Darboux coordinates for symplectic groupoid and cluster algebras
aa r X i v : . [ m a t h . QA ] F e b DARBOUX COORDINATES FOR SYMPLECTIC GROUPOID ANDCLUSTER ALGEBRAS
LEONID O. CHEKHOV ∗ AND MICHAEL SHAPIRO ∗∗ Dedicated to the memory of great mathematician and person Boris Dubrovin
Abstract.
Using Fock–Goncharov higher Teichm¨uller space variables we derive Darboux co-ordinate representation for entries of general symplectic leaves of the A n groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves foralgebras governed by the reflection equation with the trigonometric R -matrix. The obtainedresults are in a perfect agreement with the previously obtained Poisson and quantum repre-sentations of groupoid variables for A and A in terms of geodesic functions for Riemannsurfaces with holes. We realize braid-group transformations for A n via sequences of clustermutations in the special A n -quiver. We prove the groupoid relations for quantum transportmatrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. We provethe quantum algebraic relations of transport matrices for arbitrary (cyclic or acyclic) directedplanar network. Introduction
Symplectic groupoid, induced Poisson structure on the unipotent upper trian-gular matrices.
Let V denote an n -dimensional vector space, A be some subspace of bilinearforms on V . Fixing the basis in V , one can identify A with a subspace in the space of n × n matrices. The matrix B of a change of a basis in V takes a matrix of bilinear form A ∈ A to B A B T .Below we consider an important particular case when A is the space of unipotent formsidentified with the space of the unipotent matrices. The basis change B acts on A only if theproduct B A B T be unipotent itself. We thus introduce the space of morphisms identified withadmissible pairs of matrices ( B, A ) such that M = (cid:8) ( B, A ) (cid:12)(cid:12) B ∈ GL ( V ) , A ∈ A , B A B T ∈ A (cid:9) . We then have the standard set of maps:source s : M → A ( B, A ) → A , target t : M → A ( B, A ) → B A B T , injection e : A → M A → ( E, A ) , inversion i : M → M ( B, A ) → ( B − , B A B T ) , multiplication m : M (2) → M (cid:0) ( C, B A B T ) , ( B, A ) (cid:1) → ( CB, A ) ∗ Steklov Mathematical Institute, Moscow, Russia, National Research University Higher School of Economics,Russia, and Michigan State University, East Lansing, USA. Email: [email protected]. ∗∗ Michigan State University, East Lansing, USA and National Research University Higher School of Eco-nomics, Russia . Email: [email protected]. ∗ AND MICHAEL SHAPIRO ∗∗ such that the following diagram, where p and p are natural projections to the first and thesecond morphism in an admissible pair of morphisms, is commutative: MM (2) AM sp p t The crucial point of the construction is the existence of a symplectic structure: a smoothgroupoid endowed with a symplectic form ω ∈ Ω M on the morphism space M that satisfiesthe splitting (consistency) condition [27, 40] m ⋆ ω = p ⋆ ω + p ⋆ ω, which implies, in particular, that the source and target maps Poisson commute being respec-tively an automorphism and an anti-automorphism of the initial Poisson algebra. Since p ⋆ ω and p ⋆ ω are nondegenerate, they admit a (unique) Poisson structure, and because the immersionmap e is Lagrangian, this Poisson structure yields a Poisson structure on A .Identifying A with A n —the space of unipotent upper triangular matrices, in 2000, Bondal[2] obtained the Poisson structure on A n using the algebroid construction; assuming B = e g ,we obtain the Bondal algebroid using the anchor map D A to the tangent space T A A n (1.1) D A : g A → T A A n g A g + g T A , A ∈ A n , where g A is the linear subspace g A := (cid:8) g ∈ gl n ( C ) , | A + A g + g T A ∈ A n (cid:9) of elements g leaving A unipotent. Lemma 1.1. [2]
The map (1.2) P A : T ∗ A n A → g A w P − , / ( w A ) − P + , / ( w T A T ) , where P ± , / are the projection operators: (1.3) P ± , / a i,j := 1 ± sign( j − i )2 a i,j , i, j = 1 , . . . , n, and w ∈ T ∗ A n is a strictly lower triangular matrix, defines an isomorphism between the Liealgebroid ( g , D A ) and the Lie algebroid ( T ∗ A n , D A P A ) . The Poisson bi-vector Π on A n is then obtained by the anchor map on the Lie algebroid( T ∗ A n , D A P A ) (see Proposition 10.1.4 in [31]) as:(1.4) Π : T ∗ A A n × T ∗ A A n
7→ C ∞ ( A n )( ω , ω ) → Tr ( ω D A P A ( ω ))It can be checked explicitly that the above bilinear form is in fact skew-symmetric and givesrise to the Poisson bracket(1.5) { a i,k , a j,l } := ∂∂ d a i,k ∧ ∂∂ d a j,l Tr (d a i,k D A P A (d a j,l )) , having the following form in components: { a i,k , a j,l } = 0 , for i < k < j < l, and i < j < l < k, ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 3 { a i,k , a j,l } = 2 ( a i,j a k,l − a i,l a k,j ) , for i < j < k < l, (1.6) { a i,k , a k,l } = a i,k a k,l − a i,l , for i < k < l, { a i,k , a j,k } = − a i,k a j,k + 2 a i,j , for i < j < k, { a i,k , a i,l } = − a i,k a i,l + a k,l , for i < k < l. This bracket turned out to coincide with the bracket previously known in mathematical physicsas Gavrilik–Klimyk–Nelson–Regge–Dubrovin–Ugaglia bracket [22, 33, 34, 14, 39] and it arisesfrom skein relations satisfied by a special finite subset of geodesic functions (traces of mon-odromies of SL Fuchsian systems, which are in 1-1 correspondence with closed geodesics on aRiemann surface Σ g,s ) described in [6]; a simple constant log-canonical (Darboux) bracket onthe space of Thurston shear coordinates z α on the Teichm¨uller space T g,s of Riemann surfacesΣ g,s of genus g with s = 1 , a i,k . All such geodesic func-tions admit an explicit combinatorial description [15], which immediately implies that theyare Laurent polynomials with positive integer coefficients of e z α / . The Poisson bracket of z α spanning the Teichm¨uller space T g,s has exactly s Casimirs, which are linear combinations ofshear coordinates incident to the holes, so the subspace of z α orthogonal to the subspace ofCasimirs parameterizes a symplectic leaf in the Teichm¨uller space which we call a geometricsymplectic leaf .In[6], the Poisson embedding of geometric symplectic leaf into A n was constructed. Notehowever that the size n of matrix A is related to the genus and the number of holes as n = 2 g + s (with s taking only two values, 1 and 2) and that the (real) dimension of T g,s is 6 g − s increasing linearly with g whereas the total dimension of A n is obviously n ( n − / n ; for n = 3 and n = 4 these two dimensions coincide and the geometricsymplectic leaf having the dimension 6 g − s is of maximum dimension.For n = 5, the dimension of the geometric symplectic leaf has still the maximum value 8 ofdimensions of symplectic leaves in A , but we have just one central element in the correspondingTeichm¨uller space T , and two central elements in A . For all larger n the dimension ofgeometric symplectic leaf is strictly less than the maximal dimension of symplectic leaf in A n ,so the geometric systems do not describe maximal symplectic leaves in the total Poisson spaceof A n . The Darboux coordinates in geometric situation are well known to be the above shearcoordinates, but, as just mentioned, they can not help in constructing Darboux coordinates in A n for n ≥ first problem addressed in this publication is a construction of Darboux coordinatesfor a general symplectic leaf of A n and explicit expression of matrix elements a i,j in terms ofthese Darboux coordinates. It was expected for long, and we show below that these Darbouxcoordinates are related to cluster algebras, similar to the geometric cases n = 3 , A or with ageneral A gen ∈ gl n are known under the name of reflection equation algebras . A task closelyrelated to the first problem is to construct a Darboux coordinate representation for a generalmatrix A gen enjoying the reflection equation.1.2. Standard Poisson-Lie group G and its dual. Another description of the Poissonstructure on the space of triangular forms A n as a push-forward of the standard Poisson bracketon the dual group G ∗ = SL ∗ n to the set of fixed points of the natural involution was given in [1].A reductive complex Lie group G equipped with a Poisson bracket {· , ·} is called a Poisson–Liegroup if the multiplication map G × G ∋ ( X, Y ) XY ∈ G is Poisson. Denote by h , i aninvariant nondegenerate form on the corresponding Lie algebra g = Lie ( G ), and by ∇ R , ∇ L LEONID O. CHEKHOV ∗ AND MICHAEL SHAPIRO ∗∗ the right and left gradients of functions on G with respect to this form defined by (cid:10) ∇ R f ( X ) , ξ (cid:11) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( Xe tξ ) , (cid:10) ∇ L f ( X ) , ξ (cid:11) = ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( e tξ X )for any ξ ∈ g , X ∈ G .Let π > , π < be projections of g onto subalgebras spanned by positive and negative roots, π be the projection onto the Cartan subalgebra h , and let R = π > − π < . The standardPoisson-Lie bracket {· , ·} r on G can be written as(1.7) { f , f } r = 12 (cid:0)(cid:10) R ( ∇ L f ) , ∇ L f i − h R ( ∇ R f ) , ∇ R f (cid:11)(cid:1) . The standard Poisson–Lie structure is a particular case of Poisson–Lie structures correspond-ing to quasitriangular Lie bialgebras. For a detailed exposition of these structures see, e. g., [3,Ch. 1], [37] and [41].Following [37], let us recall the construction of the Drinfeld double. The double of g is D ( g ) = g ⊕ g equipped with an invariant nondegenerate bilinear form hh ( ξ, η ) , ( ξ ′ , η ′ ) ii = h ξ, ξ ′ i − h η, η ′ i .Define subalgebras d ± of D ( g ) by d + = { ( ξ, ξ ) : ξ ∈ g } and d − = { ( R + ( ξ ) , R − ( ξ )) : ξ ∈ g } ,where R ± ∈ End g is given by R ± = ( R ± Id). The operator R D = π d + − π d − can be used todefine a Poisson–Lie structure on D ( G ) = G × G , the double of the group G , via(1.8) { f , f } D = 12 (cid:0)(cid:10)(cid:10) R D ( ▽ L f ) , ▽ L f (cid:11)(cid:11) − (cid:10)(cid:10) R D ( ▽ R f ) , ▽ R f (cid:11)(cid:11)(cid:1) , where ▽ R and ▽ L are right and left gradients with respect to hh· , ·ii . The diagonal subgroup { ( X, X ) : X ∈ G} is a Poisson–Lie subgroup of D ( G ) (whose Lie algebra is d + ) naturallyisomorphic to ( G , {· , ·} r ).The group G ∗ whose Lie algebra is d − is a Poisson-Lie subgroup of D ( G ) called the dualPoisson-Lie group of G . The Poisson bracket {· , ·} D induces the Poisson bracket on G ∗ .For G = SL n the dual group G ∗ = { ( X + , Y − ) } ∈ B + × B − satisfying the additional relation π ( X + ) π ( Y − ) = Id where B + ( B − ) ⊂ SL n are Borel subgroups of nondegenerate upper (lower)triangular matrices.The involution ι G ∗ : G ∗ → G ∗ takes ( X + , Y − ) to ( Y t − , X t + ).The subgroup U + of unipotent upper triangular matrices is embedded diagonally in G ∗ . Theembedding ǫ : U + ֒ → G ∗ maps X ∈ U + to ( X, X ). The image ǫ ( U + ) is the set of fixed points ofinvolution ι G ∗ .The image ǫ ( U + ) is not a Poisson subvariety of G ∗ however the Dirac reduction induces thePoisson bi-vector Π (1.4) on U + .To remind the definition of Dirac reduction we consider a subvariety X of a Poisson variety( V, {· , ·} P B ) defined by constrains φ i = const . The second class constrains are constrains ˜ φ a whose Poisson brackets with at least one other constraint do not vanish on the constraintsurface.Define matrix U with entries U ab = { ˜ φ a , ˜ φ b } P B . Note that U is always invertible.Then, Dirac bracket of functions f and g on X is { f, g } DB = { f, g } P B − X a,b { f, ˜ φ a } P B U − ab { ˜ φ b , g } P B , see [26] for details.1.3. Braid-group action on the unipotent matrices.
The next important result con-cerning A n is that this space admits the discrete braid-group action generated by morphisms ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 5 β i,i +1 : A n → A n , i = 1 , . . . , n −
1, such that(1.9) β i,i +1 [ A ] = B i,i +1 A B Ti,i +1 ≡ e A ∈ A n , where the matrix B i,i +1 has the block form(1.10) B i,i +1 = ... ii + 1... a i,i +1 −
11 0 1 . . . 1 , and this action is a Poisson morphism [2], [39]. When acting on A , β i,i +1 satisfy the standardbraid-group relations β i,i +1 β i +1 ,i +2 β i,i +1 A = β i +1 ,i +2 β i,i +1 β i +1 ,i +2 A for i = 1 , . . . , n − β n − ,n β n − ,n − · · · β , β , A = S n A , where S n is an element of thegroup of permutations of matrix entries a i,j whose n th power is the identity transformation.Note that β i,i +1 A = A .In [7], the quantum version of the above transformations was constructed for a quantum upper-triangular matrix(1.11) A ~ := q − / a ~ , a ~ , . . . a ~ ,n q − / a ~ , . . . a ~ ,n q − / . . . ...... ... . . . . . . a ~ n − ,n . . . q − / . Here a ~ i,j are self-adjoint (cid:0)(cid:2) a ~ i,j (cid:3) ⋆ = a ~ i,j (cid:1) operators enjoying quadratic–linear algebraic rela-tions following from the quantum reflection equation (see Theorem 4.1) and coinciding withrelations obtained for quantum geodesic functions upon imposing quantum skein relations onthe corresponding geodesics and q = e − i ~ / . The analogous quantum braid-group action is A ~ → B ~ i,i +1 A ~ (cid:2) B ~ i,i +1 (cid:3) † with(1.12) B ~ i,i +1 = ... ii + 1... q / a ~ i,i +1 − q , In the geometric cases, the above braid-group morphisms are related to modular transforma-tions generated by (classical or quantum [28]) Dehn twists along geodesics corresponding to thegeodesic functions a i,i +1 (see [6]). In the absence of geometric interpretation, the only possibil-ity we may resort to is to address the second problem : to find a sequence of cluster mutationsin a quiver still to be constructed that produces the above braid-group transformation for ageneric symplectic leaf of A n . LEONID O. CHEKHOV ∗ AND MICHAEL SHAPIRO ∗∗ We solve the both formulated problems in this paper: we explicitly construct the quiver(called an A n -quiver) such that the entries a i,j of the unipotent matrix A are positive Laurentpolynomials of the cluster quiver variables, construct a quantum version of this quiver thusrealizing the representation (1.11) and finding explicitly chains of mutations of the A n -quiverthat produce the braid-group transformations.The structure of the paper is as follows:In Sec. 2, we describe quantum algebras of transport matrices in the Fock–Goncharov SL n -quiver (Theorem 2.5); this quantum algebra is based on a more general Lemma 7.17 provenin Sec. 7 for any planar (acyclic) directed network. We also prove the groupoid condition(Theorem 2.6) satisfied by quantum transport matrices in the SL n -quiver.In Sec. 3, we briefly describe general algebraic relations enjoyed by quantum transport ma-trices for SL n character variety on a general triangulated Riemann surface Σ g,s,p with p > SL n -quiver we constructa unipotent A satisfying the quantum reflection equation (Theorem 4.1). We generalize thisconstruction to solutions of quantum reflection equation that are not necessarily unipotent(Theorem 4.2).In Sec. 5, we associate the unipotent A constructed in the preceding section with a special A n -quiver and prove that special sequences of mutations at vertices of this quiver generatebraid-group transformations of elements of A (Theorem 5.4).In Sec. 6, we collect statements about Casimir elements of SL n - and A n -quivers.In Sec. 7 we consider quantum transport matrices for general acyclic planar directed net-works, establish the relation to Postnikov’s quantum Grassmannians and measurement maps,and prove the general R -matrix relation for the corresponding quantum transport matrices(Lemma 7.17).In Sec. 8, we generalize the results of Sec. 7 to arbitrary planar directed network (relaxingthe acyclicity condition) showing in Theorem 8.3 that quantum transport elements in any suchnetwork satisfy the same closed algebraic relations as elements of an acyclic planar directednetwork.Section 9 is a brief conclusion.2. sl n -Algebras for the triangle Σ , , Let Σ g,s,p denote a topological genus g surface with s boundary components and p markedpoints. In this section, we concentrate on the case of the disk with 3 marked points on theboundary Σ , , . (To simplify notations, we use Σ = Σ , , .) In this section we review thedefinition of quantized moduli space X SL n , Σ of framed SL n -local systems on the disk with threemarked points ([16]). We call disk with three marked points A, B, C on its boundary trianglewith vertices
A, B, C and use notation △ ABC (see Fig 1).2.1.
Quantum torus.
Let lattice Λ = Z m be equipped with a skew-symmetric integer form h· , ·i . Introduce the q -multiplication operation in the vector space Υ = Span { Z λ } λ ∈ Λ as follows Z λ Z µ = q h λ,µ i Z λ + µ . The algebra Υ is called a quantum torus. Fix a basis { e i } in Λ, we considerΥ as a non-commutative algebra of Laurent polynomials in variables Z i := Z e i , i ∈ [1 , N ]. Forany sequence a = ( a , . . . , a t ), a i ∈ [1 , m ], let Π s denote the monomial Π s = Z a Z a . . . Z a t . Let λ s = P tj =1 e a j . Element Z λ s is called in physical literature the Weyl form of Π s and we denoteit by two-sided colons •• Π s •• It is easy to see that •• Π s •• = Z λ s = q − P j Triangle △ 123 with vertices 1 , , Moduli space X SL n , Σ and transport matrices. Following [16], recall that a modulispace X SL n , Σ parametrizes framed SL n -local systems on Σ that is isomorphic to the triple offlags in C n in general position. Any framed SL n -local system in the triangle △ 123 determinestransport matrices.Two transport matrices M and M correspond respectively to directed paths going from oneside of Σ to the other side as on Fig 1. If M is associated to a directed path then the inversematrix M − corresponds to the same path in the opposite direction.As mentioned above, X SL n , Σ parametrizes the configurations of triples of complete flags ingeneral position ( F ) • , ( F ) • , ( F ) • in C n . Recall that a complete flag F • is a collection ofconsecutively embedded subspaces { F ⊂ F ⊂ · · · ⊂ F k ⊂ · · · ⊂ F n − ⊂ F n = C n } where F k is a linear subspace of dimension k . Denote by F a = F n − a , a = 0 , , . . . , n , the vectorsubspace of codimension a .Consider the subtriangulation of △ 123 into (cid:0) n (cid:1) white upright triangles and (cid:0) n − (cid:1) upside-down black triangles (see Fig. 5). Label all white upright triangles by triples { ( a, b, c ) | a, b, c ≥ a + b + c = n − } . Each white triangle ( a, b, c ) corresponds to a line ℓ abc = ( F ) a ∩ ( F ) b ∩ ( F ) c .Similarly, label black upside-down triangles by triples { ( a, b, c ) | a, b, c ≥ a + b + c = n − } .Each upside-down triangle ( a, b, c ) is associated with the plane P abc = ( F ) a ∩ ( F ) b ∩ ( F ) c .Note that every plane P abc of a black triangle contains all three lines ℓ ( a +1) bc , ℓ a ( b +1) c , ℓ ab ( c +1) ofwhite triangles which are neighbors of the black one. For every such plane P abc choose threevectors v ( a +1) bc ∈ ℓ ( a +1) bc , v a ( b +1) c ∈ ℓ a ( b +1) c , v ab ( c +1) ∈ ℓ ab ( c +1) such that they satisfy condition v ( a +1) bc + v a ( b +1) c = v ab ( c +1) . Hence, given a configuration of lines corresponding to triple of flags(( F ) • , ( F ) • , ( F ) • ), the choice of one vector v abc ∈ ℓ abc determines uniquely all other vectorsin the lines ℓ a ′ b ′ c ′ for all ( a ′ b ′ c ′ ) (see Fig. 2).Thus, the configuration of lines ℓ abc determines projective collection of vectors { v abc } moduloscalar scaling. Note that exactly two vectors at vertices of any gray triangle are independent.Define a snake as an oriented path running from the top black triangle containing the line ℓ n downwards to the bottom black triangle containing ℓ ab (for example, bold path in Fig 2).Any snake defines a projective basis v α , . . . , v α n of C n . Note that choosing another cornerof triangle as a top one leads to different choice of projective basis. In particular, if the basisdefined by the only snake running from ℓ n to ℓ n is v n , v n − . . . , v n then the basis definedby the only snake in the opposite direction from ℓ n to ℓ n is v n , − v n − . . . , ( − n − v n .Denote by b p the basis defined by snake p . Let b be the basis defined by the unique snakefrom ℓ n to ℓ n in the triangle △ 123 and by b the basis defined by the snake ℓ n to ℓ n .The basis b in Fig 3 is b = ( v , v , v ), the basis b = ( v , v , v ).Define T ∈ SL n as the transformation matrix from basis b to b , namely, i -th columnof T is [( b ) i ] b , i.e. coordinate vector ( b ) i with respect to basis b precomposed andpostcomposed with multiplications by diagonal matrices defined by the Fock-Goncharov coor-dinates on the sides 2 − − b and b are both definedup to multiplicative scalars T is defined up to scalar too. Hence, condition T ∈ SL n fixes T uniquely. LEONID O. CHEKHOV ∗ AND MICHAEL SHAPIRO ∗∗ ℓ ℓ ℓ ℓ ℓ ℓ Figure 2. Configuration of lines corresponding to triple of flags in C . Black trianglesare equipped with planes P abc . Plane P contains lines ℓ , ℓ , ℓ , P containslines ℓ , ℓ , ℓ , P contains lines ℓ , ℓ , ℓ . Vectors v abc ∈ ℓ abc satisfyrelations v = v + v , v = v + v , v = v + v . The bold brokenline indicates a snake. ℓ ℓ ℓ ℓ ℓ ℓ Figure 3. Configuration of lines corresponding to triple of flags in C . Black trianglesare equipped with planes P abc . Plane P contains lines ℓ , ℓ , ℓ , P containslines ℓ , ℓ , ℓ , P contains lines ℓ , ℓ , ℓ . Vectors v abc ∈ ℓ abc satisfyrelations v = v + v , v = v + v , v = v + v . Similarly, we define the transport matrix T as transformation matrix from the side 1 − − T as transformation from 2 − − M = T , M = T − (see Fig. 5).Matrix M is an upper-anti-diagonal matrix and M is a lower-anti-diagonal matrix (seeExample 2.2).Fock-Goncharov coordinates Z α parametrize X SL n , Σ . They are associated with vertices oftriangular subdivision of Σ except vertices 1 , , i, j, k ) , i + j + k = n denoted often below by Greek letters (see Fig. 5 for n = 3 andFig. 6 for n = 6). The expressions for classical transport matrices were first found in [18], (seealso [13] Appendix A.2). ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 9 Initial diagonal prefactor can be vizualized as carrying the snake from outside to inside overthe side 1 − − T corresponding to the base change from b to b is factorizable in a product of elementary basis changes corresponoding to the followingsequence of snake transformations. ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ ℓ Figure 4. Sequence of snakes factorizing transport matrix T Z Z Z Z Z Z Z T T T Figure 5. Fock-Goncharov parameters for X SL , Σ , , . Arrows shows the direction oftransport matrices. M = T , M = T − . Let χ ( k ) denote the integer step function, χ ( k ) = { , k < , , k ≥ } . Define n × n matrices H k ( t ), L k , S as follows H k ( t ) = t − n − kn diag( t χ (1 − k − , t χ (2 − k − , . . . , t χ ( n − k − ) , L k = Id n + E k for k ∈ [1 , n − 1] where Id n is the identity n × n matrix, ( E k ) i,j = δ k +1 ,i · δ k,j is the matrix whoseonly nonzero element is 1 at the position ( k + 1 , k ), ( S ) ij = ( − n − i δ i,n +1 − j . Then T = S h n − Y j =1 H n − j ( Z n − j, ,j ) i · L n − n − Y p =1 h p Y q =1 L n − q − H n − q ( Z n − − p,n − q,p + q +1 − n ) i L n − · h n − Y j =1 H j ( Z ,j,n − j ) i . (2.1)Let I = { ( a, b, c ) | a, b, c ∈ Z + , a + b + c = n } be the set of barycentric indices in the trianglewith side n , τ : I → I be the clockwise rotation by 2 π/ τ acts naturally on the sequencesof barycentric parameters and hence on sequences of Fock-Goncharov parameters: for Z =( Z α , . . . , Z α k ) the sequence τ Z = ( Z τ ( α ) , . . . , Z τ ( α k ) ), if O ( Z ) is an object depending on thecollection Z = ( Z α i ) ki =1 of Fock-Goncharov parameters then τ O = O ( τ Z ). ∗ AND MICHAEL SHAPIRO ∗∗ Note that T = τ T , T = τ T (see Fig. 5). The transport matrix M = ( τ M ) − . Example 2.1. For n = 3, we have H ( t ) = t − / t / 00 0 t / , H ( t ) = t − / t − / 00 0 t / , L = , L = , S = − .Transport matrices T from side 1 − − T from side 2 − − T from side 3 − − M = T = SH ( Z ) H ( Z ) L L H ( Z ) L H ( Z ) H ( Z ) T = SH ( Z ) H ( Z ) L L H ( Z ) L H ( Z ) H ( Z ) T = SH ( Z ) H ( Z ) L L H ( Z ) L H ( Z ) H ( Z ) .M = Z − / Z / Z − / Z − / Z / Z − / Z / ( Z − / + Z / ) Z / Z / Z / Z / Z / Z / Z / Z − / Z − / Z − / Z − / Z − / Z − / Z − / Z − / Z / Z − / Z − / Z − / Z − / Z − / Z − / ,Finally, M = T − . We can easily factorize M in the product of elementary matrices notingthat S − = ( − n − S , H k ( t ) − = H k ( t − ) = SH n − k ( t )S, L − k = Id n − E k = SL T n − k S , where L T j is the transpose of matrix L j . Then, M = H ( Z ) − H ( Z ) − L − H ( Z ) − L − L − H ( Z ) − H ( Z ) − S − = SH ( Z ) SSH ( Z ) SSL T1 SSH ( Z ) SSL T2 SSL T1 SSH ( Z ) SSH ( Z ) SS ( − n − = ( − n − SH ( Z ) H ( Z ) L T1 H ( Z ) L T2 L T1 H ( Z ) H ( Z ) .M = Z / Z / Z / Z / Z / Z / Z / Z / Z − / Z − / Z / Z / Z / Z − / Z / Z − / Z − / Z − / Z − / Z − / Z − / ( Z − / + Z / ) Z / Z − / Z − / Z − / Z / Z / Z − / Z / .To obtain quantum transport matrices we expand all entries of classical transport matrix M i in the sum of monomials m j ( Z α ) and replace all m j by the corresponding Weyl form •• m j •• .For instance, the (1 , M becomes( M ) = •• Z − / Z / Z − / Z / Z / •• + •• Z − / Z / Z / Z / Z / •• In Section 2.3 we generalize this construction to non-normalized quantum transport matricesdefined for more general class of planar quivers. Example 2.2. A toy example is the one in which all Z α are the units. Matrix entriesthen just count numbers of monomials entering the corresponding matrix elements a i,j ∈ ( − i +1 Z + [[ Z ± α ]]. Then, for the M matrix, we have the following representation:(2.2) M = − − , − − − − , etc , that is, [ M ] i,j = ( − i +1 h n − ij i for SL n . We introduce the antidiagonal unit matrix | S | = δ i,n +1 − i (to distinguish it from S = ( − i +1 δ i,n +1 − i ). ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 11 For M we have(2.3) M = M = − − 11 2 1 , − − 10 1 2 1 − − − − , etcA riddle-thirsty reader can check the following relations between these matrices: M = M = ( − n +1 | S | · M · | S | , M = ( − n +1 [ | S | · M ] = ( − n +1 I (2.4) M T1 M = A = , , etcthat is [ A ] i,j = h nj − i i .2.3. Quantum transport matrices and Fock-Goncharov coordinates. We describe nowhow quantized transport matrices are expressed in terms of quantized Fock-Goncharov param-eters (see also [13] and [17]).In the quantization of X SL n , Σ the quantized Fock-Goncharov variables form a quantum torusΥ with commutation relation described by the quiver shown on Fig. 6. Vertices of the quiverlabel quantum Fock-Goncharov coordinates Z α (we use Greek letters to indicate barycentriclabels) while the arrows encode commutation relations: if there are m arrows from vertex α to β then Z β Z α = q − m Z α Z β . Dashed arrow counts as m = 1 / 2. In particular, a solid arrow from Z α to Z β implies Z β Z α = q − Z α Z β , a dashed arrow from Z α to Z β implies Z β Z α = q − Z α Z β ,and, for the future use, a double arrow from Z α to Z β means Z β Z α = q − Z α Z β . Vertices notconnected by an arrow commute. (5 , , , , , , 0) (1 , , , , 0) (1 , , Figure 6. The quiver of Fock-Goncharov parameters in the triangle Σ , , parametrizing A SL , Σ ; note that we vertices (6 , , , , , , 6) are excluded. Consider the following planar oriented graph in the disk dual to the quiver above. Labelvertices on the left, on the right and on the bottom sides from 1 to n as shown on Figure 7.Now barycentric indices label the vertices of the quiver which correspond to the faces of thedual oriented graph. Vertices of the new dual graph are colored black and white dependingon whether there are two or one incoming arrows. Faces of G are equipped with q -commutingweights Z α . We add also three face weights Z n, , , Z ,n, , and Z , ,n with similar commutationrelations. ∗ AND MICHAEL SHAPIRO ∗∗ Any maximal oriented path in the dual graph connects a vertex on the right side 1–2 of thetriangle either with a vertex of the left side 1–3 or with a vertex on the bottom side 2–3. Weassign to every oriented path P : j i ′ from the right side to the left side or to the bottomside P : j i ′′ the quantum weight w ( P ) = •• Y face α lies to the rightof the path P Z α •• ′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′ ′′ Z , , Z , , Z , , Figure 7. The plabic graph G dual to the quiver of Fock-Goncharov parameters for X SL , Σ , , . Face weights Z , , , Z , , , Z , , are added. Definition 2.3. We define two n × n non-normalized quantum transition matrices( M ) i,j = X directed path P : j i ′ from right to left w ( P ) and ( M ) i,j = X directed path P : j i ′′ from right to bottom w ( P ) . Note that each M is a lower-triangular matrix and M is an upper-triangular matrix.In section 7 we generalize this definition. Let Γ be a planar oriented graph in the rectanglewith no sources or sinks inside (see Fig 31), m univalent boundary sinks on the left labeled 1 to m top to bottom and n univalent boundary sources on the right labelled 1 to n top to bottom.All arcs of Γ are oriented right to left, in particular, G has no oriented cycles. Note that thiscondition is in particular satisfied by the plabic graph G (see Fig 7) considered as a graph with n sources and 2 n sinks. Indeed, we can redraw G in a rectangle such that the right side of thetriangle becomes the right vertical side of the rectangle while union of the left and the bottomsides becomes the left side of the triangle.Faces of Γ are equipped with q -commuting weights Z α whose commutation relations aregoverned by the plabic graph (see Section 7 for details). We define weight of the maximaloriented path P from a source a to a sink b as(2.5) w ( P ) = •• Y face α lies to the rightof the path P Z α •• . Then, entries of a m × n non-normalized transport matrix are [ M ] ij = X directed path P : j i w ( P ) . Lemma 7.17 implies that the matrix M satisfies the quantum R -matrix relation R m ( q ) M ⊗ ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 13 M = M ⊗ MR n ( q ), where R k ( q ) is a k × k matrix(2.6) R k ( q ) = X ≤ i,j ≤ k e ii ⊗ e jj + ( q − X ≤ i ≤ k e ii ⊗ e ii + ( q − q − ) X ≤ j
2. In Sec. 8 we show thatthis algebra remains valid also in the case of a planar directed network with loops.In Fig. 7 we have an example of a directed network with 6 sources and 12 sinks. Adding faceweights Z , Z , Z supplied with “natural” commutation relations Z Z = qZ Z and Z Z = qZ Z , etc., we obtain graph Γ in the rectangle with n = 6 sources and2 n = 12 sinks; then M has a block matrix form M = (cid:18) M M (cid:19) in which we let M is the upper n × n block and M is the lower n × n block. We want to show that Lemma 7.17 implies thefollowing commutation relations for M and M :(2.8) R n ( q ) M i ⊗ M i = M i ⊗ M i R n ( q ) , i = 1 , , and(2.9) M ⊗ M = M ⊗ M R n ( q ) . Let now indices i, j run from 1 to n . We rewrite the above matrix R n ( q ) as R n ( q ) = X ≤ i,j ≤ n e ii ⊗ e jj + ( q − X ≤ i ≤ n e ii ⊗ e ii + ( q − q − ) X ≤ j
Transport matrices for the quantum space A SL n , Σ are defined as follows M = QS M D − and M = QS M D − D − where Q = diag { q − j } j =[1 ,n ] .Note that Q ⊗ Q R n ( q ) = R n ( q ) Q ⊗ Q for any diagonal matrix Q and S ⊗ S R n ( q ) = R T n ( q ) S ⊗ S for any antidiagonal matrix S. We have therefore proved the following theorem. Theorem 2.5. The above M and M satisfy the relations R T n ( q ) M i ⊗ M i = M i ⊗ M i R n ( q ) , i = 1 , , M ⊗ M = M ⊗ M R n ( q ) where (2.10) R n ( q ) = q − /n "X i,j e ii ⊗ e jj + ( q − X i e ii ⊗ e ii + ( q − q − ) X i>j e ij ⊗ e ji is the quantum trigonometric R -matrix Theorem 2.6. The quantum transport matrices T i satisfy the quantum groupoid relation T T T = Id . Remark 2.7. Recalling M = T , M = T − , M = T we have M M = M . Proof. The product T T is given by the following double sum over directed paths:(2.11) [( QS ) − T T ] ij = n X k =1 ( − k q − k X paths k → i : Y Z α : X paths j → k : Y Z β : . We now consider the pattern in the figure below. We do not indicate arrows on edges recallingthat all paths in T go from right to left and from top to bottom whereas all paths in T gofrom left to right and from top to bottom. Two paths: j → k from T and k → i from T share the common horizontal leg; if we remove this leg then the remaining part of the union of ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 15 j → k and k → i is a path that first goes from right to left, then (in a general Case I) has theleftmost vertical edge, then goes from left to right. In a very special Case II, the path does nothave the last part; this happens only for k = 1 and only if the last horizontal part of the path j → k = 1 is strictly longer than the shared horizontal leg AB C jk + 1 k i Case IIn Case I, given a path j → k encompassing the regions A and B and a path k → i encompassingthe region C we have the corresponding path j → k +1 encompassing the regions A and the path k +1 → i encompassing the regions B and C . These pairs of paths are in bijection being the onlytwo possible combinations of paths having the same union of domains A ∪ B ∪ C . Contributionsfrom these two pairs of paths have opposite signs and since •• CB •• •• A •• = q •• C •• •• BA •• they aremutually canceled in the sum (2.11).The only pairs of paths ( j → k , k → i ) that do not have counterparts are those for whichthe region C is absent (Case II): AB j i Case IIIn this case, •• B •• •• A •• = •• BA •• , k is necessarily equal 1, and after removing the common leg, allthese pairs of paths are in bijection with single paths going from right to left and encompassingthe regions A and B ; note that these paths are exactly paths constituting the matrix M ! Sothe sum in (2.11) just gives the matrix M (up to the factor QS , which we can now reconstruct).We have therefore proved that T T = M (cid:3) Note that we shall present below the second proof of the groupoid property using quantumGrassmannian. Remark 2.8. The semiclassical limit of Theorem 2.5 statement reads { M ⊗ , M } = M ⊗ M (cid:16) − n I ⊗ I + r n (cid:17) ∗ AND MICHAEL SHAPIRO ∗∗ where(2.12) r n = X i e ii ⊗ e ii + 2 X i>j e ij ⊗ e ji is the semiclassical r -matrix. Equivalently, { [ M ] ab , [ M ] cd } = − n [ M ] ab [ M ] cd + [ M ] ad [ M ] cb θ ( b − d ) , θ ( x ) = , if x > , , if x = 0 , , if x < . Remark 2.9. Note that for the trigonometric R -matrix (2.10) the quantum relation R Tn ( q ) M i ⊗ M i = M i ⊗ M i R n ( q )has an equivalent form of writing M i ⊗ M i R Tn ( q ) = R n ( q ) M i ⊗ M i for i = 1 , M and M and have the same semiclassical limit { [ M i ] ab , [ M i ] cd } = [ M i ] ad [ M i ] cb ( θ ( b − d ) − θ ( a − c )) i = 1 , , θ ( x ) = , if x > , , if x = 0 , , if x < . Example 2.10. For n = 3 direct computations show M ⊗ M = M ⊗ M R ( q ) , where R ( q ) = q − / q q − q − q q − q − q − q − q Similarly, for both i = 1 , 2, we have R T ( q ) M i ⊗ M i = M i ⊗ M i R ( q )Again, by direct computation one can check T T T = 1.3. Goldman brackets and commutation relations between transport matrices To obtain a full-dimensional (without zero entries) form of transport matrices we definetransport matrices along more general paths.Namely, let G = ( G, M ) be a disk G with four marked boundary points M = { A, B, D, C } in clockwise order, (∆ ABC, ∆ BCD ) be a triangulation of G and we also assume clockwiseorientation of all triangle sides inside every triangle. The space A SL n , G coincides with thespace of quadruple of complete flags; one flag assigned to each marked point. A snake inside atriangle determines a projective basis as explained above. Each oriented side of triangulationdetermines such a snake. It gives a projective basis in C n . The orientation of BC determines ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 17 the triangle associated with this side whose orientation is compatible with the orientation ofthe side. For instance, side B → C determines the unique snake from B to C inside triangle∆ ABC the side C → B is associated with the snake from C to B inside triangle ∆ BCD withcorresponding projective basis chosen in each case. Transition matrix from basis associated to BC to the one associated with CB is the product T BC ← CB = S ˙ H where H is the diagonal matrixdefined by the Fock-Goncharov parameters on the side BC . This allows to define transportmatrix for any pair of oriented sides as transition matrix for the pair of corresponding bases;the matrix S acts by changing the orientation of the corresponding side. Let T BC ← AB be atransport in ∆ ABC from side AB to BC . Pay attention that the sides are oriented and theorder of endpoints in side notation matters. Similarly, T CB ← DC is a transport matrix in ∆ BCD from DC to CB . Note that T DC ← CB = T − CB ← DC . Then, we define a transport T DC ← AB from AB to DC as T DC ← AB = T DC ← CB T CB ← BC T BC ← AB = T − CB ← DC T CB ← BC T BC ← AB . Similarly, T BD ← AB = T BD ← CB T CB ← BC T BC ← AB . A BC DT BC ← BA T BD ← CB T − CB ← DC Figure 8. Disk with four marked points.To describe the quantum case, we split each quantum Fock-Goncharov parameter (or quan-tum cluster parameter) on the side BC into a product of two, one inside triangle ∆ ABC theother inside triangle ∆ BCD . The quantum parameters in triangle ∆ ABC commute with thoseof triangle ∆ BCD , so the product of two Weyl-ordered monomials is itself a Weyl-orderedmonomial, in which we perform an amalgamation of variables on the side BC . Due to thedouble action of the matrix S , the amalgamation of boundary (frozen) variables in neighbortriangles respects the surface orientation, so, we amalgamate pairwise variables on the sides BC of the two triangles ordered in the same direction, from B to C . After the amalgamation,we unfreeze the obtained new variables. Therefore, in a network on a surface obtained as aunion of several triangles, the Weyl ordering of weights of any path that does not go throughany given triangle more than once is the product of Weyl orderings of weights inside each tri-angle. It was explicitly demonstrated in [38] that the weights defined by such Weyl orderingare preserved by quantum mutations which now include mutations of amalgamated variablesas well as mutations of variables in the interior of triangles.We now show that the commutation relations from Theorem 2.5 together with the groupoidcondition (Theorem 2.6) imply the commutativity relations and Goldman brackets.We begin with the pattern in the figure below. ∗ AND MICHAEL SHAPIRO ∗∗ M − l M − k M j M i S ⊗ S We use the identities M − l ⊗ M − k = R n ( q ) M − k ⊗ M − l M j ⊗ M i = M i ⊗ M j R − T n ( q ) R n ( q ) S ⊗ S = S ⊗ SR n ( q ) T , where the last identity holds for any antidiagonal matrix S whose elements commutes with allelements of quantum torus.We then have[ M j S M − l ] ⊗ [ M i S M − k ] = M i ⊗ M j R − T n ( q ) S ⊗ SR n ( q ) M − k ⊗ M − l = M i ⊗ M j S ⊗ SR − n ( q ) R n ( q ) M − k ⊗ M − l = [ M i S M − k ] ⊗ [ M j S M − l ] , so two transport matrices corresponding to nonintersecting paths commute. This is consistentwith the quantum mapping class group transformations: flipping BC edge separates the paths AB → BD and AC → CD into two adjacent triangles.Consider now the case of two intersecting paths (we consider a single intersection inside aquadrangle). M − l M − k M j M i S ⊗ S We then have q /n [ M j S M − k ] ⊗ [ M i S M − l ] − q − /n [ M i S M − l ] ⊗ [ M j S M − k ]= M i ⊗ M j S ⊗ S [ q /n R − ( q ) − q − /n R T12 ( q )] M − k ⊗ M − l =( q − − q ) M i ⊗ M j S ⊗ SP 12 1 M − l ⊗ M − k = ( q − − q ) M i S M − l ⊗ M j S M − k P n . So we have a quantum Goldman relation q /n − q − /n = ( q − − q ) P n ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 19 with P n the permutation matrix. In the semiclassical limit with q = e ~ / , the term linear in ~ gives rise to the Goldman bracket for SL n [24].Let a polygon be triangulated into collection of triangles containing triangle ∆ ABC , let EF be another side of triangulation different from sides of ∆ ABC , let γ be a path connecting EF to AB crossing any side of triangulation at most once and crossing neither AC nor BC (seeFigure 9). Denote by T γ = T BA ← EF the composition of transport matrices along the path γ ,define transport matrices M = T − AB ← CA ST γ , M = T BC ← AB ST γ . C ABT − AB ← CA T BC ← AB T γ EF Figure 9. M = T CA ← AB ST BA ← EF , M = T BC ← AB ST BA ← EF . Theorem 3.1. The transport matrices M and M in Fig. 9 satisfy the commutation relations M ⊗ M = M ⊗ M R n ( q ) ,R Tn ( q ) M i ⊗ M i = M i ⊗ M i R n ( q ) for i = 1 , .and M − M BC ← CA M = 1 . Proof . Consider a transport matrix corresponding to a path not passing twice through thesame triangle. It is given by the matrix product M ( m − i m − S · · · SM (2) i SM (1) i = T BA ← EF where i k = 1 , M ( k ) i k and M ( p ) i p commute for distinct k and p . Every such productsatisfies the relation R T n ( q ) T ⊗ T = T ⊗ T R n ( q ). Then, M ( m )1 1 S T ⊗ M ( m )2 2 S T = M ( m )1 ⊗ M ( m )2 1 S ⊗ S T ⊗ T = M ( m )2 ⊗ M ( m )1 R n ( q ) S ⊗ S T ⊗ T = M ( m )2 2 S ⊗ M ( m )1 1 SR T n ( q ) T ⊗ T = M ( m )2 2 S T ⊗ M ( m )1 1 S T R n ( q ) . (cid:3) General algebras of transport matrices in an ideal triangle decomposition of Σ g,s,p —a genus g Riemann surface with s holes and p > Reflection equation and groupoid of upper triangular matrices Representing an upper-triangular A . We assume again that both M and M aretriangular matrices.The main theorem concerns a special combination of M and M :(4.1) A := M T1 M . Note that the transposition in the quantum case if formal: the quantum ordering is preserved,only matrix elements are permuted. Also, since M and M T1 are upper-anti-diagonal matricesand M is a lower-anti-diagonal matrix, the matrix A is automatically upper-triangular. ∗ AND MICHAEL SHAPIRO ∗∗ Theorem 4.1. The matrix A = M T M satisfies the quantum reflection equation R n ( q ) A R t n ( q ) A = A R t n ( q ) A R n ( q ) with the trigonometric R -matrix (2.10), where R t n ( q ) is a partially transposed (w.r.t. the firstspace) R -matrix. The proof is a short direct calculation that uses only R -matrix relations. Note that trans-posing with respect to the first space the second relation in Theorem 2.5 we obtain M T1 ⊗ M = M R t n ( q ) M T1 and the total transposition of the first relation gives R n ( q ) M T1 ⊗ M T1 = M T1 ⊗ M T1 R T n ( q ) , so R n ( q ) M T1 1 M R t n ( q ) M T1 2 M = R n ( q ) M T1 2 M T1 1 M M = M T1 1 M T1 R T n ( q ) M M = M T1 1 M T1 2 M M R n ( q ) = M T1 2 M R t n ( q ) M T1 1 M R n ( q ) , which completes the proof.We thus conclude that we have a Darboux coordinate representation for operators satisfyingthe reflection equation. Moreover all matrix elements of A are Laurent polynomials with positivecoefficients of Z α and q . In particular, positive integers in equation (2.4) count numbers ofmonomials in the corresponding Laurent polynomials.By construction of quantum transport matrices in Sec. 2.3, all matrix elements of M and M are Weyl-ordered. For [ A ] i,j = j P k = i [ M ] k,i [ M ] k,j we obtain that for i < j , [ M ] k,i commuteswith [ M ] k,j (all paths contributing to [ M ] k,i are disjoint from all paths contributing to [ M ] k,j for i < j ), so the corresponding products are automatically Weyl-ordered,[ A ] i,j = j X k = i •• [ M ] k,i [ M ] k,j •• . For i = j , the corresponding two paths share the common starting edge, and then[ A ] i,i = q − / •• [ M ] i,i [ M ] i,i •• . This explains the appearance of q − / factors on the diagonal of the quantum matrix A ~ (see(1.11), [7]). Note that all Weyl-ordered products of Z α are self-adjoint and we assume that allCasimirs are also self-adjoint. We have that •• [ M ] i,i [ M ] i,i •• = i Y j =1 K j , where K j are special Casimirs (5.1) of the A n -quiver introduced in the next section.To obtain a full-dimensional (not upper-triangular) form of the matrix A gen let us consideradjoint action by any transport matrix: Theorem 4.2. Any matrix A gen := M T γ S T A SM γ , where M γ is a (transport) matrix satisfyingcommutation relations of Theorem 2.5 and such that its elements commute with those of A = M T M , satisfies the quantum reflection equation of Theorem 4.1. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 21 The proof is again a direct computation; note also that if we represent A = M T1 M , then M ′ = M SM γ and M ′ = M SM γ satisfy commutation relations from Theorem 3.1 and A gen := M ′ M ′ then satisfy the quantum reflection equation.5. The quiver for an upper-triangular A and the braid-group action In this section we first construct the quiver corresponding to the Darboux coordinates onthe set A n of upper-triangular matrices and, second, present the braid-group action on A n viachains of mutations in the newly constructed quiver. We would like to notice that the quasi-cluster braid group action on the transport matrices and therefore on the framed moduli space X SL n was constructed in [25]. However, our construction seems to be different.5.1. A n -quiver. Let us have a closer look on the structure of matrix entries of the product[ A ] i,j := (cid:2) M T1 M (cid:3) i,j = P k ( M ) k,i ( M ) k,j . All monomials contributing to ( M ) k,i contain thesame factor Q ki =1 Z i, ,n − i and all monomials contributing to ( M ) k,j contain the same factor Q ki =1 Z n − i,i, , so the dependence of all elements of [ A ] i,j on the frozen variables Z i, ,n − i and Z n − i,i, is via their pairwise products, and we therefore amalgamate these variables pairwisethus obtaining a single new variable¯ Z i := •• Z i, ,n − i Z n − i,i, •• , i = 1 , . . . , n − . This results in a “twisted” pattern shown in Fig. 17.Another outcome of the above amalgamation procedure is that the resulting quiver admits n − K i is a monomialexpression(5.1) K i := •• Z ,i,n − i i − Y j =1 Z j,i − j,n − i ¯ Z i n − i − Y j =1 Z j,i,n − i − j •• , which contains exactly one square of one of the remaining frozen variable Z ,i,n − i , and we haveexactly one such Casimir per every remaining frozen variable Z ,i,n − i . Let us change the systemof coordinates in A n replacing Z ,i,n − i by ( K i ) / . Clearly, it is well defined nondegeneratechange of coordinates. Fixing values of K i ’s leads to a Poisson submanifold U . The remainingelements Z a,b,c , a > U whose Poisson bracket isstill described by a quiver obtained by forgetting frozen variables Z ,i,n − i .Finally, we unfreeze the variables ¯ Z k , and then the connected part of the resulting quiver,which we call the A n -quiver , contains only unfrozen variables. A most symmetric way of vizual-izing this quiver is to “cut out” the half-sized triangle located in the left-lower corner, thenreflect this small triangle through the diagonal passing through its left-lower corner preserv-ing the incidence relations for arrows in the both parts of the quiver, then glue pairwise theamalgamated variables Z k, ,n − k and Z n − k,k, . Amalgamation operations become planar in this ∗ AND MICHAEL SHAPIRO ∗∗ n = 3 n = 4 n = 5 n = 6 Figure 10. A n -quivers for n = 3 , , , A .Note each of [ n/ 2] original Casimirs of the SL n -quiver gives rise to the corresponding Casimirelement of the A n -quiver(5.2) C k = •• ¯ Z k n − k − Y i =1 Z k,i,n − k − i ¯ Z n − k k − Y j =1 Z n − k,j,k − j •• , and in figures representing A n -quivers, cluster variables of sites of the same color contribute(all in power one) to the same Casimir. We present the A n -quivers for n = 3 , , , n/ 2] independent Casimir elements depicted in Fig. 19.Since all Casimirs of A ∈ A n are generated by λ -power expansion terms for det( A + λ A T ),we automatically obtain the following lemma Lemma 5.1. det( A + λ A T ) = P ( C , . . . , C [ n/ ) , where C i are Casimirs of the A n -quiver. Remark 5.2. In the cases n = 3 and n = 4, the constructed quivers are those of geometricsystems: these cases admit three-valent fat-graph representations in which Y-cluster variablesare identified with (exponentiated) Thurston shear coordinates z α enumerated by edges of thecorresponding graphs, and nontrivial commutation relations are between variables on adjacentedges; for n = 3 and n = 4 these graphs are the relative spines of Riemann surfaces Σ , and Σ , The corresponding polynomials were found in [12]. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 23 Γ , a , Γ , a , a , Figure 11. Fat graphs Γ , and Γ , corresponding to the respective quiversfor A and A . We indicate closed paths that produce elements a , ∈ A ∈ A and a , and a , of A ∈ A . For example, setting all Z α = 1 for simplicityand using formulas from [5] for the corresponding geodesic functions we obtain a , = tr( LR ) = 3 in A and a , = tr( LLR ) = 4 and a , = tr( LLRR ) = 6 in A , where L = (cid:2) (cid:3) and R = (cid:2) (cid:3) are matrices of the respective left and rightturns of paths occuring at three-valent vertices of a spine. It is easy to see thatthe above a i,j coincide with those in Example 2.2.depicted in Fig. 11; the Laurent polynomials for entries of A coincide up to a linear change oflog-canonical variables with the expressions obtained by identifying these entries with geodesicfunctions corresponding to closed paths on these graphs; for more details and for the explicitconstruction of geodesic functions, see [5], [7]. Note here that, likewise all a i,j constructed inthis paper, all geodesic functions for all surfaces Σ g,s are positive Laurent polynomials of e z α / .5.2. Braid-group action through mutations. Our second major goal in this paper is tofind a representation of the braid-group action from Sec. 1.3 in terms of cluster mutationsof the A n -quiver. It is well-known that for A belonging to a specific symplectic leaf in A n its matrix elements a i,j are identified with the geodesics functions. In this leaf, braid-grouptransformations correspond to Dehn twists along geodesics corresponding to geodesic functions a i,i +1 on Σ g,s ( n = 2 g + s and s = 1 , g,s is a sequence of cluster mutations since it can be presented as a chain of mutations ofshear variables on edges of the corresponding spine Γ g,s . Whereas, for n = 3 and n = 4, genericsymplectic leaves in A n are geometric and the corresponding mutation sequences are identical,for larger n the generic symplectic leaves become essentially different and we have to reinventa braid group action. Knowing the answer for n = 3 and 4 helps in guessing the answer for ageneral n . We begin with the example of A -quiver:1 24 3The following chain of mutations β , = µ µ µ µ µ = S , preserves the form of the originalquiver with the interchanged vertices 3 and 4, where µ i is a mutation at vertex i . ∗ AND MICHAEL SHAPIRO ∗∗ β , β , β , β , β , β , β , β , β , Figure 12. The braid-group action represented on the union of two triangles,each of which is the copy of the A n -quiver, for n = 5 and 6. The dotted lineindicates the line of the triangle gluing. Dashed blocks enclose cluster variablessequences of mutations at which produce elementary braid-group transformation β i,i +1 : every such sequence commences with mutating the lowest element insidea box (corresponding to a six-valent vertex), then its upper-right neighbor andso on until we reach the upper element, mutate it, and repeat mutations at allinner elements in the reverse order. So, every such braid-group transformation isproduced by a sequence of 2 n − A n -quiver.A convenient way to represent a set of elementary braid-group transformations for a general A n quiver is the process schematically depicted in Fig. 12 below: we take another copy of thetriangle representing the quiver, reflect it and glue the resulting triangle to the original onealong the bottom side of the latter in a way that amalgamated variables on the sides of twotriangles match and the colored vertices representing Casimir elements are stretched along SEdiagonals. The sequences of mutations corresponding to elementary generating elements β i,i +1 of the braid groups are indicated in the figure.Before presenting the result of the braid-group transformation, we describe contributions ofcluster variables located at sites of the A n -quiver to a normalized element a i,j ∈ A with i < j .Before normalization this element is homogeneous in frozen cluster variables ρ k ≡ Z ,k,n − k and is proportional to the product Q ik =0 ρ k Q jl = i +1 ρ l . We normalize this element by dividingit by the product Q ik =0 K k Q jl = i +1 K / l of Casimirs (5.1) thus eliminating the dependence of A on ρ i . This normalization changes the powers in which cluster variables enter sums overpaths. We have eight domains in total in the leftmost part of Fig. 13: let us describe twoof them. In the domain labeled “ a ,” nonnormalized variables enter with power 1 and each ofthem enters exactly one Casimir K k with k ≤ i and one Casimir K l with i + 1 ≤ l ≤ j , sothe normalization decreases the power by 3 / − / 2. In the uppermost domain labeled “b,” every elemententers with power two into a nonnormalized sum over paths and it enters two Casimirs K k with k ≤ i , so the normalization add power − ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 25 i jba ∼ Z ∼ Z − ∼ Z − / i j ∼ Z / ∼ Z − / i j Figure 13. Paths contributing to a i,j in the glued A n -quivers. In the left pic-ture we indicate contributions of cluster variables of SL n quiver into a normalizedelement a i,j : we schematically draw two paths: the upper path corresponds to[ M ] k,i and the lower path corresponds to [ M ] k,j . All cluster variables abovethe first path enter in power two and all variables between two path enter withpower one into a nonnormalized expression; we then normalize it by the prod-ucts of cluster variables entering the product of Casimirs Q ik =0 K k Q jl = i +1 K / l .The resulting pattern is presented in the middle picture: cluster variables enterwith powers 1 / − / A n -quiver and attach it as inFig. 12; two domains in the original triangle then constitute a parallelogram witha continuous path joining its opposite vertices.different hatchings, which overlap in the figure on the left; the resulting powers − / 2, 0, and1 / SL n triangle in the rightmost part of Fig. 13: the union oftwo domains containing cluster variables contributing into the normalized element a i,j is then aparallelogram with sides of positive lengths j − i and n + i − j , and in order to obtain the element a i,j we have to take a sum over all paths inside this parallelogram starting at the vertex of thedual lattice located “beyond” NE vertex j and terminating at the vertex of the dual latticelocated “beyond” SW vertex i (a standard exercise in combinatorics is that we have exactly h nj − i i such paths, cf. toy Example 2.2).We present a more detailed picture below. In this picture, we indicate a part of the directednetwork inside which we take a sum over paths from j to i contributing to a i,j (an example ofsuch path is shown in light color in the Figure); all contributing cluster variables are confinedinside the corresponding parallelogram. All cluster variables inside the parallelogram and abovea path enter with the power 1 / ∗ AND MICHAEL SHAPIRO ∗∗ the path enter with the power − / 2. All variables outside the parallelogram do not contribute. i j We now explore how cluster variables transform under chains of mutations β i,i +1 . Recallthat mutation µ Z transforms any variable Y at the head of an outgoing arrow Z → Y as Y Y (1 + Z ), whereas a variable X joined to Z by an incoming arrow, Z ← X transforms as X → X (1 + Z − ) − . Finally, µ Z ( Z ) = Z − and quiver mutation is standard ([20]).Till the end of this section we let r = n − β = µ B . . . µ B r µ S µ B r . . . µ B µ B (see Fig. 14). The net result of this chainof mutations is shown on the right hand side of Fig. 14. Note that the resulting quiver isisomorphic to the original one when all the mutated variables except S and S retain theirpositions, while the boundary variables S and S are permuted.The following lemma is proved by a direct calculation. Lemma 5.3. In the notation of Fig. 14, cluster variables transform as follows (recall that forbrevity we set r := n − ): B ′ k = B k η k +2 η k , k = 1 , . . . , r ; A ′ k = A k η k +1 η k +2 , k = 0 , . . . , r − C ′ k = C k η k +1 η k +2 , k = 1 , . . . , r ; A ′ r = C ′ = A r η n +1 η η ; S ′ = S S B · · · B r η r +1 η ; S ′ = η S B · · · B r , where η r +2 = 1 , η r +1 = 1 + S , η r = 1 + S + S B r ,η r − = 1 + S + S B r + S B r B r − , . . . , η = 1 + S + · · · + S B r · · · B . (5.3)Note first that Casimirs (5.2) of the A n -quiver are invariant under the transformation inLemma 5.3. This immediately follows from the equalities A ′ k B ′ k C ′ k − = A k B k C k − , k = 2 , . . . , r − ,A ′ B ′ A ′ r B ′ r C ′ r = A B A r B r C r , and(5.4) S ′ S ′ A ′ C ′ r = S S A C r . We now formulate the main statement Theorem 5.4. The cluster transformations in Lemma 5.3 generate the braid-group transfor-mations for entries a i,j of the (classical) matrix A . ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 27 A A A A r − A r S B B B B r − B r S S S C = A r C C C r − C r A ′ A ′ A ′ A ′ r − A ′ r S ′ B ′ B ′ B ′ B ′ r − B ′ r S ′ S ′ S ′ C ′ = A ′ r C ′ C ′ C ′ r − C ′ r Figure 14. The transformation of variables under a braid group transformation β i,i +1 . We indicate only variables that are transformed under the correspondingchain of mutations. Note that the cluster variables C and A r (and therefore C ′ and A ′ r ) are identified. Proof . The pivotal calculation is the transformation of quantities η k defined in (5.3) un-der transformations in Lemma 5.3. Let us denote by G the non-normalized transport matrixcorresponding to matrix A . First, let us compute the (non-normalized) element G i,i +1 of G :(5.5) G i,i +1 := η + S B r · · · B S = 1 + S + S B r + · · · + S B r · · · B + S B r · · · B S . Then, after mutation sequence β i,i +1 , we obtain η ′ k = 1 + S ′ + S ′ B ′ r + S ′ B ′ r B ′ r − + · · · + S ′ B ′ r B ′ r − · · · B ′ k = 1 + S S B r · · · B η r +1 η (cid:16) B r η r + B r B r − η r +1 η r η r − + · · · + B r B r − · · · B k η r +1 η k +1 η k (cid:17) (note that 1 + B r /η r = (1 + S )(1 + B r ) /η r = η r +1 (1 + B r ) /η r )= 1 + S S B r · · · B η (cid:16) B r η r + B r B r − η r η r − + · · · + B r B r − · · · B k η k +1 η k (cid:17)(cid:18) note that 1 + B r η r + B r B r − η r η r − = 1 η r η r − (cid:0) (1 + B r )(1 + S + S B r + S B r B r − ) + B r B r − (cid:1) = 1 η r η r − (1 + B r + B r B r − ) η r = 1 + B r + B r B r − η r − (cid:19) = · · · = 1 + S S B r · · · B η B r + B r B r − + · · · + B r B r − · · · B k η k = 1 + S S B r · · · B ( η k − η η k = G i,i +1 η − S S B r · · · B η η k . We therefore obtain that(5.6) η ′ k = G i,i +1 η − S S B r · · · B η η k , k = 1 , . . . , r + 2 , ∗ AND MICHAEL SHAPIRO ∗∗ and(5.7) G ′ i,i +1 = η ′ r + S ′ B ′ r B ′ r − · · · B ′ S ′ = G i,i +1 η . We consider several cases of matrix entries a ij ; the rest we leave for the reader. Note, first, therelations for triples of the cluster variables:(5.8) A ′ k − B ′ k C ′ k = A k − B k C k , k = 1 , . . . , r, and S ′ S ′ A ′ r = S S A r . In particular, these relations imply that the total product of all cluster variables is conserved.For all elements a i,j we take into account their normalization by taking the sum over pathsweighted by products of cluster variables (in power one) inside the corresponding parallelogramand above the path and dividing this sum by the product of all cluster variables inside theparallelogram taken with power 1 / a i,i +1 = (cid:0) S B r · · · B S (cid:1) − / G i,i +1 . Since S ′ B ′ r · · · B ′ S ′ = S B r · · · B S η − we have that a ′ i,i +1 = (cid:0) S ′ B ′ r · · · B ′ S ′ (cid:1) − / G ′ i,i +1 = a i,i +1 , so, as expected, this element is preserved by the braid-group transformation β i,i +1 .We next consider an arbitrary element a l,m with l, m = i, i +1. Note first that the normalizingfactor for any such element is a product of triples of cluster variables (5.8) taken either inpowers 1 / a l,m . Consider a contribution of cluster variables to paths that enterthe pattern in Fig. 14 from the right between elements C p − and C p and exit from the leftbetween elements A k − and A k − (with k ≤ p ). This path may cross the “ B -line” anywherebetween B k − and B p and we have to take a sum over all possible variants. The correspondingcontribution therefore has the formΠ ′ k,p = C ′ p C ′ p +1 · · · C ′ m × (cid:2) η ′ k − η ′ p +1 (cid:3) × S ′ A ′ m A ′ m − · · · A ′ k − = η p +1 C p C p +1 · · · C m × S S B r · · · B η η k η p +1 ( η k − η p +1 ) × η η k S B r · · · B A m · · · A k − = C p C p +1 · · · C m ( η k − η p +1 ) S A m · · · A k − = Π k,p , so all these elements are preserved, as well as all normalizing factors, and a ′ l,m = a l,m .Consider now a ′ i,i +2 = (cid:0) A ′ r · · · A ′ S ′ B ′ r · · · B ′ (cid:1) − / (cid:2) η ′ r +2 + A ′ r η ′ r +1 + A ′ r A ′ r − η ′ r + · · · + A ′ r A ′ r − · · · A ′ η ′ (cid:3) = (cid:0) S A r · · · A S B r · · · B η − (cid:1) − / h A r η r +1 η η (cid:16) G i,i +1 η − S S B r · · · B η η r +1 (cid:17) + . . . + ( A r · · · A k ) η k +1 η η (cid:16) G i,i +1 η − S S B r · · · B η η k +1 (cid:17) + · · · + ( A r · · · A ) η η η (cid:16) G i,i +1 η − S S B r · · · B η η (cid:17)i = h A r η r +1 + · · · + A r A r − · · · A η ( A r · · · A S B r · · · B ) / · G i,i +1 ( S B r · · · B S ) / − S + S A r + · · · + S A r · · · A ( S A r · · · A ) / i = a i,i +2 a i,i +1 − a i +1 ,i +2 . Next, a ′ i − ,i +1 = (cid:0) B ′ r · · · B ′ S ′ C ′ r · · · C ′ (cid:1) − / (cid:0) · ( G ′ i,i +1 − η ′ r +2 ) + C ′ r ( G ′ i,i +1 − η ′ r +1 )+ C ′ r C ′ r − ( G ′ i,i +1 − η ′ r ) + · · · + C ′ r · · · C ′ ( G ′ i,i +1 − η ′ ) (cid:1) ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 29 = 1( C r · · · C S ) / (cid:0) C r + C r C r − + · · · + C r C r − · · · C + C r C r − · · · C C η r +1 (cid:1) = 1( C r · · · C S ) / (cid:0) C r + C r C r − + · · · + C r C r − · · · C + C r C r − · · · C C (1 + S ) (cid:1) = a i − ,i . Proving the rest of relations we leave to the reader. (cid:3) Casimirs In this section, we derive complete sets of Casimirs for all relevant (sub)varieties of clustervariables related to regular quivers associated to SL n systems. All proofs are direct calculations:it is easy to check that indicated products of cluster variables are Casimirs, whereas theircompleteness follows from the known answers for dimensions of symplectic leaves.6.1. The full-rank SL n -quiver.Lemma 6.1. The complete set of Casimir operators for the full-rank SL n -quiver are (cid:2) n (cid:3) prod-ucts of cluster variables depicted in the figure below for the example of SL : numbers at verticesindicate the power with which the corresponding variable comes into the product; all nonnum-bered variables have power zero. All Casimirs correspond to closed broken-line paths in the SL n -quiver with reflections at the boundaries (the “frozen” variables at boundaries enter theproduct with powers two, powers of non-frozen variables can be 0,1,2, and 3, and they counthow many times the path goes through the corresponding variable. The total Poisson dimensionof the full-rank quiver is therefore ( n +2)( n +1)2 − − (cid:2) n (cid:3) . Figure 15. Three central elements of the full-rank quiver for SL . Remark 6.2. All Casimir operators from Lemma 6.1 remain Casimirs for the full-rank GL n -quiver obtained by adding three more cluster variables at the corners of the triangle (thevariables Z , , , Z , , , and Z , , in Fig. 20). If we include these three corner variables into thequiver, we have to add one more Casimir operator which is the product of all frozen (non-corner)variables along all three boundaries of the SL n -quiver taken in power one and the product ofthree corner variables taken in power three.For completeness, we also present Casimirs for a reduced quiver in which we eliminate one ofthe three sets of frozen variables. The remaining n ( n + 1) / − M (for M we have to remove another set of frozen variables).In this case, every Casimir of the full-rank quiver has its counterpart in the reduced quiverexcept the element that is represented by a triangle-shaped path in the full-rank quiver (suchan element exists only for even n ), which has no counterpart. ∗ AND MICHAEL SHAPIRO ∗∗ - - - - - - - - - - - - - - - - - - - - Figure 16. Two central elements of the reduced quiver for SL . Every element ofthe complete quiver in Fig. 15 has its counterpart in the reduced quiver except thethird element. Lemma 6.3. The complete set of Casimir operators for the reduced SL n -quiver are (cid:2) n − (cid:3) prod-ucts of cluster variables depicted in Fig. 16 for the example of SL : numbers at vertices indicatethe power with which the corresponding variable comes into the product; all nonnumbered vari-ables have power zero. All Casimirs correspond, as in Fig. 15, to closed broken-line paths inthe corresponding full-rank quiver with reflections at the boundaries (the “frozen” variables atboundaries enter the product with powers two), but now the path is split into two parts sepa-rated by two reflections at the side of the triangle that corresponds to the erased frozen variables;these two parts enter with opposite signs; the corresponding Casimir therefore contains clustervariables in both positive and negative powers. As in the case of full-rank quiver, these powerscount (with signs) how many times the path goes through the corresponding variable). The totalPoisson dimension of the reduced SL n -quiver is therefore n ( n +1)2 − − (cid:2) n − (cid:3) . In our construction below, an important role is played by the additional Casimir that appearsif we add the variable (0 , , 6) at the summit of the triangle corresponding to a reduced quiver.In this case, besides the Casimirs in Lemma 6.3, we have one more central element D describedin the following statement. Lemma 6.4. The complete set of Casimir operators for the reduced SL n -quiver with the(“frozen”) cluster variable (0 , , n ) added comprises all Casimirs described in Lemma 6.3 plusthe element D given by the following formula. Let us enumerate the plabic weights Z i,j,k as inFig. 20 by three nonnegative integers ( i, j, k ) with i + j + k = n . Then the element (6.9) D = •• n Y k =1 h Y i + j = N − k (cid:2) Z i,j,k (cid:3) k/n i •• is central for the subset of Z i,j,k with k > . Moreover, the only elements that have nonzerohomogeneous commutation relations with D are Z n, , and Z ,n, . Casimirs for the upper-triangular matrices. Entries of the matrix A := M T1 M depend on all variables of the SL n -quiver, but due to the transposition, two sets of the frozenvariables become amalgamated, that is, only their products appear in the entries of the matrix A . We explicitly show this amalgamation in Fig. 17. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 31 Figure 17. The amalgamation of the quiver corresponding to the triangle Σ , , (Theexample in the figure corresponds to SL ). Figure 18. Five new central elements of the main quiver for SL due to amalgama-tion. (We use these central elements to set all diagonal elements of the upper-triangularmatrix A = M T1 M to be the unities.) Figure 19. Three remaining central elements of the full-rank quiver for SL afteramalgamation and setting the diagonal elements of A equal to unities. ∗ AND MICHAEL SHAPIRO ∗∗ It is easy to see that all Casimirs from Lemma 6.1 remain Casimirs in the amalgamatedquiver (just four, or two, depending on the Casimir element, frozen variables become pairwiseamalgamated). More, this amalgamation results in the appearance of n − A on remaining n − A = M T1 M are particular products of these Casimirs, andwe adjust the values of these Casimir operators to make all diagonal elements of A equal tothe unities in the classsical case and q − / in the quantum case (recall that all Casimirs areassumed to be self-adjoint operators). Lemma 6.5. The complete set of central elements for the amalgamated quiver in Fig. 17comprises n − new Casimirs depicted in Fig. 18 for the case of SL and (cid:2) n (cid:3) central elements(products of old Casimirs with the new ones) depicted in Fig. 19. Quantum Grassmannian and measurement maps Non-normalized quantum transport matrices. We add additional vertices labelled( n, , , (0 , n, 0) and (0 , , n ) to the quiver of Fock-Goncharov parameters Z abc and constructdual planar bicolored (plabic) graph G ( Figure 7). Then, we define non-normalized quantumtransport matrices M and M as quantization of boundary measurement matrices of graph G introduced by Postnikov in [35]. Namely, we assign to every path P connecting a sourceof G to a sink a quantum weight w ( P ) that is element of the quantum torus Υ. We definethe boundary measurement between source p and sink q as M pq = X path P : p q w ( P ). Finally,note that G has n sources and 2 n sinks, we organize boundary measurements M pq into 2 n × n matrix that we divide into two n × n matrices M and M .Vertices of G are colored into black and white color as follows: a black vertex has twoincoming arrows and one outgoing, while a white vertex has two outgoing and one incomingarrows.We equip faces of Figure 20 with weights Z α associated with the corresponding vertices ofgraph Figure 6. We define the quantum weight of a maximal oriented path in G by formula 2.5(see Fig.20). 600 006 060051501 015105 024114204303 Figure 20. Face and path weights of G . Faces are labeled by indices i, j, k ∈ Z , i + j + k = 6, the corresponding Fock-Goncharov face weight is denoted by Z ijk .The weight w ( P ) of the blue path P is w ( P ) = •• Z Z Z Z Z Z Z •• Notethat corner faces do not carry Fock-Goncharov variables and don’t contribute to thenormalized transport matrices M and M . However, they do contribute toward non-normalized transport matrices M and M . ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 33 Example 7.1. Consider the triangular network of SL (Fig. 21) 300 210 120 030201 111 021102 012003 Figure 21. Face and path weigths of G SL . Triples i, j, k ∈ Z , i + j + k = 3 label faces. Quantum transport matrices have the following form: M = •• Z − / Z / Z − / Z − / Z / •• •• Z − / Z / ( Z − / + Z / ) Z / Z / •• •• Z / Z / Z / Z / Z / •••• Z − / Z − / Z − / Z − / Z − / •• •• Z − / Z − / Z − / Z / Z − / •• •• Z − / Z − / Z − / Z − / Z − / •• . Then, M = QSD − M , where D = •• Z Z Z Z Z Z •• and M = •• Z •• •• Z Z •• •• Z Z Z •• •• Z Z Z •• •• Z (1 + Z ) Z Z Z •• •• Z Z Z Z Z Z •• . Similarly, M = •• Z / Z / Z / Z / Z / •• •• Z / Z / Z / Z − / Z − / •• •• Z / Z / Z / Z − / Z / •••• Z − / Z − / Z − / Z − / Z − / •• •• Z − / ( Z − / + Z / ) Z / Z − / Z − / •• •• Z − / Z / Z / Z − / Z / •• Hence, M = QS •• D − D − •• M , where D = •• Z − Z − Z − Z − Z − / Z − / Z − / Z − / Z − / •• and M = •• Z Z Z Z •• •• Z Z (1 + Z ) Z Z Z •• •• Z Z Z Z Z Z Z •• •• Z Z Z Z Z Z Z •• •• Z Z Z Z Z Z Z Z •• •• Z Z Z Z Z Z Z Z Z •• Notice that both M and M are non-normalized quantum transport matrices of networkshown on Figure 21.7.2. Quantum Grassmannian and proofs of Theorems 2.5 and 3.1. We now proveTheorems 2.5 and 3.1 utilizing the notion of plabic graphs introduced by Postnikov in [35]. Remark 7.2. The semiclassical statement of Theorem 2.5 (see Remark!2.8) was proved in [23].Following Postnikov we call a planar network an oriented graph with faces equipped withweights. Let N be a network in the disk with neither sources nor sinks inside and separatedsources and sinks on the boundary. An example of such network N is drawn on Figure 22 inrectangle R with sources on the right side and sinks on the left side. ∗ AND MICHAEL SHAPIRO ∗∗ α βγ δ ǫ Figure 22. Network N in rectangle R . Denote by F aces ( N ) the set of faces of network N . Figure 22 has F aces ( N ) = { α, β, γ, δ, ǫ } .Consider the integer lattice ˜Λ generated by F aces ( N ) and vector space ˜ V = Q ⊗ ˜Λ. We equipit with the integer skew-symmetric form h , i as follows. Definition 7.3. [35] A planar bicolored graph, or simply a plabic graph is a planar (undirected)graph G , without orientations of edges, such that each boundary vertex b i is incident to a singleedge and all internal vertices are colored either black or white. A perfect orientation of a plabicgraph is a choice of orientation of its edges such that each black internal vertex v is incidentto exactly one edge directed away from v ; and each white v is incident to exactly one edgedirected towards v . A plabic graph is called perfectly orientable if it has a perfect orientation.Let us transform the oriented graph G of network into plabic graph G pl by coloring innervertices of G into black and white colors according to the rule: black vertex has two incomingarcs and one outgoing; white vertex has one incoming and two outgoing. We forget boundarysources and sinks so that any arcs connecting inner vertex to the boundary one becomes ahalf-arc (see Fig. 24). For a plabic graph G pl we define an oriented dual graph ( G pl ) ∗ as follows.Vertices of ( G pl ) ∗ are faces of G pl . For every black and white vertex x of G pl we define 3 arcsof ( G pl ) ∗ that cross half-edges attached to x in counterclockwise direction if x is black andclockwise direction if x is white (see Fig. 23). Figure 23. Dashed blue arcs are edges of the dual graph ( G pl ) ∗ around black andwhite vertex of G pl . For θ, φ ∈ F aces ( N ) let θ → φ ) denote the number of arcs from θ to φ in ( G pl ) ∗ .Define the skew-symmetric form h , i on ˜Λ by the formula(7.10) h θ, φ i = 12 ( θ → φ ) − φ → θ )) . Example 7.4. The plabic graph and its dual for the network Fig. 22 are shown on the Fig. 24. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 35 α βγ δ ǫ Figure 24. Arcs of plabic graph G pl corresponding to network N on Fig. 22 are blacksolid lines; arcs of its dual ( G pl ) ∗ are dashed blue arrows. Then, h α, β i = − / , h α, δ i =1 , h α, γ i = − / , h β, ǫ i = 1 / , h β, δ i = − , h γ, δ i = − / , h δ, ǫ i = − / Let V be the quotient space V = ˜ V / P θ ∈ F aces ( N ) θ and Λ be the induced integer lattice in V . Note that P θ ∈ F aces ( N ) θ lies in the kernel of the skew-symmetric form and its push forwardto V is well-defined. Abusing notation we will use h , i for the induced skew-symmetric form on V . Dual lattice Λ ∗ = Hom(Λ , Z ).For α ∈ F aces ( N ), we denote by Z α the corresponding quantum face weight. The quantumtorus Υ N is generated by weights Z α , α ∈ F aces ( N ) satisfying commutation relations Z α Z β = q − h β,α i Z b Z a . For a = P α ∈ F aces ( N ) c α α ∈ Λ, we define Z a = •• Q α ∈ F aces ( N ) Z c α α •• ∈ Υ N . Note thatfor a , b ∈ Λ, Z a + b = q −h a , b i Z a Z b = q −h b , a i Z b Z a . As above, for any a ∈ Λ, Z a is called thenormal (Weyl) ordering. We define weight of any vector in Λ ⊗ Q by Z a + b = •• Z a Z b •• = •• Z b Z a •• and Z r a = •• Z a •• r = •• Z ra •• for any r ∈ Q . Since every Z α is a positively defined self-adjointoperator in ℓ ( R ) having a continuous spectrum (0 , ∞ ), any rational power of it is itself apositively defined self-adjoint operator.Let p be the maximal oriented path from a source i on the right to the sink j on the left of anetwork. Complete p to an oriented loop ˜ p by following path p from i to j first and then closingthe loop following the piece of boundary of the rectangle in the clockwise direction from j to i .The oriented loop ˜ p defines a covector ˜ p ∈ Λ ∗ as follows. (We use the same notation for theloop and induced covector.) Let r be a half infinite ray with starting point inside face α anddirected towards infinity and q , . . . , q s be intersection points of r and loop ˜ p , T r be the unitdirection vector of r , T q j ˜ p is the unit tangent vector to ˜ p at q j . We assume that r is chosengeneric, i.e., for all q j vectors T r and T q j ˜ p are linearly independent.We define the intersection index ind q j (˜ p, r ) of ˜ p and r at q j to be 1 if orientation of basis( T q j ˜ p, T r ) coincides with counterclockwise orientation of the plane and − p ( α ) = P sj =1 ind q j (˜ p, r ). Note that ˜( α ) depends neither on exact position of starting point of r provided that the starting point varies inside the same connected component of complementto ˜ p nor on the particular choice of ray r with the same starting point. Since any face α liesentirely in some connected component of ˜ p we conclude that ˜ p ( α ) is well defined. Clearly,˜ p ∈ Λ ∗ .Assign to any path p a vector v p = P α ∈ F aces ( N ) ˜ p ( α ) α ∈ Λ. In the example in Section 2.3where any maximal oriented path p is non-selfintersecting the vector v p is the sum of all facesto the right from the path.Set the weight w p of the path p as w p = Z v p .Let S be the set of all sources of N , F be the set of all sinks. Define for any source a ∈ S and sink b ∈ F a quantum boundary measurement M eas q ( a, b ) = P p : a b ( − cross ( p ) w p , wherethe sum is taken over all oriented paths p from a to b where the crossing index cross ( p ) is thenumber of self-crossings of the path p . Classical boundary measurement is defined by Postnikovin [35], For the network in Section 2.3, no path is selfcrossing and M eas q ( a, b ) = P p : a b w p . ∗ AND MICHAEL SHAPIRO ∗∗ Let n = | S | , m = | F | . Define an m × n matrix Q q of quantum boundary measurements as( Q q ) ba = ( M eas q ( a, b )) a ∈ S,b ∈ F . Note that we label rows of Q q by sinks and columns by sourcesof N .In Example 22, the matrix Q q = Z α + β Z α Z α + β + γ Z α + γ Z α + β + γ + δ .Define ( m + n ) × n quantum grassmannian boundary measurement matrix Q grq of network N .Columns of Q grq are labelled by boundary sources of network; rows are labelled by all boundaryvertices. To describe matrix elements of ˜ Q q we introduce the order ord N ( b ) of boundary vertex b . Enumerate all boundary vertices of N from 1 to m + n in counterclockwise direction.Let b ∈ [1 , m + n ] be the index of boundary vertex. Let σ ( b ) be the number of sourcesamong boundary vertices with indices from 1 to b − 1. The order is defined by the formula ord N ( b ) = ( σ ( b ) , if b is not a source; σ ( b ) + , if b is a source . . Let J ( i ) ∈ [1 , m + n ] be the index of i th source, i ∈ [1 , n ]; J : [1 , n ] → [1 , m + n ] is an increasing function.We define ( Q grq ) ji = ( ( − i + ord N ( j ) q − ord N ( j ) M eas q ( i, j ) , if j is not a source; q − ord N ( j ) δ ( J ( i ) , j ) , otherwise. Example 7.5. In Example 22 , the matrix Q grq = q − / q − / − q − Z α + β q − Z α − q − Z α + β + γ q − Z α + γ − q − Z α + β + γ + δ . Remark 7.6. In [35] a boundary measurement map is defined as a map Meas from the spaceNet m × n of networks with n sources, m sinks and commutative weights to Gr ( n, m + n ). Foreach X ∈ Net m × n , boundary measurements Meas( i, j ) form an ( m + n ) × n matrix Q gr whichrepresents Meas( X ). The space of ( m + n ) × n matrices with elements in Υ N we denote byMat ( m + n ) × n (Υ N ). The group GL n (Υ N ) of invertible n × n matrices with entries from Υ N acts onMat ( m + n ) × n (Υ N ) by the right multiplication. We define the homogeneous space Gr q ( n, m + n )as the right quotient Gr q ( n, m + n ) = Mat ( m + n ) × n (Υ N ) /GL n (Υ N ). We denote by QN et m × n the space of quantum networks with n sources, m sinks and quantum weights from Υ N . Wedefine a quantization Meas q : QN et m × n → Gr q ( m, m + n ) as the composition QN et m × n → Mat ( m + n ) × n (Υ N ) → Gr q ( n, m + n ). Definition 7.7. Two networks are equivalent if they have the same boundary measurements.Simple equivalence relations (M1-M3,R1-R3) on the space of networks ( [35]) are simplelocal network transformations preserving boundary measurements. Please, note that in thefigures below we draw the plabic graph assuming that it is equipped with a perfect orientation.Different choices of compatible perfect orientation give the same result.The following claims generalize similar statements for commuting weights (cf [35]).Let e = { e i } denote a standard basis in the lattice Λ = Z n , equipped with the standarddot product · with respect to e , Z i = Z e i be the generators of a quantum torus Υ. Wesay that an infinite linear combination P λ ∈ Λ α λ Z λ is a Laurent series if there exist integers b , . . . b n ∈ Z such that α λ = 0 unless λ · e j ≥ b j ∀ j ∈ [1 , n ]. Any Laurent series u ∈ ˆ R whereˆ R = Q [ q, q − ][ Z − , . . . , Z − N ][[ Z , . . . , Z N ]]. Let m ⊂ Q [ q, q − [ Z , . . . , Z N ] be the maximal idealgenerated by Z i . Note that for any x ∈ m the expression (1 + x ) − ∈ ˆ R and (1 + x − ) − = x (1 + x ) − ∈ ˆ R . ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 37 We say that x, y ∈ ˆ R q -commute if xy = q k yx for some k ∈ Q .We call a plabic network with weights in ˆ R a quantum network.Define 6 elementary moves (M1-M3), (R1-R3) as shown below. Z α Z β Z γ Z δ Z ǫ Z ′ α Z ′ β Z ′ γ Z ′ δ Z ′ ǫ Figure 25. Elementary move M1: Z ′ ǫ = Z − ǫ , Z ′ δ = Z δ + Z δ + ǫ , Z ′ α = Z α + Z α + ǫ , Z ′ β = P ∞ j =1 ( − j − Z β + jǫ , Z ′ γ = P ∞ j =1 ( − j − Z γ + jǫ . Z α Z β Z γ Z δ Z ǫ Z α Z β Z γ Z δ Z ǫ Figure 26. Elementary move M2. Z α Z β Z α Z β Figure 27. Elementary move M3. Z α Z ǫ Z β P ∞ j = ( − ) j − Z α + j ǫ Z β + Z β + ǫ Figure 28. Elementary move R1. ∗ AND MICHAEL SHAPIRO ∗∗ Z α Z β Z ǫ Z α + β + ǫ Figure 29. Elementary move R2. Z ǫ Z ǫ Figure 30. Elementary move R3. Definition 7.8. Two networks are move equivalent if they are connected by a sequence ofelementary moves.The corresponding (move) equivalence is called quantum (move) equivalence.The following result extends the results of [35] to quantum networks. Lemma 7.9. Two quantum move equivalent networks are quantum equivalent.Proof. The proof follows [35]. Compared to the commutative case we just need to checkone additional condition that quantum parameters elementary transformations also form q-commutative family. The cases R2,R3, M2, and M3 are evident. Let’s consider M1 and R1.The case M1 is proved in [17]. We give the proof here for completeness.We want to show that Z ′ α , Z ′ β , Z ′ γ , Z ′ δ and Z ′ ǫ q-commute.Note that h α, β + ( k + 1) ǫ i = h α, β i − ( k + 1) = h α + ǫ, β + kǫ i , therefore Z ′ α Z ′ β =( Z α + Z α + ǫ ) P ∞ j =1 ( − j − Z β + jǫ = q h α,β + ǫ i Z α + β + ǫ while Z ′ β Z ′ α = q −h α,β + ǫ i Z α + β + ǫ . Hence, Z ′ β Z ′ α = q − h α,β + ǫ i Z ′ α Z ′ β . Commutation relations for all other pairs of parameters can be checked simi-larly.Different perfect orientations are in one-to-one correspondence with the almost perfect matchings(see [36]). Up to evident symmetries, there are only two essentially different almost perfectmatchings and, hence, we need to check two perfect orientations. The straightforward compu-tation shows that the elementary move M1 does not change measurements for any choice ofperfect orientation.The case R1 is similar. (cid:3) Definition 7.10. ([35]) We say that a plabic network (or graph) is reduced if it has no iso-lated connected components and there is no network/graph in its move-equivalence class towhich we can apply a reduction (R1) or (R2). A leafless reduced network/graph is a reducednetwork/graph without non-boundary leaves.The following statements are proved in [35] . Lemma 7.11. [35] Any network is move equivalent to a reduced network. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 39 Lemma 7.12. [35] Two reduced equivalent networks are (M1-M3)-move equivalent. Definition 7.13. We call a maximal simple oriented path P = ( p , p , . . . , p h ), p i = p j forall i = j unequivocal if there is no oriented path ( p k , q , . . . , q t , p ℓ ) such that q s = p r for all1 ≤ s ≤ t and 0 ≤ r ≤ h .Let P be an unequivocal path in a network N ∈ Net m,n . Reverse the orientation of P keepingface weights we obtain the new network N ′ . Lemma 7.14. Reversing the orientation of unequivocal path P does not change quantum grass-mannian measurement Meas q ( N ′ ) = Meas q ( N ) . Example 7.15. Consider networks on the Fig. 31. α βγ δ ǫ α βγ δ ǫ Figure 31. Changing orientation of the path P : transforms network N onthe left into network N ′ on the right The corresponding quantum grassmannian measurement matrices are Q grq = q − / q − / − q − Z α + β q − Z α − q − Z α + β + γ q − Z α + γ − q − Z α + β + γ + δ and (cid:0) Q grq (cid:1) ′ = q − / q − Z β q − Z δ + ǫ + β q − Z − γ q − / − q − Z α + β + γ + δ . Note that Q grq C = (cid:0) Q grq (cid:1) ′ , where C = (cid:18) q / Z β q / Z δ + ǫ + β (cid:19) . Indeed, consider for example ( Q grq C ) = ( Q grq ) + q / ( Q grq ) Z β = − q − Z α + β + q − / Z α Z β .Recall that Z α Z β = q − / Z α + β . Therefore, ( Q grq C ) = 0 = (cid:0) Q grq (cid:1) ′ . Similarly, we can proveequalities for all the entries of these 5 × q ( N ) = Meas q ( N ′ ) ∈ Gr q (2 , Proof. Let N ′ be the network obtained as a result of the change of the directions of all arrowsof the simple unequivocal path P in N from a boundary vertex a to a boundary vertex b . Wewill denote by P − the path in N ′ obtained from P by orientation reversing. We assume firstthat the boundary vertices are labelled so that 1 ≤ a < b ≤ m + n . Since path P is unequivocalthere is only one path P from a to b , and Q q ( a, b ) = w P . Moreover, any other path R from s to t where both s and t are distinct from a and b has at most one common interval [ V, W ] withpath P . The first point V (counting from s ) where two paths meet has two incoming arrowsand one outgoing and, hence, is colored black, the point W where two paths separate is white(see Figure 32). Similarly, any path from a to a sink different from b separates from P at awhite point; any path from a sink different from a to b joins path P at a black point.Let Q grq be the quantum grassmannian bounded measurement matrix of the network N , (cid:0) Q grq (cid:1) ′ be the quantum grassmannian boundary measurement matrix of N ′ . Let F P ⊂ F aces ∗ AND MICHAEL SHAPIRO ∗∗ denote the subset of all faces to the right of the path P , v P = P α ∈ F p α . Then, w P = Z v P , F P − = F aces \ F P , v P − = − v P , w P − = Z v P − = Z − v P = ( w P ) − .Consider first the case 1 ≤ a < b .Let s < a < t < b in the cyclic order of the boundary vertices (see Figure 32). Observe, w b → W → t = Z α + β + δ = •• Z β + δ Z α •• = •• w a → V → W → t · w − a → V → W → b •• = •• w a → V → W → t · ( Q ′ q ) ab •• Sincethe equality holds for any directed path from b to t , and •• ( Q ′ q ) ab ( Q q ) ta •• = •• ( Q q ) ta ( Q q ) − ba •• = q − / ( Q q ) ta ( Q q ) − ba , we conclude that ( Q ′ q ) tb = •• ( Q ′ q ) ab ( Q q ) ta •• = •• ( Q q ) − ba ( Q q ) ta •• = q − / ( Q q ) ta ( Q q ) − ba .Similarly, ( Q ′ q ) as = •• ( Q ′ q ) ab ( Q q ) bs •• = •• ( Q q ) − ba ( Q q ) bs •• = q / ( Q q ) bs ( Q q ) − ba and ( Q q ) ts = •• ( Q q ) bs ( Q ′ q ) tb •• = q / ( Q q ) bs ( Q ′ q ) tb . Therefore, ( Q grq ) ′ ts = ( Q grq ) ts + ( Q grq ) bs ( Q grq ) ′ tb = 0. Here, α, β, γ, δ are ap-propriate subsets of F aces . α βγ δ asbt VW α βγ δ asbtt VW Figure 32. Change of the orientation of the path P : a b , s < a < t < b . In the same way we investigate all the remaining mutual positions of s, t and 1 ≤ a < b which leads to the following matrix identity. Let a = J − ( a ), f be the index of source of X such that b lies between the source f and f + 1 (equivalently, J ( f ) < b < J ( f + 1). Notethat a ≤ f since a < b . Define n × n matrix C for 1 < a < b as follows C ij = δ ij , if i < a or i > f ;( − (cid:4) | j − a +1 / | (cid:5) q / ( Q q ) ja ( Q q ) − ba if i = a ; qδ i − ,j , if a < i ≤ f . Then, (cid:0) Q grq (cid:1) ′ = Q grq C .To study 1 ≤ b < a , note that (cid:0) Q grq (cid:1) ′ = Q gr · C implies Q grq = (cid:0) Q grq (cid:1) ′ · C − where C − isobtained from C by changing the signs of the off-diagonal elements and adjusting the powersof q . More exactly, define n × n matrix e C for 1 < b < a as follows e C ij = δ ij , if i ≤ f or i > a ;( − (cid:4) | j − a − / | (cid:5) q − / ( Q q ) ja ( Q q ) − ba if i = a ; q − δ i +1 ,j , if f < i < a . This observation proves Lemma 7.14 1 ≤ b < a . (cid:3) Example 7.16. Consider the networks in Figure 33.Matrices Q grq and (cid:0) Q grq (cid:1) ′ have the following form. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 41 α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α Figure 33. Change of the orientation of the unequivocal path P : , 1 < < Q grq = q − / q − Z − α − q − Z − P j =2 α j q − Z − P j =2 α j q − / q − Z − α q − / q − Z α q − / q − Z P j =3 α j − q − Z P j =5 α j q − Z α q − / q − Z P j =1 α j − q − Z P j =3 α j q − Z P j =5 α j − q − Z P j =7 α j q − Z α q − / − q − Z P j =1 α j q − Z P j =3 α j − q − Z P j =5 α j q − Z P j =7 α j − q − Z P j =9 α j q − Z α − q − Z − P j =12 α j − q − Z P j =1 α j q − Z P j =3 α j − q − Z P j =5 α j q − Z P j =7 α j − q − Z P j =9 α j q − Z P j =11 α j − q − Z − P j =13 α j − q − Z P j =1 α j q − Z P j =3 α j − q − Z P j =5 α j q − Z P j =7 α j − q − Z P j =9 α j q − Z P j =11 α j − q − Z − α q − / q − Z P j =1 α j − q − Z P j =3 α j q − Z P j =5 α j − q − Z P j =7 α j q − Z P j =9 α j (cid:0) Q grq (cid:1) ′ = q − q − Z α − q − Z − P j =2 α j q − Z − P j =2 α j − q − Z − P j =2 α j q − Z − P j =2 α j q − − q − Z P j =1 α j q − Z α q − Z − P j =4 α j − q − Z − P j =4 α j q − Z − P j =4 α j − q − Z − P j =4 α j q − Z − P j =4 α j − q − Z P j =1 α j q − Z P j =3 α j q − Z − P j =5 α j − q − Z − P j =5 α j q − Z − P j =5 α j − q − Z − P j =5 α j q − Z − P j =5 α j − q − Z P j =1 α j q − Z P j =3 α j q − Z − α − q − Z − P j =6 α j q − Z − P j =6 α j − q − Z − P j =6 α j q − Z − P j =6 α j q − / q − Z P j =1 α j q − Z − α − q − Z − P j =8 α j q − Z − P j =8 α j − q − Z − P j =8 α j q − / q − Z − α − q − Z − P j =10 α j q − Z − P j =10 α j q − / q − Z − α 00 0 0 0 0 q − / 00 0 0 0 0 q − Z α 00 0 0 0 0 0 q − / q − Z P j =11 α j − q − Z P j =13 α j q − Z α C = − q Z P j =1 α j q Z P j =3 α j q Z − P j =5 α j − q Z − P j =5 α j q Z − P j =5 α j − q Z − P j =5 α j q Z − P j =5 α j q q q ∗ AND MICHAEL SHAPIRO ∗∗ Straightforward checking proves (cid:0) Q grq (cid:1) ′ = Q grq C showing that path orientation reversion doesnot change the quantum grassmannian measurement.Consider the network for X SL n , Σ shown for n = 6 in Figure 20. The boundary measurementmatrix Q grq has size 3 n × n . The top n × n part U = ( Q grq ) [1 ,n ] is the diagonal matrix with j thdiagonal elements q − j + ; the middle part of the quantum grassmannian matrix ( Q grq ) [ n +1 , n ] = q − n M S ; and the bottom part ( Q grq ) [2 n +1 , n ] = q − n M S .Let’s change the orientation of all snakelike right to left horizontal paths of the networkFig. 20. The result is shown on the Fig. 34. Figure 34. This network is obtained by the simultaneous change of orientations ofthe snakelike horizontal bold paths (colored blue). The big arrow shows the directionof non-normalized transport matrix M . The orientation change of all bold paths (see Figure 34) leads to the new quantum grass-mannian measurement (cid:0) Q grq (cid:1) ′ . Its middle part is the submatrix ( (cid:0) Q grq (cid:1) ′ [ n +1 , n ] = U , the bottom n × n part ( (cid:0) Q grq (cid:1) ′ [2 n +1 , n ] = q − n M S . Using the fact that Q grq and (cid:0) Q grq (cid:1) ′ represent the samequantum grassmann element, we obtain q − n M SC = U , q − n M SC = q − n M S . Find C fromthe first equation: C = S − M − q n U . Substituting the expression for C into second equation weobtain M = M SU − q − n M . Note that U S = SU − q − n , then U S M = ( U S M )( U S M ).We conclude that M = M M .This is clearly equivalent to the second part of Theorem 3.1 T T T = 1 . Commutation relations between face weights induce R -matrix commutation relations betweenentries of Q q .Namely, the next lemma describes the commutation relation between elements of Q q . Lemma 7.17. R m Q q ⊗ Q q = Q q ⊗ Q q R n , where R m , R n are given by formula 2.6.Proof. We will prove this statement using factorization of matrix Q q into a product of elemen-tary matrices. Let N be a network in rectangle with m sinks on the left and n sources on theright. N can be presented as a concatenation of elementary networks of two special kinds. ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 43 Figure 35. Elementary forks Let N be a network obtained by adding the fork to j th sink, j ∈ [1 , m ] (see Figure 36). f . . . f a f f Figure 36. Adding fork to the second sink. Note, •• a f f . . . f •• = 1. Using the construction above, we write that, Q q = Q i X i , where X i = L i or X i = U i is an m i × m i +1 matrix where ( m i +1 = m i − m i +1 = m i + 1 in the second.) L i = t . . . . . . . . . t . . . . . . . . . ... . . . 0 0 . . . . . . t i . . . . . . •• t i Z a i •• . . . . . . t i +1 . . .. . . . . . , t = •• Z f •• , t = •• Z f Z f •• , etcand U i = t . . . . . . . . . t . . . . . . . . . ... . . . 0 0 . . . . . . •• t i •• •• t i Z a i •• . . . . . . t i +1 . . .. . . . . . , t = •• Z f •• , t = •• Z f Z f •• , etc . Variables t j commute with each other and commute with Z a i unless j = i . Otherwise, Z a i t i = qt i Z a i , •• Z a i t i •• = q − / Z a i t i . All entries of X i commute with all entries of X j for i = j .Clearly, it is enough to check the relations for 2 × × ∗ AND MICHAEL SHAPIRO ∗∗ For m i = 1, we have R m i L i ⊗ L i = L i ⊗ L i R m i +1 , R m i U i ⊗ U i = U i ⊗ U i R m i +1 . Indeed, q q − q − q (cid:18) ab (cid:19) ⊗ (cid:18) ab (cid:19) = q q − q − q a abbab == qa qbaqabqb = (cid:18) ab (cid:19) ⊗ (cid:18) ab (cid:19) · (cid:0) q (cid:1) Then, R m Q q ⊗ Q q = R m Q ni =1 1 X i ⊗ Q ni =1 2 X i = R m X Q ni =2 1 X i ⊗ X Q ni =2 2 X i = X ⊗ X R m Q ni =2 1 X i ⊗ Q ni +2 2 X i = · · · = Q ni =1 2 X i ⊗ Q ni =1 1 X i R m n = Q q ⊗ Q q R m n . (cid:3) Corollary 7.18. Theorem 2.5 and Remark 2.9 follow from Lemma 7.17. Proof. Indeed, it is enough to consider 2 n × n matrix B of boundary measurements of thenetwork shown on Figure 20. Denote by M the top n × n block of Q q , M is the bottom n × n block. Q q satisfies R -matrix relation 7.17. Choose subset of these relations betweentensor products of elements of M and M we obtain Theorem 2.5. Relations between tensorproducts of elements of each matrix M i give Remark 2.9. (cid:3) Directed networks with cycles In this section we generalize Lemma 7.17 to the case of planar networks containing orientedcycles and interlacing sources and sinks. R -matrix formulation of commutation relation ofelements of transport matrices is not valid for more general networks with interlacing sourcesand sinks. The corresponding statement is formulated in Theorem 8.3. For the case of networkswith separated sources and sinks commutation relation of Theorem 8.3 coincide with those ofLemma 7.17. Definition 8.1. We assign to every oriented path P : j i from a source j to a sink i the quantum weight w ( P ) = •• Y faces α lie to the rightof the path P Z α •• , where the product is taken with repetitions, Definition 8.2. For any planar directed network N , define transport elements ( α, a ) := X all paths α a ( − w ( P α a )where the sum ranges all paths from the source j to the sink i . This sum is finite for acyclicnetworks and can be infinite for networks containing cycles. In this section, we let Greekletters denote sources and Latin letters denote sinks. We draw these transport elements assimple directed paths α → a . Theorem 8.3. For any planar network, we have the algebra of transport elements: αβab [( α, a ) , ( β, b )] = ( q − q − )( α, b )( β, a ); ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 45 αβab [( α, b ) , ( β, a )] = 0 , aβαb [( α, a ) , ( β, b )] = 0 ,αab ( α, a )( α, b ) = q − ( α, b )( α, a ) , a αβ ( β, a )( α, a ) = q − ( α, a )( β, a ) . For acyclic networks these theorem was proven above; we now consider the case of networkwith cycles. The proof will be by induction. We treat in details only cases with four distinctsources and sinks (the first three cases in the theorem).We consider all possible cases corresponding to the situation in which we close the sink a and the (neighbour) source α . For a path ( β, b ) we have two possibilities: αβab αbaβ We begin with observation that in both these cases, •• ( α, b )( α, a ) n ( β, a ) •• = ( α, b )( α, a ) n ( β, a ) = ( β, a )( α, a ) n ( α, b ) , where on the right we assume the natural order of the product of operators. The effect ofclosing the line between a and α changes the transport element from β to b : in the respectivecases, we have( β, b ) = ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) , and ( β, b ) = ( β, b ) + ( β, a ) 11 + ( α, a ) ( α, b ) . Here and hereafter, we understand rational expressions as geometrical-progression expansions inpowers of the corresponding operator. We also use the standard commutation relation formulas h A, 11 + B i = − 11 + B [ A, B ] 11 + B ∀ A, B, and use the color graphics to indicate permutations of operators in formulas of this section:a pair of operators painted red produces the factor q upon permuting these operators in theoperatorial product, and a pair of operators painted blue produces a factor q − upon thecorresponding permutation; pairs of operators painted magenta commute.Below we have six cases of mutual distribution of sources { α, β, γ } and sinks { a, b, c } (Note,that planarity condition requires α and a always to be neighbour), and in each such case wehave two choices of transport elements: { ( β, b ) , ( γ, c ) } and { ( β, c ) , ( γ, b ) } , so, altogether, wehave 12 variants to be checked. Case 1 . αβγabc Variant (a): { ( β, b ) , ( γ, c ) } .[( β, b ) , ( γ, c )] = (cid:20)(cid:16) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:17) , (cid:16) ( γ, c ) − ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:17)(cid:21) ∗ AND MICHAEL SHAPIRO ∗∗ =( q − q − )( β, c )( γ, b ) + ( γ, a ) − 11 + ( α, a ) ( q − q − )( β, a )( α, b ) 11 + ( α, a ) ( α, c ) − ( q − q − )( β, c )( γ, a ) 11 + ( α, a ) ( α, b ) + ( β, a ) 11 + ( α, a ) ( q − q − )( α, c )( γ, a ) 11 + ( α, a ) ( α, b ) − ( q − q − )( β, a ) 11 + ( α, a ) ( α, c )( γ, b ) + ( β, a ) 11 + ( α, a ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c ) − ( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b )=( q − q − ) h ( β, c )( γ, b ) − ( β, c )( γ, a ) 11 + ( α, a ) ( α, b ) − ( β, a ) 11 + ( α, a ) ( α, c )( γ, b )+ ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b ) i + (cid:0) ( q − q − ) − q + q − (cid:1) ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b )=( q − q − )( β, c )( γ, b ) . Variant (b): { ( β, c ) , ( γ, b ) } .[( β, c ) , ( γ, b )] = (cid:20)(cid:16) ( β, c ) − ( β, a ) 11 + ( α, a ) ( α, c ) (cid:17) , (cid:16) ( γ, b ) − ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:17)(cid:21) =( γ, a ) − 11 + ( α, a ) ( q − q − )( β, a )( α, c ) 11 + ( α, a ) ( α, b ) + ( q − q − )( γ, a ) 11 + ( α, a ) ( β, b )( α, c ) − ( q − q − )( β, b )( γ, a ) 11 + ( α, a ) ( α, c ) − ( q − q − )( β, b )( γ, a ) 11 + ( α, a ) ( α, c ) − ( q − q − )( β, a ) − 11 + ( α, a ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c )+ ( β, a ) 11 + ( α, c ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b ) − ( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c )(two last terms mutually cancelled)=( q − q − ) (cid:20) − ( γ, a ) 11 + ( α, a ) ( β, a )( α, c ) 11 + ( α, a ) ( α, b ) − ( γ, a ) h ( β, b ) , 11 + ( α, a ) i ( α, c )+( β, a ) 11 + ( α, a ) ( γ, a )( α, b ) 11 + ( α, a ) ( α, c ) (cid:21) = − ( q − q − ) (cid:0) q − + ( q − q − ) − q (cid:1) ( γ, a ) 11 + ( α, a ) ( β, a )( α, b ) 11 + ( α, a ) ( α, c ) = 0 . Case 2 . αβcabγ Variant (a): { ( β, b ) , ( γ, c ) } .[( γ, c ) , ( β, b )] = (cid:20)(cid:16) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:17) , (cid:16) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:17)(cid:21) =( γ, a ) − 11 + ( α, a ) ( q − q − )( α, b )( β, a ) 11 + ( α, a ) ( α, c ) ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 47 − ( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b ) + ( β, a ) 11 + ( α, a ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c ) = 0Variant (b): { ( β, c ) , ( γ, b ) } .[( γ, b ) , ( β, c )] = (cid:20)(cid:16) ( γ, b ) − ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:17) , (cid:16) ( β, c ) − ( β, a ) 11 + ( α, a ) ( α, c ) (cid:17)(cid:21) =( q − q − )( γ, a )( β, b ) 11 + ( α, a ) ( α, c ) + ( β, a ) − 11 + ( α, a ) ( q − q − )( γ, a )( α, b ) 11 + ( α, a ) ( α, c )+ ( γ, a ) 11 + ( α, a ) ( q − q − )( α, c )( β, a ) 11 + ( α, a ) ( α, b ) + ( q − q − )( γ, a ) 11 + ( α, a ) ( α, c )( β, b )+ ( γ, a ) 11 + ( α, c ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b )(two last terms mutually cancelled)= − ( q − q − ) (cid:20) − ( γ, a ) h ( β, b ) , 11 + ( α, a ) i ( α, c ) + ( β, a ) 11 + ( α, a ) ( γ, a )( α, b ) 11 + ( α, a ) ( α, c ) − ( γ, a ) 11 + ( α, a ) ( β, a )( α, c ) 11 + ( α, a ) ( α, b ) (cid:21) =( q − q − ) (cid:0) ( q − q − ) + q − − q (cid:1) ( γ, a ) 11 + ( α, a ) ( β, a )( α, b ) 11 + ( α, a ) ( α, c ) = 0 . Case 3 . αcβabγ Variant (a): { ( γ, b ) , ( β, c ) } .[( γ, b ) , ( β, c )] = (cid:20)(cid:16) ( γ, b ) − ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:17) , (cid:16) ( β, c ) + ( β, a ) 11 + ( α, a ) ( α, c ) (cid:17)(cid:21) = − ( q − q − )( γ, c )( β, b ) − ( q − q − )( γ, a )( β, a ) 11 + ( α, a ) ( α, c ) − ( q − q − )( β, a ) − 11 + ( α, a ) ( γ, a )( α, b ) 11 + ( α, a ) ( α, c ) + ( q − q − )( γ, c )( β, a ) 11 + ( α, a ) ( α, b ) − ( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) + ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b )(two last terms mutually cancelled)= − ( q − q − )( γ, c ) (cid:20) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:21) − ( q − q − )( γ, a ) 11 + ( α, a ) ( α, c )( β.b ) − ( q − q − )( γ, a ) h ( β, b ) , 11 + ( α, a ) i ( α, c ) + ( q − q − )( β, a ) 11 + ( α, a ) ( γ, a )( α, b ) 11 + ( α, a ) ( α, c )= − ( q − q − )( γ, c ) (cid:20) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:21) − ( q − q − )( γ, a ) 11 + ( α, a ) ( α, c )( β.b ) − ( q − q − ) (cid:0) ( q − q − ) − q (cid:1) ( γ, a ) 11 + ( α, a ) ( β, a )( α, b ) 11 + ( α, a ) ( α, c ) ∗ AND MICHAEL SHAPIRO ∗∗ = − ( q − q − ) (cid:20) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:21) (cid:20) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:21) = − ( q − q − )( γ, c ) ( β, b )Variant (b): { ( γ, c ) , ( β, b ) } .[( γ, c ) , ( β, b )] = (cid:20)(cid:16) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:17) , (cid:16) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:17)(cid:21) = − ( γ, a ) 11 + ( α, a ) ( q − q − )( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b ) + ( β, a ) 11 + ( α, c ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c ) = 0 . Case 4 . αcbaγβ Variant (a): { ( γ, b ) , ( β, c ) } .[( γ, b ) , ( β, c )] = (cid:20)(cid:16) ( γ, b ) + ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:17) , (cid:16) ( β, c ) + ( β, a ) 11 + ( α, a ) ( α, c ) (cid:17)(cid:21) =( q − q − )( β, a ) 11 + ( α, a ) ( γ, c )( α, b ) − ( q − q − )( γ, c )( β, a ) 11 + ( α, a ) ( α, b )+ ( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b ) = 0 . Variant (b): { ( γ, c ) , ( β, b ) } .[( γ, c ) , ( β, b )] = (cid:20)(cid:16) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:17) , (cid:16) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:17)(cid:21) = − ( q − q − )( γ, b )( β, c ) − ( q − q − )( γ, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( q − q − )( γ, a ) 11 + ( α, a ) ( α, b )( β, c )+ ( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b ) − ( β, a ) 11 + ( α, a ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c )= − ( q − q − ) (cid:20) ( γ, b ) + ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:21) (cid:20) ( β, c ) − ( β, a ) 11 + ( α, a ) ( α, c ) (cid:21) = − ( q − q − )( γ, b ) ( β, c ) . Case 5 . αcβaγb Variant (a): { ( γ, b ) , ( β, c ) } .[( γ, b ) , ( β, c )] = (cid:20)(cid:16) ( γ, b ) + ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:17) , (cid:16) ( β, c ) + ( β, a ) 11 + ( α, a ) ( α, c ) (cid:17)(cid:21) =( q − q − )( β, a ) 11 + ( α, a ) ( γ, c )( α, b ) − ( q − q − )( γ, c )( β, a ) 11 + ( α, a ) ( α, b ) ARBOUX COORDINATES FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 49 + ( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b ) = 0 . Variant (b): { ( γ, c ) , ( β, b ) } .[( γ, c ) , ( β, b )] = (cid:20)(cid:16) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:17) , (cid:16) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:17)(cid:21) = − ( q − q − )( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b ) + ( β, a ) 11 + ( α, a ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c ) = 0 . Case 6 . αβcaγb Variant (a): { ( γ, b ) , ( β, c ) } .[( γ, b ) , ( β, c )] = (cid:20)(cid:16) ( γ, b ) + ( γ, a ) 11 + ( α, a ) ( α, b ) (cid:17) , (cid:16) ( β, c ) − ( β, a ) 11 + ( α, a ) ( α, c ) (cid:17)(cid:21) =( q − q − )( γ, c )( β, b ) − ( q − q − )( β, a ) 11 + ( α, a ) ( γ, c )( α, b ) − ( q − q − )( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b ) + ( q − q − )( γ, a ) 11 + ( α, a ) ( α, c )( β, b ) − ( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( β, a ) 11 + ( α, a ) ( α, c )( γ, a ) 11 + ( α, a ) ( α, b )(two last terms mutually cancelled)=( q − q − ) (cid:20) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:21) (cid:20) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:21) = ( q − q − )( γ, c ) ( β, b ) . Variant (b): { ( γ, c ) , ( β, b ) } .[( γ, c ) , ( β, b )] = (cid:20)(cid:16) ( γ, c ) + ( γ, a ) 11 + ( α, a ) ( α, c ) (cid:17) , (cid:16) ( β, b ) − ( β, a ) 11 + ( α, a ) ( α, b ) (cid:17)(cid:21) = − ( q − q − )( γ, a ) 11 + ( α, a ) ( α, b )( β, a ) 11 + ( α, a ) ( α, c ) − ( γ, a ) 11 + ( α, a ) ( α, c )( β, a ) 11 + ( α, a ) ( α, b ) + ( β, a ) 11 + ( α, a ) ( α, b )( γ, a ) 11 + ( α, a ) ( α, c ) = 0 . Concluding remarks In this paper, we have found the Darboux coordinate representation for matrices A enjoyingthe quantum reflection equation. We have also solved the problem of representing the braid-group action for the upper-triangular A in terms of mutations of cluster variables associatedwith the corresponding quiver.In conclusion, we indicate some directions of development. The first interesting problem isto construct mutation realizations for braid-group and Serre element actions that are Poissonautomorphisms in the case of block-upper triangular matrices A (the corresponding action interms of entries of a block-upper-triangular A was constructed in [8]). It is not difficult toconstruct planar networks producing block-triangular transport matrices M and M enjoying ∗ AND MICHAEL SHAPIRO ∗∗ the standard Lie Poisson algebra, then A = M T1 M will satisfy the semiclassical reflectionequation.We have shown that particular sequences of mutations lead to transformations β i,i +1 for a ij satisfying braid-group relations. We conjecture that the sequences of mutations itself providebirational transformations of Z abc that also satisfy braid group relations and we checked thisconjecture for small examples.Finally, the third direction of development is based on the semiclassical groupoid construc-tion; explicit calculations in Sec.12 of [9] show that, given that B is a general SL n -matrixendowed with the standard semiclassical Lie Poisson bracket, solving the matrix equation B A B T = A ′ , where A and A ′ are unipotent upper-triangular matrices, we obtain that en-tries of A are uniquely determined (provided all upper-right and lower-left minors of B arenonzero); the Lie Poisson bracket on B produces the reflection equation bracket on A , themapping A → A ′ is a Poisson anti-automorphism, and finally, entries of A and A ′ are mutu-ally Poisson commute. 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