Characteristic equation for symplectic groupoid and cluster algebras
aa r X i v : . [ m a t h . R T ] M a r CHARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID ANDCLUSTER ALGEBRAS
LEONID O. CHEKHOV ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ Abstract.
We use the Darboux coordinate representation found by two of the authors (L.Ch.and M.Sh.) for entries of general symplectic leaves of the A n -groupoid of upper-triangularmatrices to express roots of the characteristic equation det( A − λ A T ) = 0, with A ∈ A n , interms of Casimirs of this Darboux coordinate representation, which is based on cluster variablesof Fock–Goncharov higher Teichm¨uller spaces for the algebra sl n . We show that roots of thecharacteristic equation are simple monomials of cluster Casimir elements. This statementremains valid in the quantum case as well. We consider a generalization of A n -groupoid to a A Sp m -groupoid. Introduction
Symplectic groupoid and induced Poisson structure on the unipotent uppertriangular matrices.
Let A n ⊆ gl n be a subspace of unipotent upper-triangular n × n matri-ces. We identify elements A of A n with matrices of bilinear forms on C n . The matrix B ∈ GL n of a change of a basis in C n takes a matrix of bilinear form A ∈ A n to B A B T determininga dynamics of form transformations . A most interesting case is when the transformed form B A B T lies in the same class A n . For a fixed A ∈ A n all B ∈ GL n such that B A B T ∈ A n forma submanifold of GL n , which is not a subspace and may have a very involved structure. Weintroduce the space of morphisms identified with admissible pairs of matrices ( B, A ) such that(1.1) M = (cid:8) ( B, A ) (cid:12)(cid:12) B ∈ GL n , A ∈ A n , B A B T ∈ A n (cid:9) . In 2000, Bondal [1] obtained the Poisson structure on A n using the algebroid construction:For B = e g , we first define the anchor map D A to the tangent space T A n :(1.2) D A : g A → T A n g A g + g T A . 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. [1]
The map (1.3) P A : T ∗ A n → g A w P − , / ( w A ) − P + , / ( w T A T ) , where P ± , / are the projection operators: (1.4) P ± , / a i,j := 1 ± sign( j − i )2 a i,j , i, j = 1 , . . . , n, ∗ 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]. ∗∗∗
Xi’an Jiaotong University, Xi’an, Shaanxi, P.R. China. ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ 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 [21]) as:(1.5) Π : T ∗ A n × T ∗ A n
7→ C ∞ ( A )( 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(1.6) { 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: { a ik , a jl } = 0 , for i < k < j < l, and i < j < l < k, { a ik , a jl } = 2 ( a ij a kl − a il a kj ) , for i < j < k < l, (1.7) { a ik , a kl } = a ik a kl − a il , for i < k < l, { a ik , a jk } = − a ik a jk + 2 a ij , for i < j < k, { a ik , a il } = − a ik a il + a kl , for i < k < l. This bracket is famous in mathematical physics and is known as Gavrilik–Klimyk–Nelson–Regge–Dubrovin–Ugaglia bracket [15, 23, 24, 10, 28]. It originally arose in the context of 2Dquantum gravity; technically it governs from skein relations satisfied by a specially chosen [4]finite subset of geodesic functions a ik (traces of monodromies of SL Fuchsian systems), whichare principal observables in the theory of 2D gravity; a log-canonical (Darboux) bracket on thespace of Thurston shear coordinates z α on the Teichm¨uller space T g,s of Riemann surfaces Σ g,s of genus g with s = 1 , a ik for n = 2 g + s .All geodesic functions on any Σ g,s admit an explicit combinatorial description [11] and 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 called a geometric symplecticleaf .However the Poisson dimensions of T g,s (6 g − s ) and those of A n ( n ( n − / − [ n/ n ≥
6, so a novel Darboux coordinate construction for higher-dimensionalsymplectic leaves of A n was found in [9] where elements of A n were parameterized by clustervariables of the Fock–Goncharov framed moduli space X SL n , Σ , where Σ is a disk with threemarked points [12].Note that algebra (1.7) is universal: any choice of the subspace A ⊆ sl n for a ( B, A )-systemsubject to the form dynamics results in Poisson relations (1.7) provided A is a Lagrangiansubmanifold [6], so constructing cluster realizations of A outside the unipotent upper-triangularcase is also important. We present this construction for the first natural extension of A n , whichis A Sp m . The quantum analogue of (1.7) is known as the quantum reflection equation (seeTheorem 3.1), and the cluster coordinate realization of a general symplectic leaf of the quantumreflection equation was also constructed in [9].An important problem is to construct Casimirs of algebras (1.7) in semiclassical and quantumcases. In the semiclassical limit, Bondal showed that the algebra A n admits [ n/
2] algebraicallyindependent Casimir elements, which are coefficients of the reciprocal polynomial det[ A − λ A T ];in the geometric cases n = 3 and n = 4, these coefficients turn out to be simple functions of hole HARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 3 perimeters p i = P z α i , which are always linear functions of shear coordinates. This statementremains valid in the quantum case as well. At the same time, being expressed in a ij , thesame coefficients become inhomogeneous functions producing the quantum Markov invariantfor n = 3 and quantum invariants first obtained by Bullock and Przytycki “by brute force” in[2] for n = 4.Although we do not exploit it in this text, note that the space A n admits a discrete braid-groupaction generated by special morphisms β i,i +1 : A n → A n , i = 1 , . . . , n −
1, such that β i,i +1 [ A ] := B i,i +1 A B Ti,i +1 ∈ A n ; these transformations correspond to Dehn twists in the geometrical case,and matrices B i,i +1 differ from the unity matrix only in their diagonal 2 × i i + 1 i q / a ~ i,i +1 − qi + 1 1 0in the quantum case constructed in [5]; the matrix A then becomes a quantum matrix A ~ :(1.8) A ~ := q − / a ~ , a ~ , . . . a ~ ,n q − / a ~ , . . . a ~ ,n q − / . . . ...... ... . . . . . . a ~ n − ,n . . . q − / , where 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 relationsfollowing from the quantum reflection equation (see Theorem 3.1). The quantum braid-groupaction is β ~ i,i +1 : A ~ → B ~ i,i +1 A ~ (cid:2) B ~ i,i +1 (cid:3) † with † standing for the Hermitian conjugation. In [9],the braid-group action was realized as sequences of cluster mutations in the A n -quiver obtainedby a special amalgamation procedure (described below) from the quiver for X SL n , Σ .Our goal in this paper is to study the structure of resolvents of the quantum operators A (cid:2) A † (cid:3) − := AA −† . This combination plays a prominent role: first, assuming all operatorialexpressions be invertible, for A transforming as a form (1.1), this combination undergoes anadjoint transformation,(1.9) AA −† → B AA −† B − , so we can address the problem of solving the resolvent equation: for which λ ∈ C the operator AA −† − λ I admits null vectors? We completely solve this problem in Theorem 4.1 expressingall λ in question via Casimirs of the A n -quiver.With this solution in hands, we can address several intriguing problems: note that in thesemiclassical case, the Jordan form of AA −† determines the dimension of the correspondingPoisson leaf. Although our technique does not allow keeping control over the Jordan formstructure in case of coinciding eigenvalues λ i , we can fully treat the case where all n eigenvalues λ i are different. Then the Jordan form is diagonal and, assuming the existence of natural N ≥ n for which λ Ni = 1 for all i , we obtain that the N th power of the Jordan form is a unitmatrix, and therefore (cid:2) AA −† (cid:3) N = I . This relation opens a way to constructing minimal-modelrepresentations (containing null vectors in the sense of Kac) of related non-Abelian Verma-type modules generated by the action of a i,j on a proper vacuum vector, where the action ofcanonical cluster variables plays a role of a Dotsenko–Fatteev free-field representation. We mayalso address finite-dimensional reductions of affine Lie–Poisson systems. LEONID O. CHEKHOV ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ b n -Algebras for the triangle Σ , , Let Σ g,s,p denote a topological genus g surface with s boundary components and p > , , , which corresponds to an ideal triangle in an ideal-triangle decomposition of a generalΣ g,s,p (which comprises exactly 4 g − s + p ideal triangles).2.1. Notations.
Let lattice Λ = Z n 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 . Fixing a basis { e i } in Λ,we consider Υ as a non-commutative algebra of Laurent polynomials in variables Z i := Z e i , i ∈ [1 , N ]. For any sequence a = ( a , . . . , a t ), a i ∈ [1 , N ], let Π a denote the monomial Π a = Z a Z a . . . Z a t . Let λ a = P tj =1 e a j . Element Z λ a is called in physical literature the Weylform of Π a and we denote it by two-sided colons •• Π a •• It is easy to see that •• Π a •• = Z λ a = q − P j We describe nowhow quantized transport matrices are expressed in terms of quantized Fock–Goncharov param-eters.In the quantization of X SL n , Σ for Σ , , , the quantized Fock–Goncharov variables form aquantum torus Υ with commutation relation described by the b n -quiver ( b n stands for a Borelsubalgebra of sl n ) shown in Fig. 1 by solid and dashed lines. Vertices of the quiver labelquantum Fock-Goncharov coordinates Z α while arrows encode commutation relations: if thereare 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. Inparticular, 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 HARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 5 Z β implies Z β Z α = q − Z α Z β , and, say, a double arrow from Z α to Z β means Z β Z α = q − Z α Z β .Vertices not connected by an arrow commute.We next construct a directed network N dual to the b n -quiver. Dual directed network is adirected embedded graph in the disk (triangle) whose three-valent vertices are colored blackand white and orientation is compatible with the coloring, i.e. each white vertex has exactlyone incoming edge while every black vertex has exactly one outgoing edge. Vertices of thequiver that is dual to a directed network are located in the faces of the directed network(one vertex per each face) while directed edges of the quiver intersect black-white edges of thenetwork in such a way that quiver’s arrows travel clockwise around white vertices of networkand counterclockwise around black vertices. Since the quiver orientation depends only on apattern of black and white vertices and not on orientations of network edges, several directednetworks may correspond to the same quiver, but in the considered case orientation of the dualnetwork is determined by fixing n sources and 2 n sinks among boundary vertices. In particular,choosing all sources of the network located along one side of the triangle Fig. 1 we fix the dualdirected network uniquely.We therefore have several possible choices of directed coloring-compatible networks for the b n -quiver, two of which relevant for our studies are depicted in the figure: every solid arrowof a quiver has a white vertex on the right and a black vertex on the left; all dashed arrowshave white vertices on the right. Directed edges of the network (double lines in the figure) aredual to edges of the quiver with sources and sinks dual to dashed arrows; it is easy to see thatthe choice of sources and sinks then determines the pattern of directed double lines inside thegraph in a unique way.We adopt a barycentric enumeration of vertices and cluster variables Z ( i,j,k ) of the b n -quiver: i, j, k are nonnegative integers such that i + j + k = n (one can think about these vertices asinteger points of intersection of the lattice Z + , ⊗ Z + , ⊗ Z + , with the plane x + y + z = n ).1 2 3 4 5 61 ′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′ ′′ Z (6 , , Z (0 , , Z (0 , , ′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′ ′′ Z (6 , , Z (0 , , Z (0 , , Figure 1. Two directed networks N dual to the Fock–Goncharov b -quiver. Doublearrows are edges of the directed network, cluster variables correspond to faces of N (vertices of the b n -quiver dual to N ). On the left-hand side we present the directednetwork defining the transport matrices M (from the side { − } to the side { ′ − ′ } and M (from the side { − } to the side { ′′ − ′′ } ) and on the right-hand side wepresent the directed network defining the transport matrix M (from the side { ′ − ′ } tothe side { ′′ − ′′ } ). We also indicate three barycentrically enumerated cluster variablesat three corners of the quiver. LEONID O. CHEKHOV ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ We assign to every oriented path P : j i ′ from the right side to the left side or to thebottom side P : j i ′′ the quantum weight (2.1) w ( P ) = •• Y faces α lie to the rightof the path P Z α •• . Definition 2.1. 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 M is a lower-triangular matrix and M is an upper-triangular matrix.Theorem 9.3 of [9] states that for any planar network with separated n sources and m sinks,defining entries of m × n quantum transition matrix as in Definition 2.1, we obtain that theseentries enjoy the quantum R -matrix relation (2.2) R m ( q ) M ⊗ M = M ⊗ MR n ( q ) , where R k ( q ) denotes a k × k matrix(2.3) R k ( q ) = X ≤ i,j ≤ k e ii ⊗ e jj + ( q − X ≤ i ≤ k e ii ⊗ e ii + ( q − q − ) X ≤ j
Quantum transport matrices read M = QS M , M = QS M , and M = QS M , where Q := P ni =1 ( q ) − i +1 / e i,i ⊗ I is a diagonal matrix and S = n X i =1 ( − i +1 e i,n +1 − i ⊗ I is anantidiagonal matrix.Note that S ⊗ S R n ( q ) = R T n ( q ) S ⊗ S for any antidiagonal classical matrix S. and Q ⊗ Q R n ( q ) = R n ( q ) Q ⊗ Q for any diagonal classical matrix Q ; HARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 7 in particular, setting Q = AB with two diagonal classical invertible matrices A and B , we have(2.6) A ⊗ B R n ( q ) B − ⊗ A − = B − ⊗ A − R n ( q ) A ⊗ B. Remark 2.3. Multiplying a transition matrix by the matrix S is standard in the Fock–Goncharov approach in order to make orientations of boundaries of adjacent triangles com-patible and hence to “invert” the corresponding flags upon reaching a boundary of a network;additional q -factors collected in the matrix Q are necessary for ensuring the groupoid propertybelow; we do not normalize transport matrices (and, correspondingly, the R -matrix) as in [9]because this normalization does not affect our consideration below. We also omit the sign ofthe direct product in the subsequent text.We have the following theorem. Theorem 2.4. [9] . The quantum transport matrices M , 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 ) , M M = M R T n ( q ) M , M M = R n ( q ) M M with the quantum trigonometric R -matrix (2.3). Theorem 2.5. [9] . The quantum transport matrices satisfy the quantum groupoid condition M M = M . This condition is consistent with the quantum commutation relations in Theorem 2.4. Using relations (2.6) we obtain Lemma 2.6. The commutation relations in Theorem 2.4, as well as the groupoid condition inTheorem 2.5 are invariant under transformations M → AM C, M → BM C, M → BM A − , where A , B , and C are arbitrary nondegenerate classical diagonal matrices. The groupoid of upper triangular matrices Representing an upper-triangular A . Consider a special combination of M and M :(3.1) A := M T1 DM = M T1 DM M , where D is any classical diagonal matrix. (In notations of Lemma 2.6, D = A T B .) We can usethe freedom in choice of D to attain the canonical form (1.8) of the quantum A -matrix withthe diagonal entries equal to q − / . Then the properly normalized expression for this matrixreads(3.2) A ~ = M T1 M QS M , i.e., D − = S T Q T QS , and we use expression (3.2) when solving the problem of singular valuesof λ in Sec. 4.Note that since M and M T1 are upper-anti-diagonal matrices and M is a lower-anti-diagonalmatrix in the case of Σ , , , the matrix A is automatically upper-triangular.The following theorem was proved in [9] for A having the form M T1 M . We slightly modifythis proof to adjust it to A of the form M T1 DM M . LEONID O. CHEKHOV ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ Theorem 3.1. For the matrices M and M enjoying commutation relations in Theorem 2.4and for any diagonal matrix D , the matrix A given by (3.1) enjoys the quantum reflectionequation R n ( q ) A R t n ( q ) A = A R t n ( q ) A R n ( q ) with the trigonometric R -matrix (2.3), where R t n ( q ) is a partially transposed (w.r.t. the firstspace) R -matrix. The proof uses only R -matrix relations. Transposing relations in Theorem 2.4 with respectto different spaces, we obtain M T i R t n ( q ) M i = M i R t n ( q ) M T i , R n ( q ) M T1 2 M T1 = M T1 1 M T1 R T n ( q ) , and M T1 2 M = M M T1 R t n ( q ) . Note that R t n ( q ) retains its form if we interchange the space indices 1 ↔ 2; note also that D is a classical matrix, so, say, D commutes with all M i . We have R n ( q ) M T1 1 D M M R t n ( q ) M T1 2 D M M = R n ( q ) M T1 1 D M M T1 R t n ( q ) D M M M = R n ( q ) M T1 2 M T1 1 D D M M R T n ( q ) M M = M T1 1 M T1 R T n ( q ) D D M M M M R n ( q )= M T1 1 M T1 1 D D R T n ( q ) M M M M R n ( q ) = M T1 2 D M T1 2 M D M R n ( q ) M M R n ( q )(3.3) = M T1 2 D M M T1 R t n ( q ) M D M M R n ( q ) = M T1 2 D M M R t n ( q ) M T1 1 D M M R n ( q ) , which completes the proof.We have therefore a Darboux coordinate representation for operators satisfying the reflectionequation. Moreover, all matrix elements of A are Laurent polynomials with positive coefficientsof Z α and q .By construction of quantum transport matrices in Sec. 2.2, 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 com-mutes with [ M ] k,j (all paths contributing to [ M ] k,i are disjoint from all paths contribut-ing to [ M ] k,j for i < j ), so the corresponding products are automatically Weyl-ordered,[ A ] i,j = j P k = i •• [ M ] k,i [ M ] k,j •• . For i = j , the corresponding two paths share the common startingedge, and then [ A ] i,i = q − / •• [ M ] i,i [ M ] i,i •• . This explains the appearance of q − / factors onthe diagonal of the quantum matrix A ~ (see (1.8), [5]). Note that all Weyl-ordered products of Z α are self-adjoint and we assume that all Casimirs are also self-adjoint.To obtain a full-dimensional (not upper-triangular) form of the matrix A let us consideradjoint action by any transport matrix: Theorem 3.2. [9] Any matrix A ′ := M T γ S T A SM γ where M γ is a (transport) matrix satisfyingcommutation relations of Theorem 2.4 and commuting with A = M T M satisfies the quantumreflection equation of Theorem 3.1. Casimirs of b n -quiver. For the full-rank b n -quiver we have exactly (cid:2) n (cid:3) + 1 Casimirsdepicted in Fig 2 for the example of b : numbers at vertices indicate the power with whichthe corresponding variables come into the product; all nonnumbered variables have powerzero. All Casimirs correspond to closed broken-line paths in the b n quiver with reflectionsat the boundaries (the “frozen” variables at boundaries enter the product with powers two,powers of non-frozen variables can be 0,1,2, and 3, and they count how many times the path HARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 9 goes through the corresponding site). The total Poisson dimension of the full-rank quiver istherefore ( n +2)( n +1)2 − (cid:2) n (cid:3) − Figure 2. Four central elements of the full-rank b -quiver. Notation. Let(3.4) T i := •• n − i Y j =0 Z ( j,n − j − i,i ) •• denote products of cluster variables along SE-diagonals of the b n -quiver. Figure 3. Six mutually commuting elements K l (3.5) of the b -quiver; all theseelements commute with all variables of the A -quiver in Fig. 4 and setting all K l = 1results in the A ~ matrix of form (1.8). A n -quiver. We now construct the quiver corresponding to the system described by thereflection equation. Note that transposition results, in particular, in amalgamation of variables Z i, ,n − i and Z (0 ,n − i,i ) for 1 ≤ i ≤ n − 1. We indicate these amalgamations by dashed arrows inFig. 4. A more precise statement reads Lemma 3.3. • matrix entries of A depend only on the variables Z ( i,j,k ) with i, j, k > and amalgamated variables •• Z ( i, ,n − i ) Z (0 ,n − i,i ) •• , which are the variables of the reducedquiver , or A n -quiver, in Fig. 4, and on n variables K l , ≤ l ≤ n , depicted in Fig. 3: (3.5) K l = •• l Y i =1 Z ( i − ,n − l,l − i +1) Z l,n − l, n − l Y j =1 Z ( l,j − ,n − l − j +1) •• ; ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ • elements K l commute mutually and commute with all variables of the reduced quiver; • upon setting all K l = 1 the matrix A takes the form (1.8). We now unfreeze all amalgamated variables •• Z ( i, ,n − i ) Z (0 ,n − i,i ) •• . From now on, we set allelements K l defined in (3.5) equal the unity. This eliminates the dependence on all remainingfrozen variables and we remain with the reduced, or A n -quiver in Fig. 4. in which we indicateadditional amalgamations by dashed lines. Figure 4. The reduced quiver (the A n -quiver) obtained by amalgamating variablesof the b n -quiver and removing all remaining frozen variables. (The example in thefigure corresponds to b ) Variables of the same color constitute Casimirs (with unitpowers for all variables in every Casimir) depicted in Fig. 5. Lemma 3.4. [9] Casimirs of A n -quiver are (cid:2) n (cid:3) elements C i given by the formula: (3.6) C i = •• T i T n − i •• for ≤ i < n/ C n/ = T n/ if n/ is an integer , where T i are products (3.4) of variables of b n quiver; note that C i depend only on amalgamatedvariables. The example for n = 6 is depicted in Fig. 5. Since all Casimirs of A n are generated by λ -power expansion terms for det( A − λ A T ), weautomatically obtain a semiclassical statement that det( A − λ A T ) = P ( C , . . . , C [ n/ ), where C i are Casimirs of the A n -quiver. In the next section we explore this dependence. Figure 5. Three central elements for A -quiver. HARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 11 Solving characteristic equation The main theorem follows Theorem 4.1. Consider the A n -quiver in Fig. 4. Let λ ∈ C be an eigenvalue of the quantumoperator AA −† , i.e., the number at which the equation (cid:0) AA −† − λ I (cid:1) ψ = 0 has a nontrivial nullvector ψ ∈ V ⊗ W , where A is A ~ from (1.8). Then n admissible values of λ i are (4.1) λ i = ( − n − q − n × Q [ n/ k = i C k for ≤ i ≤ [ n/ for i = ( n + 1) / for odd n ; Q [ n/ k = n +1 − i C − k for n − [ n/ 2] + 1 ≤ i ≤ n Proof. We use representation (3.2) for the quantum matrix A ~ . Since all entries of allmatrices M i are self-adjoint, M † i = M T i , Q † = Q − and S † = S T = ( − n +1 S . Then(4.2) (cid:2) A ~ (cid:3) † = M T1 S T Q − M T3 M , so only the central block S T Q − M T3 sandwiched between M T1 and M is changed, and A ~ − λ (cid:2) A ~ (cid:3) † = M T1 (cid:0) M QS − λS T Q − M T3 (cid:1) M whereas the singularity equation becomes(4.3) (cid:0) M QS − λS T Q − M T3 (cid:1) ψ = 0for some nonzero vector ψ .The crucial observation is that both matrices: M QS and S T Q − M T3 in (4.3) are upper-anti-diagonal matrices ! All solutions to the singularity equation therefore correspond to thesituation where the combination of these matrices in (4.3) has zero element on the anti-diagonal(then the determinant of this combination becomes zero). The matrix M is lower-triangularwith the diagonal elements m = Z ( n, , , m i = •• Z ( n, , i − Y j =1 T i •• , ≤ i ≤ n, where T i are defined in (3.4). The matrices S T Q − M T3 = ⋆ ⋆ a n ⋆ ... 0 a and M QS = ⋆ ⋆ b n ⋆ ... 0 b then both are upper-anti-diagonal with(4.4) a i = ( − i +1 q i − / m i , b i = ( − n − i q − n + i − / m n +1 − i , and solutions of the singularity equation are λ i = b i /a i = ( − n − q − n m n +1 − i ( m i ) − Note that m i themselves are not Casimirs, but their quotients m n +1 − i ( m i ) − are: these quotientsare just expressions in cases in Theorem 4.1. This completes the proof. Remark 4.2. Note that representation (3.2) remains valid for any , not necessarily lower-triangular matrix M . This means that Theorem 4.1 remains valid in the case of generalquantum A -matrices, which may be not upper-triangular, provided they are presented in thecluster-variable decomposition form B † M QSB with B having the same commutation relationswith itself and with M as M . In particular, it holds for any (amalgamated) B = M QS F with F commuting with all M i and enjoying Lie–Poisson relations for its entries. ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ The factors ( − n − q − n in λ i in Theorem 4.1 are inessential and can be removed by a propernormalizaiton. A very interesting case is however when all λ i are different: in this case we havea full control over the Jordan form of AA −† , which becomes diagonal. Note also that since λ i are combinations of Casimirs, they are unit operators I in the quantum space, so we concludethat A ~ = U − Λ U, where Λ := n X i =1 λ i e i,i ⊗ I . Corollary 4.3. Assuming all λ i are distinct and we have a natural N such that λ Ni = λ Nj for all i, j (so all λ i are proportional to distinct N th roots of the unity), we have that h A ~ (cid:2) A ~ (cid:3) −† i N =const · I . 5. Sp m systems In this section, we find Casimirs for the matrix A having 2 × Sp m twisted Yangians (see [22]). Transport matrices for this systemmust have block-triangular form. A directed network corresponding to the case of 2 × m redundant sinks in thispicture and amalgamate variables of faces separated by the corresponding sinks. The resultingquiver is presented in the right part of the figure. × × × 12 34 561 ′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′ ′′ × × × Figure 6. In the left picture we present the Sp m directed network for m = 3. Weremove three sinks indicated in the light color and located at the NE side of the triangleand amalgamate cluster variables of faces separated by the corresponding directed edges(these amalgamations are denoted by dashed arcs). Note the appearance of (three)double arrows directed upward: the transport matrix ceases to be upper triangularand becomes only block-upper triangular. The resulting Sp m -quiver is presented inthe right picture. Constructing transport matrices M and M by the same rules as in Sec. 2.2 and composingthe block-upper triangular matrix A := M T1 M as in Sec. 3, we have to amalgamate frozenvariables on two sides of the Sp m quiver thus obtaining the quiver depicted in Fig. 7, whichwe call the A Sp m -quiver.Note that A Sp m -quiver resembles A n -quiver except for two rows on the right. We labelvariables of these rows by S l and Q r ; each variable S l has a corresponding variable Z ( l,n − l, inthe b n -quiver and Q r are new variables. HARACTERISTIC EQUATION FOR SYMPLECTIC GROUPOID AND CLUSTER ALGEBRAS 13 × × × S S S S S S S S Q Q Q abcde a b c d ef gk h × × × Figure 7. The A Sp m -quiver corresponding to block-upper triangular A in the caseof 2 × m = 3. Amalgamatedvariables are indicated by the same letters a , b , c , d , e , and S in the left figure. Inthe same figure we indicate cluster variables c , f , g , S , h , k stretched along the pathcorresponding to quasi-Casimir K = hkcf gS . Note that, e.g., K = edcbaS . In theright picture we present the quiver after amalgamation. We now find Casimirs for the A Sp m -quiver. Note, first, that Casimirs C i (3.6) remainCasimirs of the A Sp m -quiver. It was proved in [6] that for mk × mk block-triangular matrix A with blocks of size k × k we have (cid:2) mk (cid:3) Casimirs C i and exactly m (cid:2) k +12 (cid:3) new Casimirs dependingonly on matrix entries a i,j inside diagonal k × k blocks. For k = 2 we have exactly one suchCasimir per each diagonal block, and this Casimir is just the determinant of the corresponding2 × A Sp m -quiver.We first introduce quasi-Casimirs K l , l = 1 , . . . , m having the same content as Casimirs K l (3.5) in b n -quiver thus denoting by the same letter; we just replace Z ( l,n − l, by S l . An exampleof K is in the left picture in Fig. 7. Quasi-Casimirs K l commute with all cluster variablesexcept S i and Q r . Substituting now K l for all S l , we detach the part of the quiver containingonly variables K l and Q r ; this part looks as × × × K K K K K K Q Q Q It is easy to observe that, say, Q K K K Q is a Casimir (numbers in the figure above indicatepowers of the corresponding cluster variables). We therefore have the following statement. Lemma 5.1. Casimirs of A Sp m -quiver are m Casimirs C i (3.6) and m Casimirs R j , R j := Q j K j K j +1 K j +2 Q j +2 , with K l given by the same combinations (3.5) of cluster variables as in the case of b n -quiver. ∗ , MICHAEL SHAPIRO ∗∗ , AND HUANG SHIBO ∗∗∗ Concluding remarks In this paper, we have solved the problem of finding solutions to the eigenvalue problem( A − λ A † ) ψ = 0 for the quantum matrix A ~ from (1.8). Our solution was based on the cluster-variable realization of entries of A ~ found in [9]. We had also performed an extensive verificationof our results (in the semiclassical limit) using Mathematica computer code [30].Note that Remark 4.2 indicates that our results are by no means confined to a “triangle”Σ , , : they remain valid for any system undergoing the “form dynamics” described in the in-troduction. However, in the case of matrices A of a more general form, besides [ n/ 2] Casimirsgenerated by the same expression det( A − λ A T ) as in the upper-triangular case, we have addi-tional Casimir operators related to ratios of minors located in the lower-left corner of the matrix A (see, e.g., [6]). We constructed the corresponding Casimirs for the A Sp m -quiver correspond-ing to matrices A of block-upper triangular form with blocks of size 2 × 2. This however doesnot change the results concerning the Jordan form of AA −† and the corresponding symmetriesbecause for every such system, A = M T1 M M with the same matrix M . 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