Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers

We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with $m\geq 2$ vertices. Their global Poisson structure is characterised by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case $m=1$ corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars-Schneider system. This generalises to new spin cases recent works on classical integrable systems in the Ruijsenaars-Schneider family.


Introduction
In this paper, we continue a recent attempt initiated in [CF1] to interpret the phase spaces of classical complex integrable systems in the Ruijsenaars-Schneider (or RS) family as moduli spaces constructed from particular quivers. Before focusing on this problem, let's recall the well understood interpretation in the non-relativistic case of the Calogero-Moser (or CM) system, and its spin variant. In the pioneering work [W], Wilson unveils several structures related to the phase space for the complex CM system, one of which is the hyperkähler structure it possesses. The latter is naturally defined in the context of Nakajima quiver varieties [Na], which considers Hamiltonian reduction of representation spaces of quivers. Forgetting all but the topological structure, the phase space is nothing else than the reduced representation space of a deformed preprojective algebra associated to a Jordan quiver extended by one arrow. Therefore, it is natural to ask if one could obtain the symplectic structure also at the level of the algebra. This is indeed the case, if we consider non-commutative symplectic geometry [G, CBEG], or the analogue for noncommutative Poisson geometry [VdB1]. We refer to the review [T2] for some details. Going a step further, we can understand the spin generalisation of this model discovered by Gibbons and Hermsen [GH], by looking at a Jordan quiver consisting of a single loop-arrow, which we extend by several arrows coming from an additional vertex [BP,T1]. We obtain in this way the model in type A n , whose Weyl group W = S n determines the symmetry of the obtained system. A study of various extensions of cyclic quivers generalises the result to different complex reflection groups [CS], in particular the case W = S n ⋉ Z n m to which we shall come back. We postpone the quiver interpretation in the relativistic case, as we first need to focus on the geometric side of these systems. In a way similar to the CM case, it is possible to understand the trigonometric RS system geometrically, either as a symplectic leaf on a space defined by Poisson reduction [FR], or directly using quasi-Hamiltonian reduction [O,CF1]. While the process of Hamiltonian reduction brings down a Poisson manifold to one of smaller dimension by considering the action of a Lie group, in the quasi-Hamiltonian settings we begin with a space which has some failure to have a Poisson bracket, but we end up with a genuine Poisson manifold. These spaces that are called quasi-Poisson manifolds [AKSM], which are first introduced in [AMM] for the 'quasi-symplectic' case, find their origin in the need to get Lie group valued moment maps. In some cases, it also provides a finite-dimensional framework for infinite-dimensional symplectic reductions introduced by Atiyah and Bott [AB]. Therefore, the reduction described above provides an alternative formalism to understand earlier works of Gorsky and Nekrasov [GN, Ne]. However, if we leave the type A n , we can observe that until recent works by Fehér and collaborators (see e.g. [FG,FK2,FK3,FK4,FM] mostly in the real case), integrable systems in the trigonometric RS family are generally devised using only a suitable Lax matrix, as they originally appear in [RS], without geometric perspectives.
The lack of a specific geometrical framework to derive these models is even more apparent for spin versions. To understand what is known at the moment, let us recall how the spin RS system is introduced in the first place. In a celebrated attempt to generalise the relation between the matrix KP equation and the spin CM system [KBBT], Krichever and Zabrodin investigate solutions of the non-abelian 2D Toda chain and discover the Lax matrix for the real spin RS system, already in its elliptic form [KrZ]. This system is parametrised by n particles with positions q i , each endowed with d additional degrees of freedom a α i , for which we have d conjugate variables c α i that are function of the momentumq i . There are additional n relations, so that we have 2nd independent coordinates, which appear in the Lax matrix L (that we consider without spectral parameter) such that the Hamiltonian H 1 = tr L defines the equations of motion for the spin RS system. A striking feature of this space is the existence of a natural action of a Lie group of dimension d(d − 1), such that on the corresponding orbit space we can pick coordinates from the functions 1 (q i , f ij = α a α i c α j ). Moreover, the Hamiltonian H 1 descends to this reduced space where it becomes integrable in Liouville sense, and solutions to the equations of motion defined by H 1 can be found in terms of theta functions. In the rational and trigonometric case, a simpler form for the equations of motions corresponding to the Hamiltonians tr L k can be found [AF, RaS]. However, except in the rational case [AF] and for two particles in the elliptic case [So], the Poisson structure of the space is only known in a universal form [Kr] that is not easy to manipulate. It is the existence of a geometric formalism that allows to completely determine the Poisson brackets between the coordinates (q i , a α i , c α i ) in the rational case for the type A n [AF]. Similarly, it is the existence of a geometric interpretation that enables to prove the integrability of such system outside the type A n in the rational [Re] or trigonometric [Fe] cases. (Nevertheless, in those cases we work on the phase space where the individual spins (a α i , c α i ) are not naturally defined and we only know the collective spins (f ij ).) Thereupon, our understanding of the Hamiltonian formalism for spin RS systems remains quite limited outside the rational case, but it is natural to expect to solve this issue if we find a correct geometric framework. Now, let's come back to the algebraic interpretation with quivers. A key aspect of the work of Van den Bergh [VdB1] is that it also introduces non-commutative quasi-Poisson geometry.
To an arbitrary quiver, considering its double, one can associate a multiplicative preprojective algebra, and construct the corresponding multiplicative quiver varieties of Crawley-Boevey and Shaw [CBS]. The latter spaces are Poisson varieties, obtained after quasi-Hamiltonian reduction from the representation spaces of the quiver path algebra, and all the geometric structure can be realised at the level of the path algebra [VdB1]. Therefore, it is a reasonable guess to investigate this theory applied to the quivers studied in [CS], and it is done by Chalykh and the author for a cyclic quiver with an extra arrow in [CF1]. In that case, it is shown that any multiplicative quiver variety contains the phase space for the (non-spin) trigonometric RS system. For the simplest cyclic quiver consisting of one loop (i.e. a Jordan quiver) with d ≥ 2 arrows coming from a new vertex, this is also done by Chalykh and the author, in the companion paper [CF2]. The natural generalisation that these other multiplicative quiver varieties carry on an open subset the phase space for the spin trigonometric RS system of type A n is obtained. Henceforth it provides a crucial step to develop the geometric theory of spin RS models farther than the rational case. The next step is to turn to the application of this method on the cyclic quiver with m ≥ 2 vertices and d ≥ 2 new arrows pointing towards a chosen vertex in the cycle. This is the purpose of this work, and our most important result is that any such representation space can also be seen as the natural phase space of what we suggest to be the complex trigonometric spin RS system with W = S n ⋉ Z n m . This provides a natural generalisation of the case m = 1 corresponding to the Jordan quiver [CF2]. Interestingly, there is a Poisson isomorphism between dense open subsets of the spaces corresponding to the cases m ≥ 2 and m = 1, hence we also obtain that the representation spaces that we construct carry the system with W = S n . In particular, the flows defined by the symmetric functions of the corresponding Lax matrices can be explicitly integrated. We also study Liouville integrability in line with the original approach of Krichever and Zabrodin [KrZ], which is based on the existence of a second reduction for d ≤ n. The final step dealing with arbitrary extensions of cyclic quivers will be discussed in forthcoming works.
The paper is organised as follows. In Section 2, we recall the foundations of the theory of double quasi-Poisson brackets, how we can define such brackets from quivers, and what is the counterpart to that theory on the corresponding representation spaces, based on the work of Van den Bergh [VdB1]. Our presentation of this work relies on [CF1, Section 2], but we reproduce these results and add useful remarks to provide a self-contained exposition of this non-standard subject. In Section 3, we apply the algebraic part of the theory to the so-called spin cyclic quivers. We obtain their structure of quasi-Hamiltonian algebras and gather several results based on computations with the double quasi-Poisson bracket. All the proofs for that section are collected in Appendix A. The formalism employed being quite new, we suggest to the reader interested in the integrable systems side of this work to skip the first part of this paper, and go directly to Section 4, where we overview the multiplicative quiver varieties associated to the spin Jordan quiver (or spin oneloop quiver) recently introduced in [CF2]. In Section 5, we follow the method from [CF2] applied to the cyclic quivers on m ≥ 2 vertices, to get new multiplicative quiver varieties. They are generically isomorphic as complex Poisson manifolds to the space obtained in the case of a Jordan quiver reviewed in Section 4, which we prove in Appendix B. We get three families of functions in involution on such a space for each m ≥ 2, that we write in the set of local coordinates that exists in the Jordan quiver case. In particular, one of them contains the spin RS Hamiltonian H 1 . We count the number of independent elements in each family and perform an additional reduction to obtain integrability in Liouville sense. We can also get an explicit description of some flows, based on computations in Appendix C. We finish by explaining why we believe that this extra reduction should be avoided, and how the other two families corresponds to what should be seen as the spin RS system for W = S n ⋉ Z n m , or a modification of it. Acknowledgement. The author is grateful to O. Chalykh for suggesting the problem and for stimulating conversations, as well as for his collaboration while working on [CF2] which inspired the present paper. The author also thanks L. Fehér and V. Rubtsov for interesting discussions. Some of the results in this paper appear in the University of Leeds PhD thesis of the author, supported by a University of Leeds 110 Anniversary Research Scholarship.
1.1. Notations. The sets N, Z, C denote the non-negative integers, integers and complex numbers. We write N × , Z × , C × when we omit the zero element in those sets. Consider a finite set J, |J| = k, and totally ordered elements (a j ) j∈J such that a j1 < . . . < a j k for some j (−) : {1, . . . , k} → J.
Then the corresponding right and left products are defined as We write δ ij or δ (i,j) for Kronecker delta function. We extend this definition for a general proposition P by setting δ P = +1 if P is true and δ P = 0 if P is false. For example, δ (i =j) = 1−δ ij .

Preliminaries
We recall the necessary constructions needed in this paper as they are introduced in [CF1], with some additional remarks. Details regarding double brackets and representation spaces can be found in [VdB1,VdB2], while we refer to [CBS, Y] for generalities on multiplicative preprojective algebras and corresponding multiplicative quiver varieties.
2.1. Double brackets. We review some results of [VdB1,. We take all tensor products over C, and fix an associative unital C-algebra A. For an element a ∈ A⊗ A, we use Sweedler's notation a ′ ⊗ a ′′ to denote i a ′ i ⊗ a ′′ i . We set a • = a ′′ ⊗ a ′ . More generally, for any s ∈ S n we define τ s : A ⊗n → A ⊗n by τ s (a 1 ⊗ . . . ⊗ a n ) = a s −1 (1) ⊗ . . . ⊗ a s −1 (n) , so we can write a • = τ (12) a.
We view A ⊗n as an A-bimodule via the outer bimodule structure b(a 1 ⊗ . . . ⊗ a n )c = ba 1 ⊗ . . . ⊗ a n c. An n-bracket is a linear map { {−, . . . , −} } : A ⊗n → A ⊗n which is a derivation in its last argument for the outer bimodule structure on A ⊗n , and which is cyclically anti-symmetric: In the cases of interest, there exists a C-algebra B and a C-algebra map B → A turning A into a B-algebra, and we identify B with its image in A. Then we assume that the bracket is B-linear, i.e. it vanishes if one argument is an element of B.
We focus on 2-and 3-brackets, which we call double and triple brackets respectively. In the particular case of a double bracket, the defining relations take the form it is a derivation in the first argument for the inner A-bimodule structure on A ⊗ A given by (2.1) In a similar way to the commutative case, n-brackets can be defined from analogues of n-vector fields. Following Crawley-Boevey [CB1], we assume from now on that A is a B-algebra and we call the elements of D A/B := Der B (A, A ⊗ A) double derivations. We see D A/B as an A-bimodule by using the inner bimodule structure on be the tensor algebra of this bimodule.
In the particular case of Note also that D B A admits a canonical double Schouten-Nijenhuis bracket, which makes D B A into a double Gerstenhaber algebra [VdB1,§2.7,3.2]. This is a (graded) double bracket, that we denote by { {−, −} } SN .
For any n ≥ 2, the multiplication map m : A ⊗n → A is defined by concatenation of the factors, m(a 1 ⊗ . . . ⊗ a n ) = a 1 . . . a n .
(for the graded commutator). Then we say that A is a differential double quasi-Poisson algebra with the differential double quasi-Poisson bracket { {−, −} } P . This implies that { {−, −} } P is a double quasi-Poisson bracket using [VdB1,Theorem 4.2.3].
A multiplicative moment map for a double quasi-Poisson algebra (A, for all a ∈ A. This condition may be written explicitly as When a double quasi-Poisson algebra is equipped with a multiplicative moment map, we say that it is a quasi-Hamiltonian algebra. Combining (2.3) and (2.4), we obtain {Φ, a} = aΦ − Φa and {a, Φ} = 0 for any a ∈ A. Hence, if q 0 ∈ C and {−, −} is the left Loday bracket obtained from the double bracket on A, we get that {J 0 , A} ⊂ J 0 , {A, J 0 } ⊂ J 0 for J 0 the ideal generated by Φ − q 0 . Therefore A/J 0 is a left Loday algebra. If we consider q = s q s e s ∈ B and write J for the ideal generated by Φ − q, we only have {A, J} ⊂ J in general, so that A q := A/J is not necessarily a left Loday algebra. Nevertheless, since {J, A} ⊂ J modulo commutators, the vector space A q /[A q , A q ] is a Lie algebra for the Lie bracket obtained from {−, −} through A → A q /[A q , A q ]. This endows A q with an H 0 -Poisson structure [VdB1,Proposition 5.1.5].
Finally, assume that A is (formally) smooth. If A is a double quasi-Poisson algebra with double bracket defined by P ∈ (D B A) 2 , we say that the element P is non-degenerate if the map of A- is the double bracket defined by Proposition 2.1, and Ω 1 B A refer to the bimodule of noncommutative relative 1-forms [CQ95,Section 2]. We refer to the brilliant work of Van den Bergh [VdB2] for details and the relation to a 'double version' of [AMM].
2.3. Multiplicative preprojective algebras. Let Q = (Q, I) be a quiver with vertex set I and arrow set Q, and consider the maps t, h : Q → I that associate to every arrow a its tail and head, t(a) and h(a). We construct the doubleQ of Q by adjoining to every a ∈ Q an opposite arrow, denoted a * . We naturally extend t and h, so that t(a) = h(a * ) and h(a) = t(a * ). We define ǫ :Q → {±1} the function that takes value +1 on arrows of Q, and −1 on each arrow ofQ \ Q. We write CQ for the path algebra ofQ, whose underlying vector space is spanned by all possible paths formed onQ (including each trivial path e s associated to s ∈ I). The multiplication is given by concatenation of paths, and in particular the (e s ) s form a complete set of orthogonal idempotents. We view CQ as a B-algebra, with B = ⊕ s∈I Ce s . Finally, we extend * to an involution on CQ by setting (a * ) * = a for all a ∈ Q.
Remark 2.2. As in [CF1,VdB1], we write paths in CQ from left to right. Hence, ab means 'a followed by b', and the path ab is trivially zero if h(a) = t(b).
Let A be obtained from CQ by inverting all elements (1 + aa * ) a∈Q . For all a ∈Q, define the We consider a minor generalisation of the construction of multiplicative preprojective algebra, without the use of a total ordering on the arrows of the quiver, see the first remark at the end of [CF1,§2.5]. For each s ∈ I, we fix a total ordering < s on the arrows meeting at s, that is on all a ∈Q with h(a) = s or t(a) = s. We also assume that if two arrows a, b meet at s and r, then either we have both a < s b and a < r b or we have both b < s a and b < r a. We denote such a relation by < and refer to is as an ordering, though it is not necessarily a partial order 2 . We define 2 Strictly speaking, what we only use to get Theorem 2.3 is the total ordering <s for each s on the arrows a ∈Q with t(a) = s. However, defining an ordering as we do is easier to write the assumption in Proposition 2.4.
where the product is taken with respect to the ordering, see § 1.1. Following [CBS], given q = s∈I q s e s with q s ∈ C × , we define the deformed multiplicative preprojective algebra as the quotient Λ q = A/(Φ − q). Up to isomorphism, the algebra Λ q is independent of the ordering [CBS,Theorem 1.4].
We now explain how to see A as a quasi-Hamiltonian algebra with moment map Φ, so that we have an H 0 -Poisson structure on Λ q by § 2.2.
Theorem 2.3. ([VdB1, Theorem 6.7.1]) The algebra A has a quasi-Hamiltonian structure given by and the multiplicative moment map given by Φ = (Φ s ) s , where Φ s is defined in (2.6).
In fact, P is non-degenerate by [VdB2,Section 8]. The following result gives an explicit form to the double quasi-Poisson bracket which we denote { {−, −} } := { {−, −} } P , and that is defined by P from Proposition 2.1.
Proposition 2.4. [CF1, Proposition 2.6] Take an ordering inQ so that the arrows ofQ are ordered in such a way that a < a * < b < b * for any a, b ∈ Q with a < b. Then one has We finish by a remark on the structure of the moment map of a subquiver. assume thatQ ′ is a quiver with vertex set I ′ ⊂ I andQ ′ = {a ∈Q | t(a) ∈ I ′ and h(a) ∈ I ′ }. This means that if we look at the subset of vertices I ′ ofQ and erase all the arrows ofQ which are not both starting and ending at an element of I ′ , we getQ ′ . Moreover, we require thatQ andQ ′ are endowed respectively with the orderings <, < ′ , such that whenever a, b ∈Q ′ , a < ′ b if a < b in the initial quiverQ, and a < c when a ∈Q ′ but c ∈Q Q ′ .
We construct A ′ as A above, and we see A ′ as a subalgebra of A (after adding the removed idempotents e s for s ∈ I \ I ′ ). Define elements Φ ′ , P ′ by replacingQ withQ ′ in (2.6) and (2.7). Remark that we can write P = P ′ + P out and Φ = (Φ + s / ∈I ′ e s )Φ out for some P out ∈ (D B A) 2 and Φ out = (Φ out,s ) s∈I . Last statement is, in fact, a consequence of the fusion process which is used to endow a quiver with a quasi-Hamiltonian structure [VdB1,].
From (2.2), we get that { {b, c} } P out is a sum of terms of the form for any b, c ∈ A. By construction, P out is a sum of (double) biderivations, and each biderivation carries at least one factor ∂/∂d for d ∈Q Q ′ . Therefore, if both b, c ∈ A ′ , all terms in (2.9) must vanish, and { {b, c} } Pout = 0.
2.4. Geometric counterpart to the definitions. Fix a C-algebra A and N ∈ N. The representation space Rep(A, N ) is the affine scheme whose coordinate ring O(Rep(A, N )) is generated by symbols a ij for a ∈ A and i, j = 1, . . . , N , such that they are C-linear in a, they satisfy (ab) ij = k a ik b kj for any a, b ∈ A, and 1 ij = δ ij . Alternatively, we can see Rep(A, N ) as parametrising algebra homomorphisms ̺ : A → Mat N (C), and we get a ij (̺) = ̺(a) ij at any point ̺ ∈ Rep(A, N ). Following [VdB1,Section 7], to any a ∈ A we associate a matrix-valued function X (a) := (a ij ) ij on Rep(A, N ). Similarly, any double derivation δ ∈ Der(A, A ⊗ A) gives rise to a matrix-valued vector field We can generalise the definition in a relative setting for a B-algebra A, where B is of the form B = Ce 1 ⊕ . . . ⊕ Ce K with e r e s = δ rs e s . Representation spaces are now indexed by K-tuples α = (α 1 , . . . , α K ) ∈ N K . Given α with α 1 + . . . + α K = N , we embed B diagonally into Mat N (C) so that Id N is split into a sum of K diagonal blocks of respective sizes α 1 , . . . , α K , representing the idempotents (e s ) s . By definition, Rep B (A, α) = Hom B (A, Mat N (C)), and it can be viewed as an affine scheme in the same way as Rep(A, N ). Note in particular that for any Φ ∈ ⊕ s e s Ae s , the matrix-valued function X (Φ) on Rep B (A, α) is a block matrix X (Φ) ∈ s Mat αs (C).
Assume that A is equipped with a B-linear double bracket { {−, −} }. Then the representation spaces are endowed with an anti-symmetric biderivation as follows.
Proposition 2.6. ([VdB1, Proposition 7.5.1, §7.8]) There is a unique anti-symmetric biderivation (2.10) On Rep(A, N ) we have a natural action of GL N , induced by conjugation action on Mat N (C). Similarly, we have an action of GL α = s GL αs on Rep B (A, α). Provided that A is quasi-Hamiltonian, Rep B (A, α) is a quasi-Hamiltonian manifold [AKSM], as defined now in the smooth case (see [VdB1,] for the algebraic case).
Let G be a Lie group with Lie algebra g. Moreover, assume that g admits a non-degenerate G-invariant bilinear form (−, −). If (e a ), (e a ) are dual bases of g with respect to (−, −), we define the Cartan 3-tensor φ = 1 12 C abc e a ∧e b ∧e c , for C abc = (e a , [e b , e c ]) the tensor of structure constants. For all ξ ∈ g, write ξ L and ξ R to denote the left and right invariant vector fields on G respectively. Given a G-manifold M , the G-action gives rise to a Lie algebra homomorphism (−) M : g → Der O(M ). This can be extended to polyvector fields and we can define the 3-tensor φ M . We say that M is a quasi-Poisson manifold if there exists an invariant bivector field P on M such that [P, P ] = φ M under the Schouten-Nijenhuis bracket. We can use P to define a bracket for all functions g ∈ O (G), and we say that the triple (M, P, Φ) is a Hamiltonian quasi-Poisson manifold. In the case where the action of G on M is free and proper, for each conjugacy class C g of g ∈ G we can form the Poisson manifold Φ −1 (C g )/G. This process is called quasi-Hamiltonian reduction.
Theorem 2.7. [VdB1,§7.8,7.13] Assume that (A, P ) is a differential double quasi-Poisson algebra, which is quasi-Hamiltonian for the multiplicative moment map Φ ∈ ⊕ s e s Ae s . We have that We can note that any multiple of the identity Id α = s Id αs acts trivially on Rep B (A, α). This leads us to define is smooth, and if the action of G(α) on Y is free and the affine GIT quotient Y //G is a geometric quotient, then it is a Poisson manifold with non-degenerate Poisson bracket defined by tr(X (P)), that we denote Y /G.
Note that in the case where the conjugacy class is given by s q s Id αs with all q s ∈ C × , we know that Y //G has a Poisson bracket because A/(Φ − s q s e s ) has an H 0 -Poisson structure by [CB2,Section 4]. However, we need the quasi-Poisson formalism to conclude that the Poisson bracket is non-degenerate in Corollary 2.8. Now, fix a conjugacy class C g in Lie(GL α ) and assume that F, G ∈ O(X (Φ) −1 (C g )) are invariant under the GL α action. We can write F = tr(X (a)) and G = tr(X (b)) for some a, b ∈ A. Assuming that all spaces involved are smooth, we get from Proposition 2.6, Theorem 2.7 and (2.3) that where the bracket on the right-hand side is the associated bracket In particular, we only need to compute {a, b} modulo commutators in A/[A, A] to get the Poisson bracket between tr(X (a)) and tr(X (b)). In slightly more general settings, given arbitrary a, b ∈ A we find in the same way {tr(X (a)), X (b)} P = X ({a, b}) , (2.14) where this time we have the associated bracket {−, −} : 2.5. Multiplicative quiver varieties. From now on, fix B = ⊕ s∈I Ce s as in § 2.3. We always work in a relative setting and omit the subscript B from the notation. The matrix X (a) representing an element a ∈ A is an |I| × |I| block matrix. In the case of an arrow a ∈Q, we can use the idempotents to write a = e t(a) ae h(a) , so a is represented by a matrix with at most one non-zero block of size α t(a) × α h(a) placed in the t(a)-th block row and h(a)-th block column. Therefore, this can be viewed as a quiver representation, consisting of vector spaces V s = C αs , s ∈ I and linear maps X a : V h(a) → V t(a) for each a ∈Q. With this interpretation, we have is an affine open subset of Rep(CQ, α), so it is also smooth. It is naturally acted on by i∈I GL αs through conjugation. This induces an action of G(α) as defined in (2.12). By Theorems 2.3 and 2.7, and Proposition 2.6, Rep(A, α) is a quasi-Hamiltonian G(α)-manifold, with quasi-Poisson bracket defined by the bivector tr(X (P)) and with multiplicative moment map X (Φ). The representation space Rep(Λ q , α) corresponds to the subset such that X (Φ) = s q s Id αs , so it is a closed affine subvariety in Rep(A, α). We set q α = s∈I q αs , and note that Rep(Λ q , α) is empty when q α = 1 by [CBS,Lemma 1.5].
The points in the affine variety S α,q := Rep(Λ q , α)//G(α) are closed G(α) orbits of Rep(Λ q , α), so correspond to semi-simple representations of Λ q of dimension α. In the case where all representations in Rep(Λ q , α) are simple, we have the following description of the space.
Let Q be an arbitrary quiver with vertex set I. A framing of Q is a quiver Q with set of vertices I = I ∪ {∞} and whose arrows are the ones of Q together with additional arrows ∞ → s to the vertices of Q. We allow multiple arrows to a single vertex. Given arbitrary α ∈ N I and q ∈ (C × ) I , we extend them from I to I by putting α ∞ = 1 and q ∞ = q −α , i.e. α = (1, α) and q = q −α e ∞ + s∈I q s e s . By construction q α = 1. We can consider the multiplicative preprojective algebra of Q with parameter q, and consider the representation space Rep(Λ q , α). We refer to the quotients as multiplicative quiver varieties, that we abbreviate MQV. We say that q = s∈I q s e s is regular if q α = 1 for any root α of the quiver Q. We have the following result, which is a multiplicative analogue of [Na,Theorem 2.8], [BCE,Proposition 3].
Proposition 2.10. [CF1, Proposition 2.9] Choose an arbitrary framing Q of Q and let α and q be defined as above. If q is regular, then every module of dimension α over the multiplicative preprojective algebra Λ q is simple. Hence, the group GL α acts freely on Rep(Λ q , α) and the MQV M α,q (Q) is smooth.
When q is regular and M α,q (Q) = ∅, this implies that α = (1, α) is a positive root of Q and M α,q (Q) is a smooth affine variety of dimension 2p( α) by Theorem 2.9.

Quasi-Hamiltonian algebra structure
The developments of this section are parallel to [CF1,CF2], and can be seen as application of Van den Bergh's work [VdB1] that we recalled in § 2.1- § 2.3. Fix m, d ≥ 2 and let I = Z/mZ. Except when it is stated differently, we assume for the rest of this paper that we take the indices r, s in I, and that the Greek letters α, β, γ, ǫ placed as indices always range through 1, . . . , d.
By a spin cyclic quiver, we mean the double quiverQ of a quiver Q, where Q has m + 1 vertices labelled by I ∪ {∞}, m arrows x s : s → s + 1 and d framing arrows v 1 , . . . , v d : ∞ → 0. We write for the doubles y s = x * s : s + 1 → s and w α = v * α : 0 → ∞. We consider the following ordering at each vertex at 0 : We form the algebra A obtained by inverting all the elements (1+aa * ) a∈Q in CQ. Using Proposition 2.4, we get a double quasi-Poisson bracket on A, which satisfies the following identities between the arrows of the cycle Note the difference of signs for last two terms in (3.1c) compared to [CF1,(4.1c)], which is a consequence of the different ordering taken at each vertex s ∈ I. The double brackets involving elements of the cycle and framing arrows are determined by The remaining double brackets are nothing else than . This is because v α < v β and their heads/tails coincide. We then find (3.3a) by cyclic antisymmetry of the double bracket. Identities (3.3b) and (3.3c) are obtained in the same way.
Introduce the elements x = s x s , y = s y s and set F a = s∈I e s+a ⊗ e s ∈ A ⊗ A for any a ∈ Z. Of great help for our study are the elements (The expressions (3.6a)-(3.6b) could be written using F 1 and F −1 instead of writing the idempotents e 0 , but this form is not better suited for calculations.) Adding to A local inverses The algebra A is quasi-Hamiltonian for the double bracket given above and the multiplicative moment map Φ = s e s Φe s + e ∞ Φe ∞ where Here, we use the invertibility of 1 + x 0 y 0 and the idempotent decomposition of 1 ∈ A to get that e 0 + x 0 y 0 is locally invertible with inverse (e 0 + x 0 y 0 ) −1 := e 0 (1 + x 0 y 0 ) −1 e 0 , and the same holds for all the arrows inQ. If we further localise at x, we can write (e s + x s y s )(e s + y s−1 x s−1 ) −1 as x s z s x −1 s−1 z −1 s−1 . Following the Jordan quiver case [CF2, §3.1], we introduce the spin elements and we can define c ′ α inductively using is not a path to 0. This is due to the fact that v α z = v α z m−1 . To get the double brackets between the elements (x, z, a ′ α , c ′ α ), it remains to find the ones involving c ′ α . The next result is obtained in Appendix A : Lemma 3.1. For any α, β = 1, . . . , d, where the last sum in (3.11b) is omitted for λ = 1.
Motivated by the geometric interpretation through (2.13), we are interested in the bracket In particular, in order for the elements on which we take the bracket to be nonzero in Note also that it is sufficient to consider in the two first sums over v in (3.12b) the terms for which v = 1 mod m. All the computations are provided in Appendix A.
We work in slightly more general settings from now on, and consider u ∈ {x, y, z, s e s + xy}. We already have ǫ(x) = +1, ǫ(y) = −1, and we set ǫ(z) = −1, ǫ( s e s + xy) = +1. We can write in the three first cases The identities can be directly checked, see e.g. [CF1,Lemmas A.1,A.2] for some of them. Using these brackets, we get the following result which is proved in Appendix A.
Lemma 3.3. The C-vector space generated by the elements (u k , In fact, it is possible to write down the associated brackets for arbitrary w α v β u l as in [CF2,Lemma A.4], but we do not use this additional result and leave it to the reader. Let φ be the moment map associated to the subquiver supported at I. That is φ = s φ s for φ s = (e s + x s y s )(e s + y s−1 x s−1 ) −1 or φ s = x s z s x −1 s−1 z −1 s−1 when we localise A at x. We assume in the next result that A is localised at u. (3.14) Note that when u is not s e s + xy and K is not divisible by m, then U K +,η = 0 mod [A, A] and the result is trivial. This is because u m ∈ ⊕ s e s Ae s but u / ∈ ⊕ s e s Ae s in those cases. The proof of Proposition 3.4 is provided in Appendix A.
For a (m + 1)-uple (q ∞ , q s ) ∈ C × × (C × ) I , we set q = q ∞ e ∞ + s q s e s and the multiplicative preprojective algebra Λ q is the quotient of A by the two-sided ideal J generated by the relation Φ = q, where Φ is given by (3.8a)-(3.8c). The different equalities derived above in the Lie algebra

A natural phase space for the spin RS model
We recall the important results from [CF2] that are needed for our study, and we take the freedom to adapt the notations to suit our case.
Fix t ∈ C × not a root of unity, n ∈ N × and d ≥ 2. A dense open subset C • n,t,d of the MQV C n,t,d of dimension (1, n) defined from a spin Jordan quiver with d framing arrows is given by equivalence classes of 2d + 2 elements (A, under the equivalence defined from the action of the group GL n (C) by In fact, C • n,t,d is a smooth symplectic complex manifold of dimension 2nd, and we denote its Poisson bracket by {−, −} P . For any representative of an equivalence class, we form the matrices A ∈ Mat n×d (C) and C ∈ Mat d×n (C) by so that the moment map equation can be rewritten as Then a point of C • n,t,d is determined by the equivalence class of (A, B, V ′ α , W ′ α ) as above, or equivalently by an element (A, B, A, C) modulo identification by the group action g · (A, B, A, C) = (gAg −1 , gBg −1 , gA, Cg −1 ) for any g ∈ GL n (C). Choosing the functions the Poisson structure is determined from the identities tr(AE αβ CA r AE γǫ CA k+l−r ) + tr(AE αβ CA k+l−r AE γǫ CA r ) Define the open subspace C ′ n,t,d ⊂ C • n,t,d which is such that for any equivalence class of quadruple (A, B, A, C) ∈ C • n,t,d , the matrix A is diagonalisable with nonzero eigenvalues (x i ) i satisfying x i = x j , x i = tx j for each i = j, and when we choose a representative with A in diagonal form, the matrix A is such that the entries in each of its rows sum up to a nonzero value. Hence we can pick a representative with A in diagonal form as above, and such that α A iα = 1 in C ′ n,t,d . Note that this is uniquely defined up to action by a permutation matrix. Introduce (4.7) We can define a map ξ : h → C ′ n,t,d which associates to (x i , a α i , c α i ) iα the equivalence class of the element (A, B, A, C), where (4.8) This map is such that the following result holds.
Proposition 4.1. [CF2,Propositions 4.1,4.3] The map ξ : h/S n → C ′ n,t,d given by (4.8) defines a local diffeomorphism. It is a Poisson morphism when h/S n is equipped with the Poisson bracket {−, −} defined on coordinates by Furthermore, letting f ij = α a α i c α j , the subalgebra generated by ( In these local coordinates, the matrix B is the Lax matrix of the spin trigonometric RS system. Furthermore, the equation of motions defined by d dt = {tr B, −} P are normalised versions of the equations of motion for the spin RS Hamiltonian, as originally defined in [KrZ]. The form of the Poisson brackets between the elements (x i , f ij ) ij was conjectured in [AF].

MQV for spin cyclic quivers
Consider the multiplicative preprojective algebra for a spin cyclic quiver as in Section 3. In accordance with § 2.5, choose a dimension vector α = (1, α) with α ∈ (N × ) I , and set q ∞ = q −α = s∈I q −αs s . A representation of Λ q of dimension α is a collection of vector spaces (V ∞ , V s ) = (C, C αs ) together with linear maps representing arrows ofQ and satisfying (3.8a)-(3.8c). Denote in an obvious way the matrices representing the arrows as (X s , Y s , V α , W α ). Therefore, points of Rep(Λ q , α) are represented by 2m + 2d elements (X s , Y s , V α , W α ), 1c) and such that all factors have nonzero determinant. The group G( α) = s∈I GL αs (C) acts on these elements by and the orbits in Rep Λ q , α //G( α) correspond to isomorphism classes of semisimple representations. We are particularly interested in the cases where X = s X s is invertible at some points, hence we restrict our attention to the spaces such that α s = n ∈ N × for all s ∈ I. We now define which is the spin analogue of the space C n,q (m) introduced in [CF1, Section 4], see also [BEF,Section 5]. By construction, this is a MQV for a framed cyclic quiver, and we denote its Poisson bracket by {−, −} P . Let us identify I and {0, . . . , m − 1} to introduce the elements so that q ∞ = t −n . We also set t −1 = 1 to state the next result, which is an application of § 2.5 with the regularity condition from [CF1, Section 4].
Proposition 5.1. Suppose that t −1 s t s ′ = t k for any k ∈ Z and −1 ≤ s ≤ s ′ ≤ m − 1, with k = 0 if s = s ′ . Then C n,q,d (m) is a smooth symplectic variety of dimension 2nd.
From now on, we assume the regularity condition of the proposition. In particular, t is not a root of unity.

Local coordinates.
Consider the open subset C • n, q,d (m) ⊂ C n, q,d (m) on which the X s are invertible. Similarly to the Jordan quiver case reviewed in Section 4, introduce Z s := Y s + X −1 s , and form the matrices A (m) ∈ Mat n×d (C), C (m) ∈ Mat d×n (C) by (5.4) so that the α-th column of A (m) represents the spin element a ′ α , while the α-th row of C (m) represent t −1 c ′ α . Note the factor t −1 necessary to define C (m) . In particular, (5.1a)-(5.1b) adopt the compact form Then (5.1c) is just a corollary of these relations. Up to changing basis, a point (X s , Y s , V α , W α ) can be represented by an element of the equivalence class such that X 0 , . . . , X m−2 = Id n . Therefore, setting A := X m−1 and B := q −1 0 Z 0 , we find that the condition (5.5) gives Z s = t s B for s = m − 1, Z m−1 = tA −1 B, and A, B satisfy (5.6) Hence a point in C • n, q,d (m) is completely determined by the data (A, B, A (m) , C (m) ) up to GL n action by g · (gAg −1 , gBg −1 , gA (m) , C (m) g −1 ) seen as s g ∈ G( α), with A, B invertible and the elements Id n +W α V α (that can be reconstructed from (5.4)) invertible. Comparing with (4.4), we find Proposition 5.2. Let C • n, q,d (m) ⊂ C n, q,d (m) be as above. Let C • n,t,d be the spin Ruijsenaars-Schneider space considered in Section 4 with parameter t = s q s , so that C • n,t,d is a smooth variety. Then the map ψ : Proof. The only non-trivial identity to show is that we can recover det(Id n +W α V α ) = 0 for all α. We We can also rewrite (5.4) as We can, in fact, compare the Poisson structures on both spaces.
Proposition 5.3. The isomorphism ψ : C • n,t,d → C • n, q,d (m) from Proposition 5.2 is Poisson. The proof can be found in Appendix B. In particular, we can transfer the invariant local coordinates on C ′ n,t,d ⊂ C • n,t,d obtained in Proposition 4.1 to the open subset C ′ n, q,d (m) ⊂ C • n, q,d (m) defined by C ′ n, q,d (m) = ψ(C ′ n,t,d ). In such a case, we can always consider for a point of C ′ n, q,d (m) a gauge with representative of the form (X, Z, A (m) , C (m) ) given by Proposition 5.2, with the extra condition that X m−1 is in diagonal form with diagonal entries (x i ) i defining a point of C reg (4.7) and α (XA (m) ) iα = 1 for all i.

5.2.
New variants of the spin RS system. Set X = s X s , Y = s Y s and denote by 1 the sum of the m copies of the identity matrix on each V s = C n , s ∈ I. Let Θ = (1 + XY )(1 + Y X) −1 be the moment map restricted to the cyclic quiver without framing, so that Θ = X (φ) for φ defined in Section 3. Proposition 3.4 and (2.13) imply the following result.
Theorem 5.4. The following families of functions are Poisson commuting for any parameter η: Apart from the first family, we need j ∈ mN to have nonzero elements.
We will write down the families from Theorem 5.4 in C ′ n, q,d (m), where we can use the coordinates (x i , f ij ) ij with f ij = α a α i c α j . Hence, our first task is to use the known matrices (A, B, S = AC) instead of the matrices describing a point in C ′ n, q,d (m). Using the isomorphism from Theorem 5.2, they are given by Decomposing the moment map (restricted to the cycle) Θ = XZX −1 Z −1 as Θ = s Θ s , we get from (5.1b) that Θ s = q s Id n for s = 0, and from (5.1a) that Θ The first family contains the symmetric functions of the positions (x i ) i so we omit it. For the fourth family in Theorem 5.4, we see that (1 + ηΘ)(Y + X −1 ) is constituted of m blocks Z s−1 + Θ s Z s−1 . Thus, the block Z s−1 + ηΘ s Z s−1 is given by (1 + ηq 0 )tA −1 B + ηq 0 tA −1 S for s = 0 and t s−1 (1 + ηq s )B otherwise. We can rewrite [(1 + ηΘ)Z] m as a matrix with diagonal blocks In other words, we are interested in studying the family In particular, if we write G m j as a polynomial in η ′ under the form G m j = j l=0 (η ′ ) l G m j,l , we get that all the (G m j,l ) j,l are Poisson commuting (for fixed m) by Theorem 5.4. Now, remark that (4.4) This was obtained by developing G m j := tr (t −1 A −1 B + t −1 η ′ BA −1 )B m−1 j , which explains the two products at the end of the expression, that represent whether A −1 occurs before or after B in the (sm + 1)-th factor A −1 B + η ′ BA −1 . In particular, G j,j = G j,0 for all j = 1, . . . , n.
We now look at the third family in Theorem 5.4. We can see that for any j 1 in C ′ n, q,d (m) We get, for any s = 0, x (5.14) As noticed for the family (G m j,l ) j,l , the functions H m j,j and H m j,0 are not independent. Using that Θ = XZX −1 Z −1 , we can write (1 + ηΘ)(Z − X −1 ) as (Z − X −1 ) + ηXZ(Z − X −1 )Z −1 X −1 , so that in (5.11) it gives after expanding in η tr ((1 + ηΘ)Y ) jm = tr (Z − X −1 ) jm + . . . + η jm tr XZ(Z − X −1 )Z −1 X −1 jm , (5.15) thus the factors in η 0 and η jm agree, implying that H m j,j and H m j,0 are multiples of each other, after normalisation by the constant m −1 C −j from above.
Let's remark two results about those families. First, in the limit q 0 → ∞ where we fix the other q s , all t s → ∞ and we can see from (5.13) that H m j,0 → G m j,0 . So, by rescaling the H m j,l , we can recover all the (G m j,l ) j,l in that limit. Though it is an alternative proof of their involution, the phase space is not defined in that limit. Second, For a given m, each function H m j,l can be written as a linear combination of (G m ′ j ′ ,l ′ ) j ′ ,l ′ with smaller indices. If we allow the definition of G m 1,0 = t −1 tr B m A −1 and G m 1,1 = t −1 tr A −1 SB m−1 for m = 0, 1, we get for example, Finally, we look at the second family tr (1 + ηΘ −1 )(1 + XY ) j in Theorem 5.4, for any j ∈ N.
We first remark that in C • n, q,d (m), (1 + ηΘ −1 )XZ = XZ + ηZX. Meanwhile, X s Z s is nothing else than t s B, while for s = m − 1 Z s X s = t s B but Z m−1 X m−1 = tA −1 BA. This gives us, if we call the elements F m j , It is just a family equivalent to (G 1 j ) j with G 1 j = tr B + ηA −1 BA j , corresponding to the spin RS system, see [CF2]. Developing F m j = j l=0 η l F m j,l , we also get that F m j,0 and F m j,j are proportional. 5.3. Explicit flows. We now show that we can get explicit expressions for the flows associated to particular elements of the families in Theorem 5.4 in C n, q,d (m). Computations for the results that we use now and the general philosophy behind them are gathered in Appendix C.
Recall that the family (G m k ) k is defined in C • n, q,d (m) from the elements tr(U k η ), where U η = Z(1 + ηΘ) represents the element z(1 + ηφ) ∈ A. We get from Lemma C.1 and (2.14) while the Poisson brackets with V β or W β vanish. It does not look possible to explicitly integrate most flows. Indeed, even the matrix U η is not a constant of motion under tr U k η , though its symmetric functions certainly are. However, looking at order 0 in η, we get tr Z k which is G m k,0 up to a constant, and if we look at the flow defined by d/dt k = 1 k {tr Z k , −} P , we get the defining which can be easily integrated.
Proposition 5.5. Given the initial condition (X(0), Z(0), V β (0), W β (0)), the flow at time t k defined by the Hamiltonian tr Z k for k ∈ mN is given by In particular, the flows are complete in C • n, q,d (m) so that we can reintroduce A (m) and C (m) for all times, although some X(t k ) could be non diagonalisable. Remark also that this expression does not exactly coincide with [CF1,Proposition 4.7] when d = 1. This is due to our different choice of ordering at the vertices s ∈ I, so that C n, q,d (m) for d = 1 is isomorphic to the space in [CF1, Section 4] but this map is not the identity map. This is also true for the next flows.
For the other family (H m k ) k , expressed from trŪ k η if we setŪ η = Y (1 + ηΘ), Lemma C.2 and (2.14) give and the Poisson brackets with V β or W β vanish. Again, we can write the flows for the functions which are the coefficients at order 0 in η. If we want to obtain the flow of tr Y k which is a multiple H 1 k,0 , we get by writing d/dτ Proposition 5.6. Given the initial condition (X(0), Y (0), V β (0), W β (0)) the flow at time τ k defined by the Hamiltonian tr Y k for k ∈ mN is given by The expression for X(τ k ) is analytic in Y (0) so does not require its invertibility. The dynamics take place in C n, q,d (m).
Reproducing this scheme for the family (F m k ) k using Lemma C.3 and (2.14), this yields for and the Poisson brackets with V β or W β vanish. It is easier to work with 1 + XY instead of Y in this case, because when we look at order 0 in η we obtain for the flow of tr(1 + XY ) k after setting Proposition 5.7. Given the initial condition (X(0), Y (0), V β (0), W β (0)), setting T = 1 + XY , the flow at timet k defined by the Hamiltonian tr T k for k ∈ N is given by In particular, assuming that X(0) is invertible, this completely determines the solution The extra assumption that X(0) is invertible is satisfied in our interpretation of this family as being the one containing the spin RS Hamiltonian. In that case, the flows take place in C • n, q,d (m) and using the isomorphism of Proposition 5.2, we can see that they can be related the corresponding flows derived in [CF2,RaS].
A natural question to ask is to obtain locally the flows that we could not compute explicitly. As we see in § 5.5, we can define integrable systems from these families in order to compute them by quadrature. However, this requires the additional assumption d n, which we comment in § 5.6.1.

Linear independence.
We assume from now on that d n and look at the number of independent functions in each family. Since the family (F m j,l ) j,l corresponds to functions for the spin RS system, we already know that this contains nd − d(d − 1)/2 linearly independent elements [KrZ].
For the families (G m j,l ) j,l and (H m j,l ) j,l , remark that there can be at most n(n + 1)/2 linearly independent functions. Indeed, we look at the symmetric functions of n × n matrices (see (5.9), (5.12)), so we have by developing in η, n + n(n + 1)/2 functions for j = 1, . . . , n, with n constraints coming from the relation between the terms (j, 0) and (j, j). It now suffices to remark that we can write these functions generically in the form {tr(C +ηT ) j | 1 j n} for C invertible with distinct eigenvalues, and T of rank d, with distinct nonzero eigenvalues. This yields (n − d)(n − d + 1)/2 additional constraints, as we now explain with the family (G m j,l ) j,l . The other case is similar. We adapt the method introduced for the spin Calogero-Moser family [BBT, KBBT], and we write G m k = tr(C + ηT ) k for C = A −1 B m , T = SB m−1 A −1 and η = η ′ /(t −1 + η ′ ), see (5.9). Remark that C, T take the form stated above. Consider the spectral curve We can expand each r i (η) in terms of η as r i (η) = n−i s=0 I n−i,s η s . Hence the set of linearly independent functions is contained in the {I n−i,s }, which are functions of the n + n(n + 1)/2 functions {G m k,K }. It remains to see how many relations exist on the {I n−i,s }. In a neighbourhood of η = ∞, we can write (5.19) for (ν i ) i the eigenvalues of T . At a generic point, we can order the (ν i ) i so that ν 1 < ν 2 < . . . < ν d are nonzero, and ν d+1 = . . . = ν n = 0. Thus near η = ∞ we write Γ(η, µ) = µ n−d d i=1 (µ − ην i ). From this behaviour at infinity, we require that if we write Γ(η, µ) ≡ n i=0 Γ i (µ)η i , then Γ i (µ) = 0 for all i = d + 1, . . . , n. Each Γ i (µ) has order n − i as a polynomial in µ whose coefficients are functions of the {I n−i,s }. Hence we get the expansion Γ i (µ) = n−i s=0 J n−i,s µ s , for some functions J n−i,s (I k,t ). Their vanishing for i > d is equivalent to imposing relations, which proves our claim. Imposing the initial n relations, which are independent from the ones just obtained, we have a total of nd − d(d − 1)/2 independent functions. 5.5. Additional reduction. We would like to construct a space of dimension 2nd − d(d − 1) into which the different families descend. Introduce the d(d − 1)-dimensional algebraic group (5.20) whose elements are invertible d × d matrices such that the vector (1, . . . , 1) ⊤ is an eigenvector with eigenvalue +1. This is precisely the algebraic group H needed to get Liouville integrability of the RS system in the original work [KrZ]. Define the action of H on (X, Z, A (m) , C (m) ) by h·(X, Z, A (m) , C (m) ) = (X, Z, A (m) h, h −1 C (m) ). By definition of C ′ n, q,d (m) at the end of § 5.1, we can always take a representative (X, Z, A (m) , C (m) ) on this subspace such that α (XA (m) ) iα = 1 for all i. This condition is preserved under the action of H. Hence, we define the reduced space C H n, q,d (m) as the affine GIT quotient C H n, q,d (m) = C ′ n, q,d (m)//H. It has dimension 2nd − d(d − 1) and is generically smooth as we will shortly see. The coordinate ring O(C H n, q,d (m)) is generated by elements of the form tr γ, where γ is a word in the letters X, Z, S = A (m) C (m) . If we write such functions in coordinates by lifting them to C ′ n, q,d (m), they become invariant polynomial in the elements (x i , x i x −1 j − t, f ij ) ij , which form a Poisson subalgebra of {−, −} by Proposition 4.1. Thus the Poisson bracket {−, −} P descends to C H n, q,d (m). It is such that the projection Proof. We show the existence of a non-empty open subset of C ′ n, q,d (m) where H acts properly and freely in Lemmas 5.9 and 5.10, so that the corresponding space of H-orbits defines a smooth complex manifold of dimension 2nd−d(d−1) inside C H n, q,d (m). In particular, a point (X, Z, A, C) in the subspace is characterised by the fact that all the d-dimensional minors of A are nonzero. This is the complement of the Zariski closed subsets defined by having a vanishing minor of dimension d. Thus this subspace is dense in C ′ n, q,d (m), and so does its reduction in C H n, q,d (m). The elements in each family are H-invariant, and also linearly independent by the argument developed in § 5.4. Thus the proof follows for the first two families. For the last family, remark that we also need to restrict to the open subset where Y is invertible before performing the reduction, but this is dense again.
Using the isomorphism of Proposition 5.2, a point (X, Z, A (m) , C (m) ) of C ′ n, q,d (m) can be equivalently characterised by a quadruple (A, B, A, C) satisfying (4.8). By abuse of notation, we denote this point by (X, Z, A, C) and assume that it has the form just stated. We remark that the H-action is given by h · (X, Z, A, C) = (X, Z, Ah, h −1 C) so that α (Ah) iα = 1 for all i.
Lemma 5.9. The action is free on the subset of C ′ n, q,d (m) where, given a point (X, Z, A, C), either A or C has rank d.
Proof. Assume A has rank d, the proof being the same if we assume the latter for C. By definition, there exists K = (k 1 , . . . , k d ) ⊂ {1, . . . , n} such thatĀ = (A kαβ ) is a d × d matrix which has rank d, so is invertible. If we take some h in the stabiliser of the point (X, Z, A, C), then in particular Ah = A and thusĀh =Ā. Indeed, Lemma 5.10. The action is proper on the subset S ⊂ C ′ n, q,d (m) where, given a point (X, Z, A, C), all the minors of dimension d of A are invertible.

Proof. The claim follows if we can show that given sequences (h
For any choice of K = (k 1 , . . . , k d ) ⊂ {1, . . . , n}, we can formĀ as in Lemma 5.9. We also use the notationD for the d × d matrix obtained in that way from some n × d matrix D. We see that for some T m ∈ GL d (C). Forming h (K) and h (L) from them, we get Next, remark that h ∈ GL d (C): as h =Ā −1 A ′ and both elements on the right hand-side have nonzero determinant, so too has h. Finally, h ∈ H because Here we use that α A iα = 1 for all i, implies that we have αĀ γα = 1 for all γ. That isĀ ∈ H, which in turn yieldsĀ −1 ∈ H.
We summarise the projection from the representation space of the multiplicative preprojective algebra to the space we have just constructed as Let us formulate one last comment on the reduced space C H n, q,d (m). We can integrate some equations of motions for the families in Theorem 5.4, thus defining flows in C n, q,d (m). If flows quit the subspace C ′ n, q,d (m), then the last projection given in (5.22) can not be defined, so the flows are not complete in C H n, q,d (m). This suggests that C H n, q,d (m) is not the natural phase space for our systems in the complex case, and it motivates a search for other first integrals, see § 5.6.1. 5.6. Final remarks. We finish by some additional comments that could lead to new investigations about these models. 5.6.1. The case d = 2 and Liouville integrability. Using Lemma 3.3, we get the following result.
Theorem 5.11. The following families of functions on C n, q,d (m) are linearly independent and in involution tr X jm , tr W 1 V 1 X jm j = 1, . . . , n , tr(1 + XY ) j , tr W 1 V 1 (1 + XY ) j j = 1, . . . , n , where the last family is viewed on the subspace C • n, q,d (m) ⊂ C n, q,d (m) where X is invertible. In particular, these define completely integrable systems for d = 2.
Though we do not prove it, we have as in the Jordan quiver case [CF2] that we could replace the matrix W 1 V 1 by W α V β , for any α, β ∈ {1, . . . , d}. Moreover, taking the first family as an example, the elements (tr X jm ) j are in the centre of the Poisson subalgebra generated by the elements (tr X jm , tr W α V β X jm ). This suggests that even in the cases d > 2 we should get Liouville integrability directly on C n, q,d (m). In fact, the method presented in [CF2,§5.2] can be adapted to our case, and we conjecture that we can construct a Liouville integrable system containing tr Z in that way. We will return to this question for an arbitrary framing of a cyclic quiver. 5.6.2. Degenerate integrability. As we explained in § 5.6.1, if we consider U = X, Y, Z, it is possible to show that the functions tr U jm and tr W α V β U jm , where j ∈ N and α, β = 1, . . . , d, define a Poisson subalgebra of O(C n, q,d (m)) and that any tr U jm Poisson commute with these other functions. This is also true for U = 1 + XY with tr W α V β U j . This suggests that all these systems can be integrated in the sense of degenerate integrability (also called non-commutative integrability or superintegrability). More specifically, the proof of degenerate integrability presented in [CF2, §5.1] for the Jordan quiver can be adapted to the present situation. This property was first remarked for the spin RS system by Reshetikhin in the real rational case [Re]. 5.6.3. Self-duality. The work of Reshetikhin [Re] considers the duality between the spin hyperbolic CM system and the spin rational RS system. This was discovered in the non-spin case by Ruijsenaars [R], together with the self-duality of the hyperbolic RS system. In the complex settings where the hyperbolic and trigonometric cases are the same, the latter self-duality can be obtained by noticing that, with the notations of Section 4 in the non-spin case m = d = 1, the transformation ̟ : (A, B) → (B, A) is an (anti-)symplectic mapping [CF1,Proposition 3.8]. We can make a step in that direction for the spin case, though this requires the additional reduction of § 5.5. Hence, we assume d ≤ n.
To work in full generalities, let A be the algebra localised at x constructed in Section 3. Consider the quasi-Hamiltonian algebraÂ obtained from A by removing the elements v α , w α , e ∞ , i.e.Â = A/ e ∞ . This can be seen as the analogue of A obtained by construction from the non-framed cyclic quiver, that is the subquiverQ ′ ⊂Q supported at I = Z/mZ. We can easily see that the algebra homomorphism ι :Â →Â defined by ι(e s ) = e m−s , ι(x s ) = z m−s−1 , ι(z s ) = x m−s−1 , (5.23) satisfies ι 2 = idÂ. It corresponds to flippingQ ′ such that the vertex 0 is fixed. Moreover, from (3.5a), with z instead of y, and (3.7) we can show that so that ι is an anti-morphism of quasi-Poisson algebras. (We can check this equivalently on { {x s , x r } }, { {z s , z r } } and { {x r , z s } }.) In particular, ι(φ) = ι(xzx −1 z −1 ) = φ −1 , so ι maps the moment map ofÂ to its inverse. Remark that from § 5.5 and (5.5), the coordinate ring O(C H n, q,d (m)) is generated by elements of the form tr(Γ), where Γ is a sum of matrices whose factors are either X or Z, so that the Poisson structure on C H n, q,d (m) is completely defined from the double brackets { {x, x} }, { {z, z} } and { {x, z} }. Indeed, they define the quasi-Poisson brackets in Rep(Λ q , α) for the elements (X ij , Z ij ), which determine the Poisson bracket on C H n, q,d (m) by construction. This yields the following result.
As indicated in Proposition 5.12, ̟ is only defined at a generic point, e.g. in C H n, q,d (m) there are points where the product Z 0 . . . Z m−1 is not semisimple. Hence, we do not have self-duality of the system in the strict sense of [R,FK1,FK3,FA] which requires a global Poisson isomorphism. Nevertheless, the underlying interpretation on the quiverQ ′ is easy to understand, and works also for m = 1, where it extends the geometric approach to the self-duality for the trigonometric RS to the spin case. Given the natural appearance of the self-duality for the non-spin case in gauge theory [FGNR, GR], it would be interesting to understand the interpretation of the spin case within this framework.
Let us formulate two final remarks. Firstly, if we replace z by y in the definition of ι (5.23), this also gives an anti-morphism of quasi-Poisson algebras. Hence the integrable system containing tr X is dual to the integrable system containing tr Z, but also to the one containing tr Y by adapting the above argument. Secondly, note that this isomorphism ι :Â →Â does not directly extend to A itself. We will return to this issue in further work, in order to lift the map ̟ to a well-defined map on C • n, q,d (m).
5.6.4. Relation to generalised symmetric group. It is remarked in [CF1] that in the case d = 1, the new Hamiltonians obtained for a cyclic quiver on m vertices correspond to W = S n ⋉ Z n m . In the study of (non-multiplicative) quiver varieties, the Hamiltonians of Calogero-Moser type obtained in the spin case have also that particular symmetry [CS, Section VI]. Thus, we would expect that the elements of the families (G m j,l ) j,l and (H m j,l ) j,l are also related to the generalised symmetric groups G(m, 1, n) = S n ⋉ Z n m . To establish this link, consider p ∈ C ′ n, q,d (m) ⊂ C • n, q,d (m) determined by a point of C ′ n,t,d as in Proposition 5.2. Using the local coordinates of C ′ n,t,d (4.8), the point p = (X s , Z s , A (m) , C (m) ) is characterised by X s = Id n for s = m − 1, X m−1 = diag(x 1 , . . . , x n ), and the matrices (Z s , A (m) , C (m) ) are given by In particular, the matrices (X s ) s take two different forms : either the identity or a diagonal matrix whose entries are interpreted as particle positions. To set them all to the same diagonal matrix, recall that an element g ∈ G( α) acts as in (5.2), so we can write this action as g.(X s , Z s , A (m) , C (m) ) = (g s X s g −1 s+1 , g s+1 Z s g −1 s , C (m) g −1 0 , g 0 A (m) ) , g ∈ G( α) . (5.24) Choose elements (λ i ) i such that λ m i = x i . They are nonzero distinct and satisfy λ m i λ −m j = t for all i = j. We can form the element s g s ∈ G( α) with g s = diag(λ m−s 1 , . . . , λ m−s n ) for s = 0, . . . , m−1, and acting on p in its above form yields for any s = 0, . . . , m − 1. Hence, the choice of a representative (X s , Z s , A (m) , C (m) ) in C ′ n, q,d (m), such that all the X s are in the same diagonal form and α (A (m) ) iα = 1, is unique up to acting by S n ⋉ Z n m . Here, the action of an element (σ, M ) ∈ S n ⋉ Z n m , is represented by the matrix g = s g σ g −s M , where g σ is the permutation matrix corresponding to σ while g M = diag(ζ M1 , . . . , ζ Mn ) for M = (M 1 , . . . , M n ) and ζ is a primitive m-th root of unity.
In the case m = 2, write q 0 = e −2γ0 and q 1 = e −2γ1 so that t = e −2γ for γ = γ 0 + γ 1 . We get We can write down tr Z 2 and tr Y 2 , which are multiples of G 2 1,0 and H 2 1,0 respectively, in C ′ n, q,d (m) as Comparing last two expressions with tr B 2 and tr(B − A −1 ) 2 obtained from Section 4 strengthens our claim that the (G m j,l ) j,l correspond to a spin RS system for W = S n ⋉ Z n m (and (H m j,l ) j,l to a modification of it).
Appendix A. Calculations for the spin cyclic quivers Remark A.1. Note that the proofs of Lemmas 3.1, 3.2 and Proposition 3.4 also apply for their analogues in the case m = 1 considered in [CF2]. We make a comment on the changes that are needed in the latter case at the beginning of each of these proofs. The elements a ′ α , c ′ α are denoted by a α , b α in [CF2].
A.1. Double bracket with spin variables : Proof of Lemma 3.1. We will show that (3.11a) and (3.11c) holds, while we replace (3.11b) by This is nothing else than (3.11b) because the first term vanishes. Indeed, note that c ′ β = e ∞ c ′ β e m−1 , so that c ′ β γ = 0 for any γ which is a path beginning by x, z or some a ′ α = w α since then γ = e 0 γ. However, we will carry on such terms of the form c ′ β γ, because our proof also applies in the case of a Jordan quiver (see [CF2,Lemma 3.1]) where it does not vanish. Indeed, if we allow the case m = 1 and set F b = e 0 ⊗ e 0 for any b ∈ Z, The double brackets between the elements x, z, v α , w α given in Section 3 exactly match the double brackets in [CF2,§3.1.2].
We prove the results by induction using (3.10). Knowing the double brackets in Section 3, if we want to compute the bracket Γ, c ′ β for some Γ ∈ A, we first find { {Γ, c ′ 1 } } = { {Γ, v 1 z} } and then show our statement by induction using To get (3.11a), we first compute { {x, c ′ α } } and show how to deal with the idempotents. Recalling that F 1 = s e s ⊗ e s−1 , we get from the double brackets in Section 3 In order to simplify the F 1 to go from the second to the third equality, we used that v α = v α e 0 , x ∈ ⊕ s∈I e s Ae s+1 and z ∈ ⊕ s∈I e s+1 Ae s . For example, v α zxF 1 = v α zxe 0 F 1 = v α zxe 0 ⊗ e m−1 . In particular, if we use c ′ 1 = v 1 z we get the expression for { {x, c ′ 1 } } given in (3.11a) as our basis for the induction.
Meanwhile, we compute so that if we assume that the first equality in (3.11a) is true for any λ < α, we get from (A.2) which is exactly the first equality in (3.11a) by using (3.10). For the bracket and { {z, c ′ 1 } } given by the second equality in (3.11a) holds. We can find { {z, v α w λ } } = 0 so that the general case follows by induction, in a way similar to .
recalling that o(α, β) = δ (α<β) − δ (α>β) . In particular, this yields which is exactly the case β = 1 in (A.1) (and c ′ 1 a ′ α = 0 as we mentioned at the beginning of the proof to get in fact (3.11b)). Next, we can compute , and this implies that In the case α β this gives since Otherwise, we just write Now, assume by induction that (A.1) holds for any λ < β, which we can write In the first case, α ≥ β, we find from (A.2) and (3.10) which coincide with (A.1). In the second case, we get which, after some easy manipulations on the sums, yields as expected from (A.1) since the first term is zero.
As an intermediate result for (3.11c), we need Proof. Note that the first term vanishes, but we keep it for the case m = 1 as explained at the beginning of the proof. We compute We keep the last vanishing term for computations. In particular, we get which agrees with our statement for β = 1. Now, we compute Assume by induction that for all λ < β, In the case α > β we find In the case α = β we have We can finish the proof of Lemma 3.1 by showing (3.11c). It is easier to use the induction in the first variable, that is with Γ = c ′ β in our case. By doing so, we can repeatedly use (3.11a), (A.1) and Lemma A.2. We first get using that the first and third terms vanishes (they would cancel out in the Jordan quiver case).

This gives in particular
The first and third terms cancel out, so we can write Now, assume by induction that for all λ < α, and let us show that this holds for λ = α. Note that it is exactly (3.11c) since in the case λ = β the two terms cancel out. We find by (A.3) and our previous computations If α > β we find In the other cases, and this is trivially . A.2. Proof of Lemma 3.2. This proof can be applied without change in the case m = 1 treated in [CF2]. Indeed, we use (A.1) instead of (3.11b) when computing {{a ′ γ c ′ ǫ , a ′ α c ′ β }} below. In that way, all the double brackets that we use during this proof are the ones in [CF2] if we set m = 1.
We assume that the integers k, l ≥ 1 satisfy the conditions for the elements to be nonzero, otherwise the proof is trivial. The first equality is an easy computations, or can be obtained as a consequence of [CF1,Lemma A.3]. Next, we compute from (3.6a) and (3.11a) Combining this result with (3.5a), If we apply the multiplication m, we have in the last two terms that the nonvanishing terms are for s ∈ I such that e m−1 x τ +1 e s = x τ +1 and e m−1 x τ −1 e s = x τ −1 respectively, and we get that only the third and fourth terms do not cancel out. We find that {x k , a ′ Note that the middle line can be decomposed as can easily obtain in all cases We show that for any fixed α = 1, . . . , d, the elements (u k , w α v α u l ) form a commutative Lie subalgebra in A/[A, A]. It is just an application of [CF1,Lemma A.3] to show {u k , u l } = 0, or we can use Proposition 3.4 at order η, η ′ = 0. Next we compute We find in a similar way to (A.5) in the proof of Lemma 3.2 , To have nonzero terms modulo commutators, recall that we consider k, l = 0 mod m if u = x, y, z, while k, l ∈ N otherwise. In particular we do not need to write down the idempotents after multiplication, and all terms cancel out.
On one hand, we get again by adapting the argument in the proof of Lemma 3.2 because we can get rid of the idempotents modulo commutators, after careful analysis. After simplification, all terms vanish modulo commutators. On the other hand, we compute Hence m • ({ {w α v α , w α v α } } x l * x k ) = 0 modulo commutators and we can conclude.
A.4. Proof of Proposition 3.4. This proof can be applied without change in the case m = 1 treated in [CF2], after setting e s = e 0 for each s ∈ I, and F b = e 0 ⊗ e 0 for all b ∈ Z.
We begin with the first identity, and let U α = u(1 + ηφ) instead of U +,α to ease notations.
where γ is any word in the letters {e s , x s , y s } (with possible inverses). The second equation is obtained by combining (2.4) and Lemma 2.5 applied to the subquiver based at I, the set of all vertices in the cycle. We see that we can write because φ ∈ ⊕ s e s Ae s , so φ commutes with any e s . As u ∈ ⊕ s e s Ae s+θ(u) we have ue s = e s−θ(u) u, so that { {u, φ} } = 1 2 φ(uF θ(u) − F θ(u) u) + 1 2 (uF θ(u) − F θ(u) u)φ .
Lemma A.3. Fix some r ∈ N and let a ∈ ⊕ s e s Ae s , b 0 , b 1 ∈ ⊕ s e s Ae s+r and c ∈ ⊕ s e s Ae s+2r . Then (b 0 F r b 1 ) • = b 1 F r b 0 and (cF r a) • = aF r c.
Proof. We compute (b 0 F r b 1 ) • = s e s b 1 ⊗ b 0 e s+r = s b 1 e s+r ⊗ e s b 0 = b 1 F r b 0 . The second equality follows similarly.

Appendix B. Poisson isomorphism between MQVs
Before proving Proposition 5.3, let's remark that Lemma 3.2 together with (2.13) give {tr X k , tr X l } P = 0 , {tr X k , tr(A (m) E αβ C (m) X l )} P = k tr(A (m) E αβ C (m) X k+l ) , (B.1a) Note the appearance of two constants t −1 in the last two terms. Indeed, we used that (A (m) ) iα is the i-th component of the covector representing a ′ α , while (C (m) ) βj is the j-th component of the vector representing t −1 c ′ α , while X and Z respectively represent x, z. Furthermore, for the last equality, it is an easy exercise to see that both matrices Z in the right-hand side can be replaced by Z m−1 after using that k, l = 1 modulo m in that expression. The second set of Poisson brackets that we need are given by (4.6a)-(4.6b) for the coordinates defined in (4.5), and we write in our case for k 0 , l 0 ≥ 1 {tr(A k0 ), tr(A l0 )} P = 0 , {tr(A k0 ), tr(AE αβ CA l0 )} P = k 0 tr(AE αβ CA k0+l0 ) , Proof. [Proposition 5.3]. It suffices to show that the map ψ : C • n,t,d → C • n, q,d (m) is a Poisson map with respect to a basis of functions on C • n, q,d (m). It is not hard to see that we can pick the functions F k := tr(X k ) and G γǫ l := tr(A (m) E γǫ C (m) X l ), which are nonzero for k = k 0 m and l = l 0 m + 1 with k 0 , l 0 ≥ 1. We always assume such a choice for the indices from now on (which thus depends on the function F k or G γǫ l ). Using the notations from (4.5), we can see that ψ * F k = m tr(A k0 ) = mf k0 , ψ * G γǫ l = tr(A −1 AE γǫ CA l0+1 ) = g γǫ l0 .
Appendix C. Calculations for the dynamics Our method goes as follow : the Hamiltonians come from functions of the form tr(X (u η ) K ), for some u η ∈ A. Then, defining the derivation d/dt K := {tr(X (u η ) K ), −} P , the evolution of a matrix X (c) representing an element c ∈ A is governed by the ODE dX (c) dt K = X ({u K η , c}) , X (c)| t=0 := C 0 , (C.1) using 2.14, for some initial condition C 0 . Thus, we are interested in computing the left Loday bracket {u K η , c} = m • u K η , c , which can be found by after using the derivation property in the first variable then multiplying. Hence we need to compute { {u η , c} }, then substitute the result back into (C.2). Note that from the discussion at the end of § 2.2, we get for the ideal J = (Φ − q) that {u K η , J} ⊂ J, hence (C.1) defines flows in Rep Λ q , α that we can project in C n, q,d (m). The data of (C.2) for a set of generators in A can be seen as an analogue in the quasi-Poisson case to an Hamiltonian ODE on A as defined in [DSKV,Section 2.4] for a double Poisson algebra.
First, we look at the family (G m k ) k , which are the symmetric functions of the matrix representing the element u η := z(1 + ηφ). We need the double brackets