space is a Lie subalgebra of the 'realification' G C R of G C (i.e. G C viewed as a real Lie algebra), and it gives rise to the direct sum decomposition Correspondingly, we may write any X ∈ G C R as We equip G C R with the invariant, non-degenerate, symmetric bilinear form −, − I , defined as the imaginary part of the complex Killing form −, − of G C . The decomposition (3.4) represents a so-called Manin triple [27,49], since G and B are isotropic subalgebras of G C R with respect to −, − I . Let G C R be a connected and simply connected real Lie group whose Lie algebra is G C R , and denote G and B its connected subgroups associated with the Lie subalgebras G and B. These subgroups are simply connected and G is compact. Later we shall also need the connected subgroup G C 0 < G C R associated with G C 0 as well as the subgroups G 0 < G and B 0 < B associated with G 0 and iG 0 . Occasionally, we view G C R as the realification of the corresponding complex Lie group, G C . Now, we recall [48,49] that the group manifold carries the following two natural Poisson brackets: where π G and π B are the projections from G C R onto G and B, respectively, defined by means of (3.4). Here, we use the G C R -valued 'left-and right-derivatives' of F, H ∈ C ∞ (M ): The minus bracket makes M into a Poisson-Lie group, of which G and B are Poisson-Lie subgroups. Their inherited Poisson brackets take the form and 10) The derivatives are G-valued for ϕ i ∈ C ∞ (B) and B-valued for f i ∈ C ∞ (G), reflecting that these subalgebras are in duality with respect to −, − I . To be sure, we write the definitions where ϕ ∈ C ∞ (B) and f ∈ C ∞ (G). We shall also use the G-valued derivatives of f ∈ C ∞ (G), 13) and note that the 14) and thus the two kinds of derivatives of f ∈ C ∞ (G) are related by 16) the relation of the derivatives can also be written as Of course, analogous relations hold for the right-derivative D ′ f , too. With these relations at hand, one can prove the identity In terms of the decomposition X = X > + X 0 + X < , one has (3.19) and the right-hand side of (3.18) has the familiar form of a Sklyanin bracket. The Poisson bracket {−, −} + (3.7) corresponds to a symplectic form [2], and (M, {−, −} + ) is called [48] the Heisenberg double of the Poisson-Lie groups G and B. It is a Poisson-Lie analog 2 of the cotangent bundle T * G (and of T * B). Any element K ∈ M admits unique (Iwasawa) decompositions [26] into products of elements of G and B, which we write as These decompositions give rise to the maps Ξ L , Ξ R : M → G and Λ L , Λ R : M → B, These are all Poisson maps from the (M, {−, −} + ) onto the respective Poisson-Lie groups, and the same is true for the products of any two of these maps into the same group. Without going into details, we recall that any Poisson map into a Poisson-Lie group serves as a moment map that generates a (possibly only infinitesimal) Poisson-Lie action of the corresponding dual group [31]. In particular, the Poisson 22) generates the so-called quasi-adjoint action of G on the Heisenberg double. As was shown in [25], the corresponding action map, A 1 : G × M → M , is given by Said more directly, the pair ( It is shown in Appendix B that the map m is a Poisson diffeomorphism if M is endowed with the following Poisson bracket: The derivatives on the right-hand side are taken at (g, b) ∈ G × B, with respect to the first and second variable, according to the definitions (3.12) and (3.11), respectively. In particular, An alternative form of (3.26) results by employing G-valued derivatives with respect to the first variable, defined like in (3.13).
In terms of the model M, the quasi-adjoint action (3.21).) Here, we use the dressing action of G on B, defined by whose infinitesimal version reads where the decomposition (3.5) is applied to where A i η denotes the map of the relevant manifold obtained by fixing the first argument of A i . We observe that the G-action A 2 (3.27) has the same orbits as the simpler action given by the map A : G × M → M: (g, b)) := (ηgη −1 , Dress η (b)). (3.31) Since the orbits of A are the same as those of the Poisson-Lie action A 2 , these two G-actions share the same invariant functions, and thus are equivalent from the point of view of Poisson reduction. The real Lie algebra G C R carries the Cartan involution, θ, that fixes G pointwise and multiplies the elements of iG by −1. It lifts to a corresponding involutive automorphism Θ of G C R , of which G < G C R is the fixed point set. Referring to (3.1), θ maps G C > onto G C < . We shall use the notations Z τ := −θ(Z), The maps Z → Z τ and K → K τ are anti-automorphisms satisfying This operation is often denoted simply by dagger, since for the classical Lie groups one can choose the conventions in such a way that X τ = X † and K τ = K † with dagger denoting the matrix adjoint [26]. Later we shall also need the closed submanifold which is diffeomorphic not only to G but also to B. Note that P is a connected component of the fixed point set of the anti-automorphism K → K τ of G C R , and a diffeomorphism with B is provided by the map ν : B → P, ν(b) := bb τ . (3.35) The map (3.35) intertwines the dressing action with the obvious conjugation action of G on P, since we have (3.36) This implies that any element of B can be transformed into B 0 = exp(iG 0 ) by the dressing action. As an alternative to G × B, one may also take G × P as a model of the Heisenberg double.
Remark 3.1. After small notational changes, all considerations of the paper apply to reductive compact Lie groups as well. For example, one can take G = U(n), G C = gl(n, C), and G C = GL(n, C), in which case B can be taken to be the upper triangular subgroup whose diagonal elements are positive real numbers. Then, K τ = K † , and P is the space of positive definite, Hermitian matrices. The reader may keep this example (or the example of G = SU(n)) in mind when reading the text. We restricted ourselves to simple Lie algebras just in order have a shorter presentation.

3.2.
Two degenerate integrable systems on the Heisenberg double. Now we present two degenerate integrable systems. For this, we let π 1 and π 2 be the projections from M onto G and B, respectively, π 1 : (g, b) → g, π 2 : (g, b) → b.
(3.37) Then, consider the following families of functions on M, where the superscript refers to invariance with respect to the conjugation and dressing actions of G on G and on B, respectively. When presented in terms of the model M , these become  .27) and (3.30). In order to see that they yield two Abelian Poisson algebras and to identify their constants of motion, let us describe the flows generated by these Hamiltonians. For this, we notice from (3.7) that the Hamiltonian vector field belonging to H ∈ C ∞ (M ) generates the evolution equatioṅ (3.42) The solution K(t) corresponding to the initial value K(0) is provided by Proof. We begin by pointing out that φ ∈ C ∞ (B) G satisfies which are both G-valued. Formula (3.41) follows by putting these derivatives into (3.40), where we applied (3.45) and the decomposition K = g L b −1 R . By taking K = b L g −1 R and using that Dφ(b R ) is G-valued, (3.41) impliesġ R = Dφ(b R )g R andḃ L = 0. It follows that b L remains constant. The moment map Λ (3.22) is also constant along the flow, for H ∈ C ∞ (M ) G , and therefore b R stays constant as well.
Hence we obtain the formula for g R (t).
The formula for the time development of g L then follows directly from (3.42), or alternatively from the identity where we took (3.45) into account. This also provides a consistency check on our calculations.
Since all smooth functions depending on b L and b R are constants of motion, we see in particular that the elements of Λ * R C ∞ (B) G Poisson commute. 3 The number of independent constants of motion is 2 dim(B) − ℓ. This is a consequence of the identity that leads to ℓ relations between the functions of b R and b L . We here used that G acts by conjugations on the model P of B (3.35), and thus The ring C ∞ (P) G ≃ C ∞ (B) G is generated by ℓ = rank(G) basic invariants, which equals the functional dimension of Λ * R C ∞ (B) G (3.39) as well. In conclusion, these Hamiltonians form a degenerate integrable system on M . Of course, the same is true for the equivalent Hamiltonians π The constituents in the decompositions (3.52) The solution can be written as where β(t) and γ(t) are determined by the following factorization problem in G C R : Proof. Lemma B.2 now gives Any function h ∈ C ∞ (G) satisfies Dh(g) = (gD ′ h(g)g −1 ) B , and Dh(g) = D ′ h(g) holds for h ∈ C ∞ (G) G . Thus we get and ρ(∇H(K)) = 1 2 b L (Dh(g R ))b −1 L . Inserting these into (3.40) leads to (3.51): where the last equality relies on writing 58) and [g R , i∇h(g R )] = 0 because of the invariance property of h. By using these relations, the formula foṙ g R is derived from (3.59) Turning to the solution, we first note that the curve (g R (t), b R (t)) defined by (3.53) satisfies the differential equationsġ (3.60) Moreover, the equality (3.54) implies From here, we get where first equality holds because of the G-invariance of h. We see from (3.62) that Inserting these relations into (3.60), we obtaiṅ Since (i∇h(g R )) B = Dh(g R ), comparison with (3.52) shows that the proof is complete.
It is clear that Ξ * R C ∞ (G) G is generated by ℓ = rank(G) functionally independent Hamiltonians, which are in involution, since they remain constant along the flows (3.53). To show their degenerate integrability, we observe that any smooth real function of is a constant of motion. Indeed, Λ(K) = b L b R and Ξ L (K) = g L are both constants of motion by (3.52). We see from (3.65) that the set of the elements W (K) is the union of those conjugacy classes in G C that have representatives in G 0 < G. Generically, the elements of this set can be parametrized by (N − ℓ) real variables, where N = 2 dim(G) is the dimension of the Heisenberg double. This holds since the generic elements of G 0 < G C R are fixed precisely by G C 0 with respect to conjugations. It follows that the functional dimension of the ring of joint constants of motion of the Hamiltonians belonging to Ξ * R C ∞ (G) G is (N −ℓ), and thus these Hamiltonians form a degenerate integrable system. We denote by C ∞ (M) G the ring of invariant functions. Any F ∈ C ∞ (M) G satisfies the identity as follows by the taking derivative of F • A e tX = F with respect to t, for every X ∈ G. Here we utilized the decomposition (3.4), but below we shall also use the alternative decomposition whereby we may write Consider the connected subgroup G C 0 < G C corresponding to G C 0 . By definition, the subset and for any g 0 ∈ G C 0,reg we extend the definition (2.22) by putting We introduce the following subsets of M (3.24), and observe that the restriction of functions provides an isomorphism using the normalizer N (2.17). For any F ∈ C ∞ (M reg 0 ), we introduce the derivative D 1 F (Q, b) ∈ B 0 by The G-valued derivatives D 2 F and D ′ 2 F are determined analogously to (3.11).
where the subscript B refers to (3.4), the derivatives are taken at (Q, b), and R(Q) is given by (3.69).
where we use the decomposition (3.1). Then the identity (3.66) implies that and This is solved by where we use the triangular decomposition (3.1).
We have to substitute the above relations into On the other hand, we get    can be taken to bė because φ ∈ C ∞ (B) G , as was noted before (3.45). Moreover, we have D 1 H(Q, b) = 0 and, due to the antisymmetry of R(Q), Proof. Let us put X := R(Q)(Dφ(L)), which belongs to G, and note that Then, starting from (3.83), we geṫ It is an interesting exercise to recast the Poisson bracket (3.26) and its reduced version (3.79) in terms of the models G × P and G reg 0 × P of M and M reg 0 . 3.3.1. Reduced integrability and interpretation as deformed spin Sutherland models. Let us define the smooth, G-equivariant map Ψ 3 : M → P × P by where η ∈ G acts on P × P by conjugating both components of (a, b) ∈ P × P. Then, for any function χ ∈ C ∞ (P × P) G , Ψ * 3 (χ) ∈ C ∞ (M) G gives a smooth, G-invariant constant of motion. Recalling that P = exp(iG), we see the close analogy with the constants of motion observed in the cotangent bundle case (cf. equation (2.38)). Thus, degenerate integrability on generic symplectic leaves of M/G should hold in our present case as well. We do not repeat the arguments of Section 2.1, only make two remarks.  [38,50,51].) By using the Poisson-Lie version of symplectic reduction, we developed a detailed description of these subspaces in [12]. Now, we translate the result into our present Poisson reduction setting.
Let and this induces the identification In this parametrization of the Poisson quotient the components of q in Q = e iq and p form canonically conjugate pairs, and S + ∈ B > is a 'collective spin degree of freedom' that decouples from q and p under the reduced Poisson bracket. The space B > /G 0 represents the reduction B with respect to G 0 < G, at the zero value of the classical moment map that generates the conjugation action of G 0 on B. The 'main reduced Hamiltonians' are obtained by taking the trace of L(Q, p, S + ) = e p b + (Q, S + )b + (Q, S + ) τ e p (3.96) in the fundamental irreducible representations of G C . In [12], the structure of b + (Q, S + ) was elaborated (for G = U(n) even its fully explicit formula was given), and by using this it was shown that the Lax matrices L(Q, p, S + ) and the main Hamiltonians of the models at issue are deformations of the Lax matrices (1.9) and main Hamiltonians of the spin Sutherland models (2.37). For the details of these results, one can consult [12].
3.4. The duals of the deformed spin Sutherland models. Now, we describe the reduction of the integrable system of and denote by B reg the union of the G-orbits in B that intersect B reg 0 . Then define We remark that all powers of Γ ∈ B reg 0 belong to B reg 0 . Similarly to (3.71), the restriction of functions provides the isomorphism (3.99) Our aim is to find the formula for the Poisson bracket on C ∞ (M ′ reg 0 ) N induced by this isomorphism. The derivation follows the steps of the previous section, but it is slightly more complicated. In preparation, we introduce ̺(Γ) ∈ End(G C ) by This is well-defined 5 due to the definition of B reg 0 . Note that ̺(Γ) vanishes on G C 0 and it maps B > to itself. Below we shall apply the operator R(Γ 2 ) (3.69), which can also be written as For any F ∈ C ∞ (M ′ reg 0 ), the derivative D 2 F (g, Γ) ∈ G 0 is determined by and the B-valued derivatives D 1 F and D ′ 1 F are determined analogously to (3.12). Let F ∈ C ∞ (M ′ reg 0 ) N be the restriction of F ∈ C ∞ (M ′ reg ) G . Then we obviously have , (3.103) and the residual G 0 -invariance of F implies The full expression of D 2 F (g, Γ) through the derivatives of F is given by the next lemma.

106)
and this implies the formula Furthermore, we have 108) 5 In fact, G C ⊥ is spanned by the root vectors Eα and α(γ) is a nonzero real number for e γ ∈ B reg 0 . See Appendix A. 19 and Here, Γ = e γ ∈ B reg 0 and the operators (3.100), (3.101) are employed. For the definition of Y τ , see equation (3.32) and Appendix A.
Proof. Let us note that any V ∈ G C can be decomposed as and then According to (3.106), we need to solve an equation of the form for X, where Y = Y > ∈ B > and X = (X > + X < + X i 0 ) ∈ G. By using that Γ τ = Γ and X > = −(X < ) τ , we get From here, we get Then we find By taking X = D ′ 2 F (g, Γ), the proof is finished.
Let us recall that the functions f on G have the G-valued derivatives ∇f and ∇ ′ f defined in (3.13), and (as seen from (3.17)) these are related to the B-valued derivatives Df and D ′ f by In consequence of (3.16) and (3.17), one also has

121)
where ∇ 1 F and ∇ 1 H denote the G-valued derivatives defined similarly to (3.13), and R(Γ 2 ) (3.101) is used. 20 Proof. We have to evaluate the formula (3.26) at (g, Γ) ∈ G × B reg 0 for invariant functions. By using the equalities (3.103) and Lemma 3.9 we find (3.122) We took into account that R(Γ 2 ) maps B to itself and that X, Y I = 0 for any X, Y ∈ B. Next, direct substitution gives We now collect terms, and in doing so employ the antisymmetry of R(Γ 2 ) together with the properties which follow from the definitions. This gives Referring (3.119), we can write where the last step holds since D ′ 1 F, X I = ∇ ′ 1 F, X G for all X ∈ G, and R(Γ 2 )(iX) ∈ G for all X ∈ G. Therefore, the first 4 terms of (3.125) yield the right-hand side of (3.121). The rest of the terms cancel, because (by (3.119)) we have and this is just the opposite of the remaining term − (D ′ 1 F, g −1 (D 1 H)g I of (3.26). This holds due to the identity (3.18).

(3.129)
One can also use the identity (3.128) to get an alternative formula the Poisson-Lie struture (3.18) on G.
As a consistency check, we verified that the reduced evolution equation (3.132) results also by applying the projection method to the corresponding unreduced evolution equation (3.64). Of course, the evolution equation on M ′ reg 0 is unique only up to infinitesimal gauge transformations that do not change its eventual projection on M ′ reg /G ≃ M ′ reg 0 /N. In the G = SU(n) case the reduced system characterized by Theorem 3.10 and Proposition 3.12 gives [15] a special real form of the trigonometric Ruijsenaars-Schneider system on a small symplectic leaf of dimension 2(n − 1) in M ′ reg 0 /N. Thus one may expect to obtain spin Ruijsenaars-Schneider type systems on generic symplectic leaves. However, it is not known how to introduce positions, momenta and spin variables in such a way that would endow the reductions of the pullback invariants π * 1 (C ∞ (G) G ) with a many-body interpretation. This is analogous to the open problem that exists in relation to the second kind of reduced systems obtained from T * G.
Toward integrability after reduction. We here explain that the mechanism described around equation (1.11) is applicable in the present case, too. For this, we use the original model (M, {−, −} + ) (3.7) of the Heisenberg double, and define the map Ψ 4 : M → G C by using the formula (3.65) and the definitions (3.21). We have seen that this map is constant along the Hamiltonian flows generated by the pullback invariants Ξ * R (C ∞ (G) G ). The conjugation action of G on G C is defined by the maps C η , We wish to show that Ψ 4 is equivariant with respect to the quasi-adjoint action (3.23) and the conjugation action (3.134), In order to derive this, notice from (3.23) that Therefore, we obtain Let us then consider the ring of G-invariant real functions Finally, let us note that the two kinds of reduced systems described in this section are subject to a similar duality relation that we outlined at the end of Section 2.
Remark 3.13. Let us consider the function χ ρ (K) := tr ρ (K) on G C , where ρ is some finite dimensional irreducible representation of G C . Then we obtain tr ρ (W (K)) = tr ρ (Ξ R (K)). This shows that the constants of motion F • Ψ 4 contain the basic pullback invariants associated with the real and imaginary parts of the characters of the irreducible representations of G.

Reduction of the quasi-Poisson double G × G
Quasi-Hamiltonian manifolds [3] and quasi-Poisson manifolds [1] were introduced primarily in order to provide a purely finite dimensional construction of the symplectic and Poisson structures of moduli spaces of flat connections, which were originally obtained by infinite dimensional symplectic reduction. Since then, interesting applications of these concepts came to light in several fields, including the construction of finite dimensional integrable Hamiltonian systems [8,11,16,18]. The content of this section is closely related to the work Arthamonov and Reshetikhin [4], who constructed degenerate integrable systems on moduli spaces of flat connections, working mostly in a complex holomorphic setting.
Let us recall that a quasi-Poisson manifold is a G-manifold, here denoted S, equipped with a Ginvariant bivector, Π, whose key property is that the formula The vector field V H descends to S/G if H is G-invariant. This means that the process of taking the quotient by the G-action works for quasi-Poisson manifolds in the same way as it does for Poisson manifolds. The quotient space is known to be a disjoint union of smooth symplectic manifolds, just as for reductions defined by Hamiltonian actions of compact Lie groups [38,50,51]. For general functions, the quasi-Poisson bracket (4.1) violates the Jacobi identity in a specific manner. For this and further details, one may consult [1].
We see from Proposition 4.1 that the ring π * 2 C ∞ (G) G forms an Abelian Poisson algebra, and g 2 as well asg 1 := g 1 g 2 g −1 1 are constant along all of the corresponding integral curves (4.13). This shows that the functional dimension of the joint constants of motion for the evolution equations in (4.13) is dim(D)−rank(G). In conclusion, the family of Hamiltonians π * 2 C ∞ (G) G , of functional dimension rank(G), behaves basically in the same way as a degenerate integrable system on a symplectic manifold. Quite similar observations apply to the Poisson algebra π * 1 C ∞ (G) G . We merely note that the relevant constants of motion are now provided by arbitrary smooth functions of g 1 andg 2 := g 2 g 1 g −1 2 . Mimicking the reduction procedure of Section 2, we introduce the submanifolds and Using the normalizer N (2.17), restriction of functions engenders the isomorphisms and We next point out that the bracket (4.9) simplifies considerably for invariant functions.
Proposition 4.2. If F , H ∈ C ∞ (D) G , then the formula (4.9) can be rewritten as 21) and alternatively also as Proof. The derivatives of the G-invariant functions F and H satisfy leads to the formula gives Here, R(Q) is given by (2.22), and the derivatives are taken at (Q, g) and at (g, Q), respectively. 24 Proof. By taking advantage of the identity (4.23) at (Q, g) ∈ D reg 0 (4.17), we can express the derivatives of F in terms of the derivatives of F as follows: g)) . (4.29) By inserting this and the similar formula for the derivatives of H into (4.21), we obtain (4.25). The details of this straightforward calculation are omitted. The derivation of (4.27) is fully analogous, and can also be obtained from the previous case by exchange of the subscripts 1 and 2, accompanied by applying an overall minus sign.

30)
and if H is the restriction of H = π * 1 φ with φ ∈ C ∞ (G) G , then the reduced Poisson bracket (4.27) gives Therefore, the reduced evolution equations associated with H can be written, respectively, aṡ   Remark 4.7. The investigations reported in [16,18] are equivalent to studying particular Poisson subspaces of D/G for G = SU(n). They can be obtained by fixing the values of the functions h in (4.34) so that they define a minimal conjugacy class in G, of dimension 2(n − 1). The Poisson subspaces in question were shown to be smooth symplectic manifolds, and the reduced integrable system was interpreted as a compactified trigonometric Ruijsenaars-Schneider model.
We end by recalling [17] that the group SL(2, Z) acts on D/G as follows. Define the diffeomorphisms S D and T D of the double by S D (g 1 , g 2 ) = (g −1 2 , g −1 2 g 1 g 2 ) and T D (g 1 , g 2 ) := (g 1 g 2 , g 2 ). (4.35) These maps descend to mapsŜ andT of D/G that satisfy the identitieŝ

Summary and outlook
In this paper, we performed a systematic study of Poisson reductions of 'master integrable systems' carried by the classical doubles of any compact (connected and simply connected) Lie group G associated with a simple Lie algebra G. Informally, using the terminology of matrix Lie groups, the outcome of our analysis can be summarized as follows. The starting phase space always consists of a pair of matrices, and the action of G is equivalent to simultaneous conjugation of those two matrices by the elements of G. We proceeded by bringing one of those matrices to a 'diagonal' normal form, and letting the other matrix serve as a Lax matrix that generates commuting Hamiltonians. The Lax matrix then satisfies reduced evolution equations of the formL We explained that the unreduced master systems possess the characteristic properties of degenerate integrability. Then, we presented convincing arguments indicating that these properties are inherited by the reduced systems, on generic symplectic leaves of the reduced Poisson space. A fully rigorous proof of integrability after reduction is hindered by the fact that the orbit space of the G-action is not a smooth manifold. We conjecture that reduced integrability holds on all symplectic leaves of the quotient space, generically degenerate integrability, and only Liouville integrability on exceptional symplectic leaves.
On special symplectic leaves of the reduced Poisson spaces associated with G = SU(n), one recovers the trigonometric Sutherland and Ruijsenaars-Schneider models, which are known to be (only) Liouville integrable [44]. These special cases and the changes of variables discussed around equations (1.9), (2.37) and (3.96) motivated us to call the reduced systems spin Sutherland and spin Ruijsenaars-Schneider type models. This terminology was also used in the papers by Reshetikhin [41,42] dealing with related complex holomorphic systems.
A very interesting open problem concerns the generalization of our analysis to doubles of loop groups. The investigation of quantum Hamiltonian reductions corresponding to our classical reductions appears to be a worthwhile project for the future, too. As far as we know, such a reduction treatment is so far available (see, e.g., [20]) only for the spin Sutherland models descending from T * G.
since every element Z ∈ G C R can be decomposed uniquely as By definition, the complex conjugation on G C R with respect to G is the map θ defined by θ(X + iY ) := X − iY. (A.4) The complex conjugation θ is an involutive automorphism of the real Lie algebra G C R . It is a Cartan involution, since −, − R is negative definite on its fixed point set, G, and is positive definite on its eigensubspace with eigenvalue −1, iG. When regarded as a map of G C to itself, θ is conjugate linear, i.e., θ(λZ) =λθ(Z) for all λ ∈ C. Notice also from the definitions that We also need the real bilinear form on G C R provided by the imaginary part of the complex Killing form, As a result of (A.5), this invariant, non-degenerate, symmetric bilinear form enjoys the equality A crucial fact is that G C R can be presented as the vector space direct sum of two isotropic subalgebras with respect to the bilinear form −, − I : B is a suitable 'Borel' subalgebra. We next recall how these subalgebras can be described using the root space decomposition of G C . For this, let us pick a maximal Abelian subalgebra G 0 of G. Its complexification G C 0 is a Cartan subalgebra of G C . Then consider the corresponding set of roots, R, and decompose R into sets of positive and negative roots R ± . Moreover, let ∆ be the associated set of simple roots. It is easily seen that the Cartan involution θ maps any root subspace G C α (α ∈ R) to G C −α . We then choose a Weyl-Chevalley basis of G C , which consists of root vectors E α for which E α , E −α = 2/|α| 2 for all α ∈ R + , and Cartan elements H αj := [E α , E −αj ] for α j ∈ ∆. The root vectors are chosen in such a way that all structure constants are real and E −α = −θ(E α ) holds. (Then, if α, β and (α + β) are roots, one has [E α , E β ] = N α,β E α+β and [E −α , E −β ] = −N α,β E −α−β ; and all structure constants are integers [46].) Using any such basis, G is given by and one can take 10) It is worth noting that there are as many choices for B as systems of positive roots, but all of them are equivalent by the action of the Weyl group of the root system.
Next, we explain why the map ν : B → P (3.35) is a diffeomorphism. To start, define the maps Recall that G C R is diffeomorphic to B × G and to P × G by the Iwasawa and global Cartan decompositions, respectively, and P = exp(iG) is diffeomorphic to iG by the exponential map [26]. It follows that µ 1 and µ 2 are (real analytic) diffeomorphisms with the inverses Therefore the composed map ν = µ 2 • µ −1 1 : B → P is a diffeomorphism, with the inverse operating as ν −1 : P → Λ L ( √ P ). At the end, we present some remarks on the rings of G-invariant functions on which our integrable systems are based. Here, the following isomorphisms are fundamental: where W is the Weyl group. These are generalizations [32,39] of the Chevalley isomorphism theorem between G-invariant polynomials on G and W-invariant polynomials on the Cartan subalgebra G 0 . The isomorphisms result from the pertinent restrictions of functions, and they readily imply that both C ∞ (G) G and C ∞ (G) G have functional dimension ℓ = rank(G). By combining a theorem of [47] on smooth invariants with the fact that the ring of G-invariant polynomials on G is freely generated by ℓ homogeneous polynomials, σ 1 , . . . , σ ℓ , one obtains that C ∞ (G) G consists of the functions φ of the form φ = f (σ 1 , . . . , σ ℓ ) with arbitrary f ∈ C ∞ (R ℓ ). This gives the structure of the ring C ∞ (B) G , too, by utilizing the isomorphisms 15) which arise from the G-equivariant diffeomorphism ν (3.35) and the exponential parametrization of P = exp(iG).
Let ρ : G → GL(V ) be an irreducible unitary representations of G, and ̺ the corresponding representation of G. Then, the character G ∋ g → trρ(g) is a G-invariant (in general complex) function on G, and P ∋ e iX → tre i̺(X) is a G-invariant real function on P. By taking suitable real or imaginary parts, it should be possible to obtain ℓ functionally independent elements of C ∞ (G) G from the fundamental irreducible representations of G. In the case of G, the real trace functions G ∋ X → tr(i̺(X)) k , with k ≥ 2, provide convenient invariants.
is a real analytic diffeomorphism.
Proof. For any K ∈ M , the unique Iwasawa decompositions This shows that b L ∈ B and g L ∈ G, and thus also K, can be recovered from b R ∈ B and g R ∈ G. Hence, the map m is injective. The surjectivity of the map m is also clear, since by re-decomposing b −1 R g R in the other order we can construct K such that (g R , b R ) = m(K). The real analytic nature of the relevant decompositions is well known [26].
Let π 1 and π 2 denote the obvious projections from M = G × B onto G and B, respectively. Then we have the identities We start with two useful lemmas.
Lemma B.2. For any f ∈ C ∞ (G) and ϕ ∈ C ∞ (B), consider the functions f • Ξ R and ϕ • Λ R on M . Then, the derivatives of these functions obey the identities and where the decompositions K = b L g −1 R = b L g −1 R (3.20) are used. Proof. Denote F := ϕ • Λ R and use the decompositions of K ∈ M defined in (3.20). Then, for any X ∈ B and K ∈ M , we have where the second equality reflects the relation of the left-and right-derivatives of ϕ. Next, taking X ∈ G, notice from the definitions (3.21) and (3.28) that and therefore The second equality in (B.11) follows from formula ( Based on the above definitions and the relations of the various derivatives, the following statement is easily verified. Lemma B.3. For arbitrary smooth functions ϕ i on B and f i on G (i = 1, 2), we have (B.14) and Proof. For example, let us consider arbitrary f ∈ C ∞ (G) and ϕ ∈ C ∞ (B). Then, due to Lemma B.2, the first term in the formula (B.1) gives (B.17) On the other hand, the second term gives Combining these terms, we obtain the first identity in (B.16). The rest of the identities follows by similar, but shorter, calculations. holds for all ϕ i ∈ C ∞ (B) and f i ∈ C ∞ (G). These statements follow also from the general theory of the Heisenberg double [48,49]. Proof. Notice that the equality (B.5) follows for all smooth functions on M if we prove it for those functions that are of the form f • π 1 and ϕ • π 2 for arbitrary smooth functions f on G and ϕ on B. In order to see this, it is enough to remark that the exterior derivatives of such functions span the cotangent space to M at any point. For the types of functions that feature in Lemma B.3, using also (B.4), we can write (B.20) The other cases of functions are handled in exactly the same way.
Appendix C. On the construction of G-invariant constants of motion via averaging Let X be a G-manifold and V a G-invariant vector field on X, where A η denotes the diffeomorphism of X associated with η ∈ G. The G-invariance of the vector field is equivalent to the property that if x(t) is an integral curve of V , then A η (x(t)) is also an integral curve, for each η ∈ G. Suppose now that G is compact and denote by d G the Haar measure normalized so that the volume of G is 1. For any real function F ∈ C ∞ (X) define the function F G by averaging the functions A * η F over G, It is clear that F G ∈ C ∞ (X) G . Moreover, if F is a constant of motion for the vector field V , then F G is also a constant of motion for V . Indeed, for any integral curve x(t) since A η (x(t)) is an integral curve for all η. In [23,53] this mechanism was used for arguing that, generically, degenerate integrability survives Hamiltonian reduction. In these papers the starting point was a Hamiltonian action on a symplectic manifold, in which case the Hamiltonian vector fields of the G-invariant Hamiltonians are G-invariant.
The averaging of the constants of motion is applicable to the unreduced integrable systems of our interest if the relevant unreduced evolution vector fields are G-invariant. This obviously holds for the two degenerate integrable systems on T * G considered in Section 2, and is also easily checked for the unreduced evolution vector fields on the quasi-Poisson double D studied in Section 4. The Hamiltonian vector fields belonging to our master systems on the Heisenberg double M enjoy the relevant invariance property with respect to the quasi-adjoint action A 2 (3.27). Indeed, Hamiltonians invariant with respect to a Poisson action of G always generate invariant Hamiltonian vector fields. This follows, for example, from Proposition 5.12 in [49].
For completeness, we below answer the question whether the invariance property holds for the Hamiltonian vector fields associated with the two sets of pullback invariants on M, with respect to 'conjugation action' A of G on M defined by equation (3.31).
Proposition C.1. The derivatives of any φ ∈ C ∞ (B) G satisfy the relations As a consequence, if (g(t), b(t)) is an integral curve of the Hamiltonian vector field of H = π * 2 φ ∈ C ∞ (M) G , then A η (g(t), b(t)) is also an integral curve (with the G-action defined in (3.31)).
We see from Proposition C.1 that taking the G-average of an arbitrary constant of motion using the conjugation action A (3.31) yields a G-invariant constant of motion for the degenerate integrable system on the Heisenberg double whose Hamiltonians arise from C ∞ (B) G . However, as we shall see below, the Hamiltonian vector fields stemming from C ∞ (G) G do not have the relevant invariance property with respect to this action.
Remark C.3. Proposition C.2 shows that A η (3.31) does not map the pertinent integral curves (C.16) onto integral curves. At the same time, it confirms that changing the initial value by the G-action (3.31) does not effect the projection of the integral curve to the quotient space M/G. This is equivalent to the fact that the Hamiltonian vector field V of H = π * 1 h, for h ∈ C ∞ (G) G , satisfies where the vector field Z is tangent to the G-orbits. One could find Z explicitly, if desired. It is worth observing from this state of affairs that the action (3.31) on M is not a Poisson action.