Quantum superintegrable spin systems on graph connections

In this paper we construct certain quantum spin systems on moduli spaces of $G$-connections on a connected oriented finite graph, with $G$ a simply connected compact Lie group. We construct joint eigenfunctions of the commuting quantum Hamiltonians in terms of local invariant tensors. We determine sufficient conditions ensuring superintegrability of the quantum spin system using irreducibility criteria for Harish-Chandra modules due to Harish-Chandra and Lepowsky&McCollum. The resulting class of quantum superintegrable spin systems includes the quantum periodic and open spin Calogero-Moser spin chains as special cases. In the periodic case the description of the joint eigenfunctions in terms of local invariant tensors are multipoint generalised trace functions, in the open case multipoint spherical functions on compact symmetric spaces.

Dedicated to the memory of Gerrit van Dijk 1. Introduction 1.1.Let Γ be a connected oriented finite graph with vertex set V , edge set E, and source and target maps s, t : E → V .Let G be a connected compact Lie group.The product group G E = {g = (g e ) e∈E | g e ∈ G} of graph G-connections (or lattice gauge fields) on Γ consists of colorings g of the edges of Γ by group elements g e ∈ G (e ∈ E).We view G E both as compact Lie group and as algebraic group (via Tannaka duality).
The group G V = {k = (k v ) v∈V | k v ∈ G} of lattice gauge transformations acts on G E by 1(1) (k • g) e = k s(e) g e k −1 t(e) .The resulting space G E /G V of G V -orbits in G E is the moduli space of graph Gconnections on Γ introduced by Fock and Rosly [11] to describe moduli spaces of flat connections on surfaces, see also [2,3].See [1,4,5,22] and references therein for the associated quantization problem.In this paper we construct quantum systems with Hamiltonians being differential operators on the moduli space G E /K of graph G-connections modulo gauge groups K of the form K = v∈V K v ⊆ G V , with K v arbitrary subgroups of G.
1.2.Let D(G E ) be the algebra of algebraic differential operators on G E .The contragredient K-action on the space of algebraic functions on G E induces a Kaction on D(G E ) by algebra automorphisms.We denote by D(G E ) K ⊆ D(G E ) the subalgebra of K-invariant differential operators.In this paper we consider its subalgebras with D inv (G E ) ⊆ D(G E ) the subalgebra generated by the left and right G Einvariant differential operators and D biinv (G E ) ⊆ D(G E ) the subalgebra of G Ebiinvariant differential operators.
1.3.Let σ : K → GL(S) be a finite dimensional K-representation.Functions in the space H = H Γ,G,S of algebraic sections of the associated vector bundle over G E /K are called spin graph functions.They are algebraic functions f : Here spin refers to the interpretation of S as spin space for the associated quantum spin system, see §1.4 and §1.9.
1.4.The algebra D(G E ) K acts as scalar valued K-invariant differential operators on the space H of spin graph functions.The resulting homomorphic image of the inclusions (2) of algebras gives rise to an inclusion (3) I ⊆ J ⊆ A of subalgebras of End(H).We view (3) as a quantum spin system with quantum state space H, and with the homomorphic image A = A Γ,G,S ⊆ End(H) of D(G E ) K as algebra of quantum observables, the homomorphic image J = J Γ,G,S of D inv (G E ) K as algebra of quantum integrals, and the homomorphic image I = I Γ,G,S of D biinv (G E ) as the algebra of quantum Hamiltonians.
1.5.The quantum state space H breaks up in an algebraic direct sum of finite dimensional simultaneous eigenspaces for the action of the quantum Hamiltonians, with I ∧ the set of characters of I. We say that the quantum spin system is superintegrable if for all χ ∈ I ∧ , the simultaneous eigenspace H χ is either {0} or an irreducible J-module.Equivalently, the quantum spin system is superintegrability when eigenstates f, g ∈ H χ with the same energy eigenvalues χ ∈ I ∧ are related by a quantum integral: g = D(f ) for some D ∈ J.
We say that the superintegrable quantum spin system is integrable when I = J.In this case dim(H χ ) ≤ 1 for all χ ∈ I ∧ , i.e., an eigenstate is determined by its energy eigenvalues up to normalisation.1.6.The main result of this paper is as follows.
Theorem.The quantum spin system on H = H Γ,G,S is superintegrable when the following three conditions are satisfied: (a) G is simply connected, (b) the local gauge group K v is a closed connected subgroup of G for all v ∈ V , (c) σ : K → GL(S) is irreducible.
We will prove this theorem using a result of Harish-Chandra [13] and Lepowsky & McCollum [14] relating irreducible g-modules to irreducible U(g) K -modules for appropriate compact Lie groups K (this result plays an important role in the proof of the subquotient theorem for Harish-Chandra modules).1.7.We will say that a spin graph function f is elementary if it is a simultaneous eigenfunction of the quantum Hamiltonians, i.e., when f ∈ H χ for some χ ∈ I ∧ .For tensor product K-representations S we construct spanning sets of H χ using the data of the following colored version of Γ.
The colors at the vertices v ∈ V are the local representations σ v : K v → GL(S v ) of the tensor product representation S. To obtain the colors of the edges, we use the fact that H χ = {0} if and only if χ is the central character of an irreducible representation of G E .The irreducible G E -representation provides the colors of the edges of Γ by local irreducible G-representations.
We construct the spanning set of the elementary spin graph functions in H χ in terms of local invariant tensors (local in the sense that they only depend on the star of some vertex v of the colored graph Γ).
1.8.If Γ is the directed cycle graph with n edges with K v = G for all v ∈ V , then we show that the resulting elementary spin graph functions are essentially the generalised (or n-point) trace functions from Etingof & Schiffmann [8].
If Γ is the linearly ordered linear graph with n edges and the local gauge groups K v are G (resp.K) for 2-valent (resp.1-valent) vertices v ∈ V , then we show that the resulting elementary spin graph functions are the n-point spherical functions from [21].For n = 1 these are the usual elementary σ-spherical functions on G, see, e.g., [12,24].
In both cases the local invariant tensors may be identified with topological degenerations of vertex operators, cf.[24,23].
1.9.The explicit desciption of the elementary spin graph functions as multipoint trace functions and multipoint spherical functions for the two special cases in §1.8, connects the associated quantum spin systems to the periodic and open quantum spin Calogero-Moser chains from [8,21,20] and [24,21], respectively.This can be made concrete on the level of quantum Hamiltonians.It requires a parametrisation of the moduli space G E /K of G-graph connections in terms of an appropriate subtorus T of G, as well as Harish-Chandra's radial component techniques to describe the action of the edge-coordinate quadratic Casimirs on H Γ,G,S in terms of explicit second-order End(S)-valued differential operators H e (e ∈ E) on T (which are the quadratic Hamiltonians of the quantum Calogero-Moser spin chain up to a gauge).The differences H e − H e ′ for neighboring edges e and e ′ then form an explicit commuting family of first order differential operators, called asymptotic Knizhnik-Zamolodchikov-Bernard operators.See [8] and [24,21] for the details.1.10.Combining the results from §1.6 and §1.9 we obtain explicit conditions ensuring the superintegrability of the periodic and open quantum spin Calogero-Moser chains.For the special case of the directed cycle graph Γ with one vertex and K = G, the superintegrability of the associated periodic quantum spin Calogero-Moser system was considered before in [18].The classical superintegrability of the periodic and open Calogero-Moser spin chains is discussed in [19].1.11.The contents of the paper is as follows.
In Section 2 we describe the type of quantum systems that we consider in this paper, and discuss the concept of superintegrability in this context.
In Section 3 we formulate a result of Harish-Chandra [13] and Lepowsky & Mc-Collum [14] (Corollary 3.17) that will play the key role in establishing the superintegrability of the quantum spin systems on moduli spaces of graph connections.This involves the concept of reductive extensions of Lie algebras, which we discuss in detail.
In Section 4 we introduce the space of spin graph functions on graph connections, and provide a spanning set in terms of local invariant tensors (Theorem 4.15).For the directed cycle graph and the linearly ordered linear graph we relate the spin graph functions to multipoint trace functions and multipoint spherical functions (see §4.17 and §4.18).
In Section 5 we provide a detailed introduction of the quantum spin systems on moduli spaces of graph connections.We state the conditions ensuring superintegrability of the quantum spin system and discuss the superintegrability of the quantum periodic and open spin Calogero-Moser chains (see §5.12 and §5.13).
In Section 6 we give the proof of the main result (Theorem 1.6/5.11).The crucial intermediate step, which will allow us to use the result of Harish-Chandra and Lepowsky & McCollum in this context, is the translation of the condition of superintegrability in terms of irreducibility conditions of local intertwining spaces at the stars of the vertices of the graph (see Corollary 6.9).

Conventions:
The ground field will be C unless explicitly stated otherwise.Lie algebras are finite dimensional unless stated explicitly otherwise.We use Hom(V, W ) for the Hom-space in the category of complex vector spaces.For G a group, A an associative algebra and g a Lie algebra we write Hom G (V, W ), Hom A (V, W ), Hom g (V, W ) for the Hom-space in the category of G-representations, left A-modules and g-modules, respectively.We write U(k) for the universal enveloping algebra of a complex Lie algebra k, and Z(k) for its center.
For sets X, I with I finite, we write X I for the direct product of #I-copies of X.In case X is a Lie group/algebra, we endow X I with the direct product Lie group/algebra structure.
For a finite family {M i } i∈I of vector spaces M i with index set Acknowledgements: both authors were supported by the Dutch Research Council (NWO), project number 613.009.1260.In addition, the work of N.R. was supported by the NSF grant DMS-1902226, by the RSF grant 18-11-00-297 and by the Changjiang fund.

Centraliser algebras
In this section we derive some elementary properties of centraliser algebras, with an eye towards the application to quantum superintegrable systems.The starting point is a complex vector space H and an inclusion of unital algebras, with I being commutative.In applications to quantum mechanics H is the quantum state space, A the algebra of observables, and I its subalgebra of quantum Hamiltonians.We do not fix a particular H ∈ I as the quantum Hamiltonian of the system, since we are not considering quantum dynamics at this point.
2.1.Denote by I ∧ the set of characters of I.For an element χ ∈ I ∧ , i.e., for an unital algebra homomorphism χ : I → C, we write for the corresponding joint eigenspace (it may be zero).

Denote by
2.3.Suppose that J ⊆ End(H) is a sub-algebra stabilising H χ ⊆ H. Then H χ is a J-module, and J χ := {x| Hχ | x ∈ J} is a sub-algebra of End(H χ ).If H χ is a finite dimensional irreducible J-module, then J χ = End(H χ ) by the density theorem.If J = I then we have I χ = Cid Hχ .
2.4.Let J be a sub-algebra of A containing I. Then If in addition J stabilises H χ for some χ ∈ I ∧ , then C A (J) stabilises H χ in view of §2.2.The fact that both J and C A (J) stabilise H χ implies that C A (J) χ is contained in the commutant of J χ in End(H χ ).
If in addition H χ is an irreducible finite dimensional J-module (in particular, H χ = {0}), then C A (J) χ = C id Hχ = I χ by Schur's lemma.
2.5.Suppose that J ⊆ A is a subalgebra satisfying For such an algebra J the joint eigenspace H χ is J-stable for all χ ∈ I ∧ , in view of §2.2.
If in addition H is a semisimple I-module (i.e., H = χ∈I ∧ H χ ), then the map is an injective algebra homomorphism, with χ∈I ∧ End(H χ ) the direct product of the family {End(H χ ) | χ ∈ I ∧ } of algebras.Its image is contained in χ∈I ∧ J χ .
2.6.Suppose that J ⊆ A is a subalgebra satisfying (4) Assume furthermore that the following two spectral properties hold true: (a) H is a semisimple I-module.
(b) For χ ∈ I ∧ , either H χ = {0} or H χ is an irreducible finite dimensional J-module.By §2.3 and §2.4 we then have for all χ ∈ I ∧ .Informally speaking, J is "locally" of maximal size and equal to C A (I), and I is "locally" the center of J.
2.7.The setup of §2.6 provides the mathematical framework for superintegrability of quantum systems in this paper.From this perspective (4) is defining a quantum system with quantum state space H, algebra of quantum observables A, algebra of quantum Hamiltonians I and algebra of quantum integrals J.
Definition.The quantum system (4) is said to be superintegrable if the two spectral conditions 2.6(a)&(b) hold true.
The resulting properties (5) for the quantum superintegrable system provide the link with the notion of a core structure of a quantum superintegrable system considered in [18, §2].
A quantum superintegrable system is said to be quantum integrable if I = J.In this case dim(H χ ) ≤ 1 for all χ ∈ I ∧ , i.e., the eigenvalues of the quantum Hamiltonians determine the corresponding joint eigenvector up to a multiplicative constant.
For quantum superintegrable systems eigenstates this is no longer true.But when f, g ∈ H χ then there exists a quantum integral D ∈ J such that g = D(f ).Here we use that by the density theorem, the irreducibility of the J-module H χ is equivalent to For the examples of quantum superintegrable systems we thus have the weaker condition that simultaneous eigenspaces are finite dimensional, but two joint eigenvectors with the same eigenvalues can always be related through the action of a quantum integral.
2.8.From the perspective of quantisation, quantum superintegrability requires the algebras I, J and A to be quantisations of the Poisson algebras of Hamiltonians, integrals and observables for a classical superintegrable system (which is also sometimes called a degenerate integrable system).This is known in the case of periodic and open quantum spin Calogero-Moser chains [17,19].For a discussion of classical superintegrability, see [17] and references therein.

Preservation of irreducibility
In this section we focus on a purely representation theoretic result due to Lepowsky and McCollum [14, Thm.5.5] (in special cases it goes back to Harish-Chandra [13,Thm. 2]).It will be the crucial ingredient in proving superintegrability of the quantum spin systems on graph connections in Section 6.
3.1.Let g be a Lie algebra.Recall that a g-module M is said to be semisimple if M is the sum of its irreducible g-submodules.If furthermore all the irreducible gsubmodules of M are finite dimensional, then we say that M is a finitely semisimple g-module.
3.2.Let G be a real Lie group and K ⊆ G a connected compact Lie subgroup.Denote by g 0 the Lie algebra of G, and g its complexification.If π is a Hilbert space representation, then its (dense) subspace M of smooth K-finite vectors becomes a (g, K)-module with x ∈ g 0 acting by (see, e.g., [25, §3.3.1] for the definition of a (g, K)-module).The (g, K)-module M is finitely semisimple as a k-module.Furthermore, if π is irreducible and unitary, then the associated (g, K)-module M is irreducible as g-module.This in fact holds true under the weaker assumption that π is irreducible and admissible (see, e.g., [25, §3.3-4] for further details).
3.3.Let k ⊆ g be an inclusion of Lie algebras and M a g-module.Denote by k ∧ the set of isomorphism classes of finite dimensional irreducible k-modules.
and only if M is finitely semisimple as a k-module.

3.4.
A Lie subalgebra k ⊆ g is said to be reductive in g when g is a semisimple ad(k)-module.Note that if k is reductive in g, then k is a reductive Lie algebra.On the other hand, if k is a semisimple Lie subalgebra of g, then k is reductive in g by Weyl's complete reducibility theorem.

3.5.
Let G be a real Lie group with Lie algebra g 0 , and K ⊆ G a connected compact Lie subgroup.Denote by k and g the complexified Lie algebras of K and G, respectively.Then k is reductive in g.
3.6.Let g be a Lie algebra and θ ∈ Aut(g) an automorphism of finite order n.The associated fix-point Lie algebra is Proposition.Suppose that θ is an automorphism of a semisimple Lie algebra g of finite order m.Then g θ is reductive in g.

Proof.
The proof is a rather straightforward adjustment of the proof of the statement for involutions, see [6,Prop. 1.13.3].We give the proof for convenience of the reader.
Denote by g r (r ∈ Z/mZ) the eigenspace of θ with eigenvalue e 2πi/r .Then g θ = g 0 and g r are ad(g θ )-invariant subspaces of g.Since θ is of order m, the assignment 1 → θ defines a representation Z/mZ → GL(g) of the finite abelian group Z/mZ.By Maschke's theorem, we conclude that g = r∈Z/mZ g r .
Write p := r =0 g r , so that g = g θ ⊕ p.Let κ : g × g → C be the Killing form of g.Then κ(θ(x), θ(y)) = κ(x, y) for all x, y ∈ g, hence κ(g θ , p) = 0. Since g is semisimple, we conclude that κ| g θ ×g θ is nondegenerate.Furthermore, if x ∈ g θ and x = s + n is the abstract Chevalley decomposition of x in g, with s ∈ g (resp.n ∈ g) the semisimple (resp.nilpotent) part of x, then s, n ∈ g θ (this holds true for any automorphism θ of g).Then [6, Prop.1.7.6]implies that g θ is reductive in g.

3.7.
If k is reductive in g and M is a finitely semisimple g-module, then M is finitely semisimple as a k-module by [6, Prop.1.7.9(ii)].In particular, suppose that we have inclusions of Lie algebras where m is reductive in g and l is reductive in m, then l is reductive in g.

3.
8. For a homomorphic image of a Lie algebra k ⊆ g which is reductive in g, we have the following result.
Lemma.Suppose that k is reductive in g.Let φ : g ։ l be an epimorphism of Lie algebras.Then φ(k) is reductive in l.
Proof.Let g = m i=1 S i be a decomposition as a direct sum of finite dimensional irreducible ad(k)-modules.Then l = m i=1 φ(S i ).Either φ(S i ) = {0} or φ(S i ) is an irreducible ad(φ(k))-module.By a straightforward induction argument it follows that l = i∈I φ(S i ) for some subset I ⊆ {1, . . ., m}.This completes the proof.
3.9.For m ∈ Z >0 denote by δ (m) g : g → g ×m the Lie algebra homomorphism mapping x ∈ g to the m-tuple (x, . . ., x).If k is a Lie subalgebra of g, then we denote by k (m) ⊆ g ×m its image under δ (m) g .Proposition.Suppose that g is semisimple and that k is reductive in g.Then k (m) is reductive in g ×m .Proof.By Lemma 3.8, k (m) is reductive in g (m) .Note that with θ m the automorphism of g ×m of order m defined by θ m (x 1 , . . ., x m ) := (x m , x 1 , . . ., x m−1 ).Proposition 3.6 then shows that g (m) is reductive in g ×m .Hence k (m) is reductive in g ×m by §3.7.
3.10.Let k ⊆ g be an inclusion of Lie algebras.We say that g is a reductive extension of k when the inclusion map k ֒→ g is a section of ad(k)-modules and the quotient module g/k is a semisimple k-module.We then typically write p for a choice of an ad(k)-invariant complement of k in g (which is finitely semisimple as ad(k)-module).3.12.The following result should be compared to the transitivity property in §3.7.
Lemma.Let l ⊆ m ⊆ g be inclusions of finite dimensional Lie algebras.Suppose that g is a reductive extension of m and that l is reductive in m.Then l is reductive in g.

Proof.
Let p ⊆ g be an ad(m)-invariant subspace such that g = m ⊕ p.By the assumptions, m is a finite dimensional semisimple ad(l)-module and p is a finite dimensional semisimple ad(m)-module.It then follows from [6, Prop.1.7.9(ii)](see also §3.7) that p is also semisimple as an ad(l)-module.Since l is reductive in m, it follows that l is also reductive in g.

3.13.
The following lemma is the analog of Lemma 3.8 for reductive extensions.
Lemma.Let g be a reductive extension of k.Let φ : g ։ l be an epimorphism of Lie algebras.Then l is a reductive extension of φ(k).
Proof.Let p ⊆ g be an ad(k)-invariant subspace such that g = k ⊕ p.Let p = m i=1 S i be a decomposition as a direct sum of finite dimensional irreducible ad(k)modules.Then l = φ(k) + m i=1 φ(S i ), and either φ(S i ) = {0} or φ(S i ) is an irreducible ad(φ(k))-module.A straightforward induction argument then shows that l = φ(k) ⊕ i∈I φ(S i ) for some subset I ⊆ {1, . . ., m}.This completes the proof.
3.14.Let k ⊆ g be an inclusion of Lie algebras and g a reductive extension of k.Lepowsky and McCollum [14, Prop.4.2] obtained the following criterion to detect whether an irreducible g-module is finitely semisimple as a k-module.
Proposition.Let g be a reductive extension of k, and M an irreducible g-module.
Then M is finitely semisimple as a k-module unless M α = 0 for all α ∈ k ∧ .
3.15.Let k ⊆ g be an inclusion of Lie algebras.Using the associated canonical inclusion U(k) ⊆ U(g) of universal enveloping algebras, we set for the centraliser subalgebra of U(k) in U(g) (which equals the centraliser of k in U(g)).
Let M be a g-module, and view it as an U(g)-module.The corresponding homomorphism U(g) → End(M) restricts to an algebra map As a consequence, for a k-module S the space Hom k (S, M) of k-linear maps S → M becomes a left U(g) k -module, with U(g) k acting on the codomain M.
3.16.Let k ⊆ g be an inclusion of Lie algebras.Let S α be a finite dimensional irreducible k-module of isomorphism class α ∈ k ∧ .For a g-module M, the space Hom k S α , M) models the multiplicity space of S α in M. In fact, Hom k S α , M) is isomorphic to Hom k S α , M α ) as a complex vector space, and the k-module M α is isomorphic to an algebraic direct sum of dim(Hom k (S α , M)) copies of S α (see [6, §1.2.8]).In particular, Hom k S α , M) = 0 if and only if M α = 0. Hence Proposition 3.14 can be restated as follows: Proposition.Let g be a reductive extension of k, and M an irreducible g-module.
Then M is finitely semisimple as a k-module unless Hom k (S α , M) = 0 for all α ∈ k ∧ .
The multiplicity space Hom k S α , M) "remembers" the g-action on M through the left U(g) k -action from §3.15.Up to isomorphism of U(g) k -modules, the multiplicity space Hom k (S α , M) does not depend on the choice of S α .3.17.Let g be a reductive extension of k and α ∈ k ∧ an isomorphism class of a finite dimensional irreducible k-module.Lepowsky and McCollum [14,Thm. 5.5] showed that M → Hom k (S α , M) gives rise to a bijective correspondence between the isomorphism classes of irreducible g-modules M with M α = 0 and the isomorphism classes of irreducible modules over the quotient algebra U(g) k /(U(g) k ∩ U(g)J α ), where J α ⊆ U(k) is the annihilator of S α in U(k).In the context of §3.2, this correspondence goes back to Harish-Chandra [13].
In view of §3.16, we have the following immediate consequence of this correspondence.
Corollary.Let g be a reductive extension of k.Let M be an irreducible g-module.
For each α ∈ k ∧ , the multiplicity space Hom k (S α , M) is either {0} or it is an irreducible U(g) k -module.

Spin graph functions
In this section we introduce the space H = H G,Γ,S of spin graph functions and construct spanning sets of H using local tensor invariants.Here G is a connected compact Lie group, and Γ = (V, E, s, t) is a finite oriented graph with vertices V = {v 1 , . . ., v r }, edges E = {e 1 , . . ., e n } and source and target maps s, t : E → V .4.1.Let C(G) be the space of continuous complex-valued functions on G, viewed as G ×2 -representation by the left-and right-regular G-action, is an isomorphism of G ×2 -representations, where M * ⊗ M is endowed with the natural tensor product action of G ×2 .
4.3.Let G ∧ be the set of isomorphism classes of irreducible finite dimensional continuous G-representations.We denote the isomorphism class of an irreducible finite dimensional G-representation simply by its representation map π.The associated representation space is then denoted by M π .The Peter-Weyl theorem yields the decomposition 4.5.We write G E for the compact product Lie group G E .Its elements are denoted either by g = (g e ) e∈E or by g = (g 1 , . . ., g n ), with g j = g e j the group element attached to the edge e j .The group G E is called the group of graph G-connections on Γ.
4.7.Let σ : K → GL(S) be a finite dimensional representation.The space of global algebraic sections of the associated vector bundle over G E /K is denoted by In other words, H consists of the S-valued representative functions f on G E satisfying the equivariance property for k ∈ K and g ∈ G E .We call functions f ∈ H spin graph functions (spin refers to the interpretation of S as spin space for the associated quantum spin system, see §5.9, §5.12 and §5.13).
4.8.Fix π ∈ (G E ) ∧ an isomorphism class of a finite dimensional irreducible representation of G E and fix a finite dimensional representation σ : K → GL(S).
Then the subspace We call a spin graph function f ∈ H elementary if f is π-elementary for some π ∈ (G E ) ∧ .We denote for the space (R π (G E ) ⊗ S) K of π-elementary spin graph functions.Note that the elementary spin graph functions span H, since by the Peter-Weyl theorem (see §4.3).

4.9.
Let π e j = π j : G → GL(M j ) be finite dimensional G-representations, attached to the edges of Γ.The associated tensor product representation π : where It is convenient to think of the local representations π e (e ∈ E) as a choice of coloring of the edges of Γ.
We will identify n as in §4.9.In particular, for pure tensors When the π j : G → GL(M π j ) are all irreducible, then π is irreducible and its representation space will be denoted by M π .The assigment (π e ) e∈E → π induces a bijection If π ∈ (G E ) ∧ then we call the π e ∈ G ∧ such that π ≃ π the local components of π.
The local components of π * are π * e .

Similarly we denote tensor product representations of the gauge group
and S := S 1 ⊗• • •⊗S r , where σ v i = σ i : K v i → GL(S i ) are finite dimensional representations of the local gauge groups K v i .We now think of the local representations σ v (v ∈ V ) as a choice of coloring of the vertices of Γ.If σ : K → GL(S) is a finite dimensional irreducible representation then σ is isomorphic to a tensor product representation with finite dimensional irreducible local K v -representations σ v : K v → GL(S v ).

The product group G
4.13.The star S(v) of v ∈ V is the set of edges e ∈ E with source and/or target equal to v. Then S(v) = S(v|s) ∪ S(v|t) where S(v|s) is the set of edges oriented outward of v and S(v|t) the set of edges oriented toward v.Note that the union may not be disjoint since we allow loops in the graph Γ.
We consider the K v -representation M πe and K v acting diagonally, Here the tensor factors are ordered using the total order on S(v|s) and S(v|t) induced from the total order on E. We will identify the dual K v -representation M π S(v) * with e∈S(v|s) with K v acting diagonally, cf.§4.4.

Fix finite dimensional K
) for e ∈ E, thus providing the vertices and the edges of Γ with representation colors.At vertex v ∈ V we assign to the colored graph Γ the K v -representation with K v acting diagonally, and we endow v∈V with the tensor product action of the gauge group v) and u v ∈ S v .

4.15.
For π e ∈ G ∧ and σ v : K v → GL(S v ) finite dimensional K v -representations, consider the linear map Theorem.The linear map Ψ π defines a K-linear isomorphism It restricts to a linear isomorphism Proof.By §4.12 it is clear that Ψ π is a linear isomorphism.It is K-linear, since for k = (k v ) v∈V ∈ K and g ∈ G E , Ψ π v∈V e∈S(v|s) The second statement of the theorem follows now immediately from the fact that v∈V M π S(v) ⊗ S v is endowed with the tensor product action of K = v∈V K v , see §4.14.  v) .We then have a linear isomorphism Combined with Theorem 4.15 we thus obtain a parametrisation of the space R π (G E ) ⊗ S K of π-elementary spin graph functions in terms of spaces of local intertwiners (local in the sense that they only depend on the colors of Γ at the star of a vertex v).
4.17.Let n ≥ 1 and consider the directed cycle graph Γ with n edges.We enumerate the vertices v i and edges e j (i, j ∈ Z/nZ) in such a way that s(e i ) = v i and t(e i ) = v i+1 for i ∈ Z/nZ.The order 1 < 2 < • • • < n on Z/nZ provides a total order on V and E. We take K = G V as gauge group.
With these conventions G E ≃ G ×n by (g e ) e∈E → (g e 1 , . . ., g en ), and K ≃ G ×n by (k v ) v∈V → (k v 1 , . . ., k vn ).The left gauge action of K on G E then reads as The partial trace Tr S M πn (B) of B ∈ Hom(M πn , M πn ⊗ S) is the unique vector in S satisfying where Tr M πn is the usual trace on End(M πn ).The elementary spin graph functions now have the following description in terms of partial traces of compositions of intertwiners.
Proposition.Endow M π i−1 ⊗ S i with the diagonal G-action.We have a linear isomorphism Proof.In the current situation we have 9) and the fact that where Under these identifications the intertwiner Φ i ∈ Hom G M π i , M π i−1 ⊗ S i corresponds to the local invariant tensor Combined with Theorem 4.15 we thus obtain a linear isomorphism and applying (7), we obtain the explicit expression For i = n the sum over k i ∈ I e i can be simplified using the identity Hence f π Φ = f π Φ by the cyclicity of the partial trace: The study of n-point trace functions originates from the paper [8].Intertwiners Hom G M π i , M π i−1 ⊗ S i may be viewed as a topological degenerations of vertex operators.The class of (elementary) n-point trace functions and its generalisation to the affine and quantum group level are particularly well studied, see, e.g., [8,10,9].4.18.Let n ∈ Z ≥1 .As a next example we consider the linearly ordered linear graph Γ with n edges.We denote the ordered vertex and edge sets by V = {v 1 , . . ., v n+1 } and E = {e 1 , . . ., e n }.The source and target maps are s(e i ) = v i and t(e i ) = v i+1 for i = 1, . . ., n.As local gauge groups we take ( 11) where H, K ⊆ G are subgroups.
The action of the associated gauge group Let S i (1 < i ≤ n) be finite dimensional G-representations, L a finite dimensional H-representation and N a finite dimensional K-representation.Denote by the resulting tensor product spin representation of K. We also write for the "bulk" G ×(n−1) -representation associated to S, so that S = L ⊗ S ⊗ N.

Denote by
Q L⊗S,N : Hom(N * , L ⊗ S) ∼ −→ S the linear isomorphism defined by with {n j } j a basis of N and {n * j } j the corresponding dual basis of N * .Let π : G ×n → GL(M π ) be an irreducible tensor product representation, with local components In the following proposition we will also view M π 1 (resp.M πn ) as H-representation (resp.K-representation) by restriction.
Proposition.We have a linear isomorphism Proof.At vertices v i with 1 < i ≤ n the analysis of the local space of invariants M π S(v i ) ⊗ S i Kv i is as in the proof of Proposition 4.17.For i = 1 we have with the isomorphism as in §4.16.For i = n + 1 we analogously have Under these isomorphisms the intertwiner Θ ∈ Hom H M π 1 , L) corresponds Combined with Theorem 4.15 we thus obtain a linear isomorphism where Φ i is given by (10).
A direct computation now shows that the spin graph function f π Θ,Φ,Ξ (g) is explicitly given by Contracting the bulk intertwiners Φ i (i = 2, . . ., n) as in the proof of Proposition 4.17, we obtain the expression , which is easily seen to be equal to f π Θ,Φ,Ξ (g).This completes the proof.
For H = K the modified spin graph functions Q −1 L⊗S,N f π Θ,Φ,Ξ are the elementary n-point spherical functions from [24,21].For n = 1, they reduce to the elementary spherical functions on compact symmetric spaces.

Quantum spin systems on graph connections
Let Γ be a finite oriented graph and G a connected compact Lie group.We write V = {v 1 , . . ., v r } and E = {e 1 , . . ., e n } for the totally ordered vertex and edge set of Γ.We denote by g the complexification of the Lie algebra g 0 of G.In this section we introduce a quantum spin system on the spaces H = H Γ,G,S of spin graph functions.5.1.We call a linear differential operator D on G algebraic if it preserves the space R(G) of representative functions on G (see §4.1).We have an inclusion of algebras with D(G) the algebra of algebraic differential operators on G, with D inv (G) ⊆ D(G) the subalgebra generated by the left and right G-invariant differential operators on G, and with D biinv (G) ⊆ D inv (G) the subalgebra of G-biinvariant differential operators on G.

5.2.
We have a surjective algebra map We then have the balancing condition for X, Y ∈ U(g) and Z ∈ Z(g), where ι is the antipode of U(g) (i.e., ι is the unique anti-algebra automorphism of U(g) such that x → −x for x ∈ g).Hence the algebra map (12) descends to an isomorphism of algebras ( 13) with the balanced tensor product over Z(g) relative to the ι-twisted right regular Z(g)-action on U(g) and the left regular Z(g)-action on U(g) (injectivity of the map (13) was shown in [15]).

The algebra
In particular, D biinv (G) is contained in the center of D inv (G).
5.4.Consider the algebra D(G E ) of algebraic differential operators on the connected compact Lie group G E , and recall the gauge action of K on G E (see §4.6).
The corresponding contragredient K-action on R(G E ) is ) and g ∈ G (this is the special case of the K-action on R(G E ) ⊗ S from §4.7 when S is the trivial K-representation).This action induces an K-action 5.5.As in §5.2, we identify U((g E ) ×2 ) ≃ U(g E )⊗U(g E ).Furthermore, we identify U(g E ) ≃ U(g) ⊗#E as algebras, with the isomorphism induced by for (x e ) e∈E ∈ g E .It restricts to an isomorphism Z(g E ) ≃ Z(g) ⊗#E .We will use the notation for X ∈ U(g) and i ∈ {1, . . ., n}, and a pure tensor in U((g E ) ×2 ) will be denoted by X ⊗ Y with and X e , Y e ′ ∈ U(g).
Lemma.The formula defines an action of K on U((g E ) ×2 ) by algebra automorphisms.Furthermore, Proof.The first statement is immediate.For the second statement, it suffices to check (16) when X = x (i) and Y = 1 U (g E ) and when X = 1 U (g E ) and Y = y (i) , where x, y ∈ g 0 .When X = x (i) and Y = 1 U (g E ) we have as desired.A similar computation proves ( 16) when X = 1 U (g E ) and Y = y (i) .
5.6.Lemma 5.5 shows that D inv (G E ) is a K-invariant subalgebra of D(G E ).Denote by the subalgebra of K-invariant differential operators in D inv (G E ).By §5.3 and Lemma 5.5 we then have the inclusion Then formula (14) remains true in this more general context, of subalgebras of End(H Γ,G,S ).We omit the labels Γ, G, S if they are clear from context.Note that I is contained in the center of J, in view of §5.3.
5.9.Following §2.7 we view the inclusion of algebras as a quantum spin system with quantum state space H, algebra of quantum observables A, algebra of quantum integrals J, and commutative algebra of quantum Hamiltonians I.For i ∈ {1, . . ., n} and Ω ∈ Z(g) the quadratic Casimir element, the action of on H is a quantum Hamiltonian H i ∈ I of the quantum spin system.We call H i (i ∈ {1, . . ., n}) the edge-component quadratic Hamiltonians of the quantum spin system.
for i = 1, . . ., n and Z ∈ Z(g) (here the π j are the local components of the tensor product representation π, see §4.9).It follows from §4.12 that the space H π of πelementary spin graph functions can alternatively be described as the simultaneous I-eigenspace for the one-dimensional I-module χ π ∈ I ∧ , Hence condition (a) from §2.6 always holds true for the quantum spin system, with H χπ = H π the finite dimensional space of π-elementary spin graph functions.
5.11.The following is the main result of the paper.
Theorem.Let Γ be a finite connected oriented graph, G a connected compact Lie group, K = v∈V K v with K v ⊆ G subgroups, and σ : K → GL(S) a finite dimensional representation.The quantum spin system on H = H Γ,G,S as defined in §5.9 is superintegrable if the following three conditions hold true: (a) G is simply connected.(b) For each v ∈ V , the local gauge group K v ⊆ G is closed and connected.
We give the proof of the theorem in §6.10.
5.12.Consider the quantum spin system with Γ the oriented cycle graph with n edges, K = G V and σ : K → GL(V ) a finite dimensional representation, see §4.17.By Theorem 5.11 it is superintegrable when G is simply connected and σ is irreducible.The condition on σ implies that σ is equivalent to a tensor product representation σ with its local representations σ v : G → GL(S v ) irreducible for all v ∈ V .This quantum spin system can be made more explicit using the parametrisation of its moduli space M of graph G-connections in terms of a maximal torus T ⊂ G.The edge-component quadratic Hamiltonians H i then become explicit second-order End(S)-valued differential operator on T of Calogero-Moser type.The differences H i − H i−1 are first-order commuting differential operators called asymptotic Knizhnik-Zamolodchikov operators, which can be entirely described in terms of Felder's classical trigonometric dynamical r-matrix (see [8,23,20]).This provides the interpretation of this quantum spin system as a quantum periodic spin Calogero-Moser chain [21].
For the special case n = 1, the superintegrability of the quantum periodic Calogero-Moser spin system was discussed in [18].5.13.Consider now the quantum spin system with Γ the linearly ordered linear graph with n edges, local gauge groups of the form (11) with H, K ⊆ G closed connected subgroups, and σ : K → GL(S) a finite dimensional representation of the associated gauge group K (see §4.18).By Theorem 5.11 this quantum spin system is superintegrable when G is simply connected and σ is irreducible.The condition on σ implies that S ≃ L⊗S 2 ⊗• • •⊗S n ⊗N with L an irreducible H-representation, N an irreducible K representation and S j irreducible G-representations.This quantum spin system can be made more concrete when H = K is the connected component of the identity of a fix-point subgroup G Θ of an involution Θ of G, using an appropriate parametrisation of its moduli space M of graph G-connections in terms of an appropriate subtorus A ⊂ G.The edge-component quadratic Hamiltonians H i then become second-order End(S)-valued differential operator on A of Calogero-Moser type and H i − H i−1 are asymptotic boundary Knizhnik-Zamolodchikov operators, which are first order differential operators involving folded classical dynamical r-matrices and associated dynamical k-matrices (see [24,21,23,20]).This provides the interpretation of this quantum spin system as a quantum open spin Calogero-Moser chain [21].

Conditions for superintegrability
In this section we provide a proof of the sufficient conditions ensuring superintegrability of the quantum spin systems defined in §5 (see Theorem 5.11).We retain the notations and conventions of §5.In particular, Γ is an oriented finite graph, G is a connected compact Lie group, and K = v∈V K v with subgroups K v ⊆ G.
We take as finite dimensional K-representation of the quantum system a tensor product representation σ : K → GL(S) (see §4.10).We furthermore fix an irreducible finite dimensional tensor product representation π : G E → GL(M π ), with local irreducible G-representations π e : G → GL(M πe ).
Finally, we write g 0 for the Lie algebra of G, and g for its complexification.
6.1.For v ∈ V consider the linear isomorphism ( 18) where {u tv ) in terms of the tensor product basis of M π S(v) (see §4.16).Its expansion coefficients will be denoted by φ tv .

Turn Hom(S
The linear map τ v (see (19)) is K v -linear, with the K v -action on the codomain of τ v as defined in §4.14.Hence τ v restricts to a linear isomorphism A direct computation using (19) then leads to the formula for φ v ∈ Hom(S * v , M π S(v) ).
A pure tensor in U(g) (v) is denoted by X v|s ⊗ Y v|t with M π e ′ , as K v -representation space relative to the diagonal K v -action π S(v) .Differentiating the G-action turns M π * e and M π e ′ into irreducible U(g)-modules, and hence M π S(v) into an irreducible U(g) (v) -module via the diagonal U(g) (v) -action.
We view the linear space Hom(S * v , M π S(v) ) as U(g) (v) -module, with U(g) (v) acting on its co-domain, for X v|s ⊗ Y v|t ∈ U(g) (v) , T ∈ Hom(S * v , M π S(v) ) and ξ ∈ S * v .
6.6.The local gauge group K v acts by algebra automorphisms on U(g) (v) via the diagonal adjoint action, Ad(k v )X e ⊗ e ′ ∈S(v|t) We then have (v) and B ∈ M π S(v) .Let (U(g) (v) ) Kv be the algebra of K v -invariant elements in U(g) (v) relative to the K v -action • v .It follows from §6.2 and §6.5 that the space Hom Kv (S * v , M π S(v) ) of K v -intertwiners is a (U(g) (v) ) Kv -module, with the action on Hom Kv (S * v , M π S(v) ) given by (21).Consider the tensor product action of the gauge group K = v∈V K v on the co-domain v∈V U(g) (v) of the algebra isomorphism (22), with the K v -action on U(g) (v) as defined in §6.6.A direct check shows that the algebra isomorphism ( 22) is K-linear, with K acting on U((g E ) ×2 ) according to Lemma 5.5.The algebra isomorphism (22) thus restricts to an algebra isomorphism (23) U((g E ) ×2 ) K ∼ −→ v∈V (U(g) (v) ) Kv .
By §5.2 and §5.7 the universal enveloping algebra U((g E ) ×2 ) also acts on the space R π (G E ) ⊗ S of S-valued representative functions on G E by (X ⊗ Y) • (f ⊗ u) := D X⊗Y (f ) ⊗ u for X, Y ∈ U(g E ), f ∈ R π (G E ) and u ∈ S.
Proof.Using the notations from §6.7 we have The result now follows from the fact that i ,m j ).

3. 11 .
Let k ⊆ g be a Lie subalgebra.The following two statements are equivalent: (a) k is reductive in g.(b) k is a reductive Lie algebra and g is a reductive extension of k.In particular, §3.5, §3.6 and §3.7 provide examples of reductive extensions.
be the subalgebra of representative functions on G.In other words, R(G) consists of the functions f ∈ C(G) which generate a finite dimensional G ×2 -subrepresentation of C(G).

4. 4 .
For two G-representations M and N we identify M * ⊗ N * ≃ (M ⊗ N) * as G ×2 -representations, where M ⊗ N and M * ⊗ N * are endowed with the tensor product G ×2 -action.Under this correspondence, φ ⊗ ψ for φ ∈ M * and ψ ∈ N * corresponds to the linear functional on M ⊗ N satisfying m ⊗ n → φ(m)ψ(n) for m ∈ M and n ∈ N. In particular, if {m i } i and {n j } j are bases of M and N and {m * i } i and {n * j } j are the respective dual bases of M * and N * , then the basis m e ⊗ u v := c π φ,m ⊗ u with φ := e∈E φ e , m := e∈E m e and u := v∈V u v .This is well defined due to the disjoint union decompositions v∈V S(v|s) = E = v∈V S(v|t) of the edge set E of Γ.

4. 16 .
We keep the setup as in §4.15.For e ∈ E let {m e,ie } ie∈Ie be a basis of M πe .For v ∈ V set I(v|s) := {i = (i e ) e∈S(v|s) | i e ∈ I e } and write for i ∈ I(v|s), m i (v|s) := e∈S(v|s) m e,ie .In a similar way we define m i (v|t) for indices i from I(v|t) := {i = (i e ) e∈S(v|t) | i e ∈ I e }.Then (8) { m i (v|s) * ⊗ m j (v|t) | (i, j) ∈ I(v|s) × I(v|t) } is a basis of M π S

5. 7 .
Let S be a finite dimensional K-representation.The space R(G E ) ⊗ S of S-valued representative functions on G E becomes a D(G E )-module by D(h ⊗ u) := D(h) ⊗ u for D ∈ D(G E ), h ∈ R(G E ) and u ∈ S. In addition we have the restricted gauge group K acts on R(G E ) ⊗ S by the twisted K-action (k, f ) → k • f from §4.7.

5. 8 .
As a consequence of §5.7, the algebra D(G E ) K of K-invariant algebraic differential operators on G E acts on the space H Γ,G,S = (R(G E ) ⊗ S) K of spin graph functions.The resulting homomorphic image of the inclusions(17) of algebras in End(H Γ,G,S ) gives rise to the inclusion

∼−→ H π . 6 . 4 .
where the domain of Υ π is viewed as K-representation relative to the tensor product action of K = v∈V K v .The map Υ π restricts to a linear isomorphismΥ π : v∈V Hom Kv (S * v , M π S(v)) Let I be the set of sequences (i e ) e∈E with i e ∈ I e .Consider the tensor product basis {m i } i∈I of M π , wherem i := e∈E m ie,e ,and write m * i := e∈E m * ie,e for the corresponding dual basis elements of (M π ) * ≃ ⊗ e∈E M π * e , cf. §4.9.For i ∈ I write i v|s := (i e ) e∈S(v|s) ∈ I(v|s), i v|t := (i e ) e∈S(v|t) ∈ I(v|t).
j ⊗ v∈V tv φ v [t v ; i v|s , j v|t ] u (v) tv

X
v|s = e∈S(v|s) X e , Y v|t = e ′ ∈S(v|t)Y e ′ ,where we order the tensor products along the total orders on S(v|s) and S(v|t) induced by the total order on E. In §4.13 we considered the spaceM π S(v) = e∈S(v|s) M π * e ⊗ e ′ ∈S(v|t) ⊗ Y v|t ) • T (ξ) := (X v|s ⊗ Y v|t ) • (T (ξ))

6. 7 .X
Consider the algebra isomorphism (22) U((g E ) ×2 ) v|s ⊗ Y v|t for X = e∈E X e and Y = e ′ ∈E Y e ′ , where X v|s = e∈S(v|s) X e , Y v|t = e ′ ∈S(v|t) Y e ′ .
4.6.For each vertex v ∈ V we fix a Lie subgroup K v ⊆ G.It will play the role as local gauge group at the vertex v.The product group K := v∈V K v is the associated gauge group.It is a subgroup of the group G V of lattice gauge transformations.A group element in K is denoted by k i and Y • m j refer to the U(g E )-action on (M π ) * and M π , obtained by differentiating the G E -action).The second equality follows from the expansion formula(X v|s ⊗ Y v|t ) • φ v [t v ; i ′ v|s , j ′ v|t ] = = i v|s ,j v|t X v|s • m i v|s (v|s) * (m i ′ v|s (v|s))m j ′ v|t (v|t) * (Y v|t • m j v|t (v|t))φ v [t v ; i v|s , j v|t ]with the sums in the left hand side taken over i v|s ∈ I(v|s) and j v|t ∈ I(v|t), and the fact that v [t v ; i v|s , j v|t ] u (v) tv (here X • m * i ′ ∈I v∈V X v|s • m i v|s (v|s) * (m i ′ v|s (v|s)) m * i ′