Quantization of deformed cluster Poisson varieties
Man-Wai Mandy Cheung, Juan Bosco Frías-Medina, Timothy Magee
aa r X i v : . [ m a t h . QA ] J u l Quantization of deformed cluster Poisson varieties
Man-Wai Mandy Cheung, Juan Bosco Fr´ıas-Medina, and Timothy Magee
Abstract
Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Pois-son varieties ) in [FG09]. Meanwhile, families of deformations of cluster X -varieties were introduced in[BFMN18]. In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of X -varieties to the families of [BFMN18]. As a corollary, we obtain that thesefamilies and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A -varieties ([BZ05]). Finally, inspired by the counter-example to quantumpositivity of the quantum greedy basis in [LLRZ14], we compute a counter-example to quantum positivityof the quantum theta basis. Cluster varieties come in two flavors, A and X . Both types are a union of tori glued via a special class ofbirational maps, known as mutation maps . The input of this construction is a collection of combinatorialdata Γ, which we review in Section 2.1. The output is a pair of schemes ( A Γ , X Γ ), together with maps p : A Γ → X Γ . Up to a certain rescaling of character lattices, the tori in the atlas of A Γ and the tori in theatlas of X Γ are dual to one another. Replacing Γ with Langlands dual data Γ ∨ (reviewed in Section 2.1)gives a new pair ( A Γ ∨ , X Γ ∨ ) where this duality is precise– the tori in the atlas of A Γ are dual to the tori inthe atlas of X Γ ∨ , and the tori in the atlas of X Γ are dual to the tori in the atlas of A Γ ∨ .While closely related, A - and X -varieties differ in some important ways. A -varieties exhibit the Laurentphenomenon ([FZ02]) and as a result come with a natural collection of linearly independent global regularfunctions ( theta functions ). A consequence of this fact is that A -varieties naturally (partially) compactifywith projective geometry constructions. These constructions are reminiscent of the construction of (relative)projective toric varieties via convex polyhedra. On the other hand, X -varieties are generally non-separatedand so cannot possibly be compactified in this way. Instead, they come with an atlas naturally described interms of a fan, and partially compactify in much the same way that a fan defines a toric variety partiallycompactifying a torus. In fact, a single object ( a scattering diagram ) encodes both the theta functions on A Γ and this partial compactification (known as the special completion ) b X Γ ∨ of X Γ ∨ . The special completionwas initially explored by Fock and Goncharov in [FG16].In both cases, the partial compactifications can be put into families by introducing coefficients to the mu-tations. For A , these coefficients already appeared in the first work on cluster algebras [FZ02], and theresulting toric degenerations were explored in [GHKK18]. For X , the coefficients and corresponding toricdegenerations were given in [BFMN18].Next, Fock and Goncharov describe in [FG09] a quantization X q of the X -variety. In fact, they show that X -varieties have a Poisson structure– one of their fundamental features– by taking a semi-classical limit of X q . Meanwhile, Berenstein and Zelevinsky address the quantization of A varieties in [BZ05]. The Berenstein-Zelevinsky A -quantization admits coefficients, and thus admits families of quantum A -varieties.1n this paper we show that the quantization of cluster Poisson varieties due to Fock and Goncharov ([FG09])extends to the family of cluster Poisson varieties with coefficients of [BFMN18]. To do so, we introducecoefficients to the quantum mutation formulas given by Fock and Goncharov in [FG09, Section 3.3.1] andobtain Proposition 1.1 (Proposition 3 . . The quantum X -mutation with coefficients µ qk, t ; G : K (cid:16) A | I | M ; G ′ ,q ( R ) (cid:17) → K (cid:16) A | I | M ; G ,q ( R ) (cid:17) is given in cluster coordinates by µ qk, t ; G (cid:16) e X i ; G ′ (cid:17) = e X − i ; G if i = k, e X i ; G | ǫ ik | Y ℓ =1 (cid:16) t [sgn ( ǫ ik ) c k ; G ] + + t [ − sgn ( ǫ ik ) c k ; G ] + q ℓ − k e X − sgn ( ǫ ik ) k ; G (cid:17) − sgn ( ǫ ik ) if i = k. . Setting the coefficients t i to , we recover the usual Fock-Goncharov quantum X -mutation. Meanwhile, takingthe quantum parameter q to , we recover the classical X -mutation formula with coefficients of [BFMN18]. We further obtain a Poisson structure on the family c X of [BFMN18] and its fibers through a semi-classicallimit: Proposition 1.2 (Proposition 5 .
3, Corollary 5 . . c X is a Poisson scheme over C [ t i : i ∈ I ] , and its fibersover closed points are Poisson schemes over C . Finally, we show that this quantization of X -varieties with coefficients is compatible with the Berenstein-Zelevinsky quantization of A -varieties with coefficients. Proposition 1.3 (Proposition 4 . . With the identification q FG = q − lcm( d i : i ∈ I uf )BZ , a p ∗ -map on the levelof character lattices of cluster tori induces a ∗ -algebra homomorphism from the quantum X -torus algebrawith coefficients to the quantum A -torus algebra with coefficients which commutes with mutation. That is,it induces a map from the Berenstein-Zelevinsky quantum A -variety with coefficients to the Fock-Goncharov X -variety with coefficients. To complete the story, we highlight a key difference between the classical and quantum settings. Specifically,inspired by an example of [LLRZ14] for the quantum greedy basis, we illustrate a wall in a rank two quantumscattering diagram whose scattering function is non-positive. This leads to the failure of “quantum positivity”for the quantum theta basis, which is expected from the corresponding result for the quantum greedy basisin [LLRZ14].
Structure of the paper
Section 2 is an extended background section, split into several parts. First, we recall basic notions for(classical) cluster varieties and cluster varieties with principal coefficients in Section 2.1. This is followed bya review of the formulation of scattering diagrams in Section 2.2. We then move on to the quantum settingfor both X and A cluster algebras in Section 2.3. We conclude the background section with a sketch of theconstruction of quantum scattering diagrams in Section 2.4, where we also provide an example of a quantumscattering diagram which has wall function with a non-positive coefficient– a key difference from the classicalcase. The detail computation used in this example is provided in Appendix B.2n Section 3, we construct the quantization of the family c X . For the convenience of the reader, we reviewsome properties of the quantum affine algebra in Section 3.1. The quantum mutation of the c X family isintroduced in Section 3.2.Section 4 is devoted to relating quantum X -mutation with coefficients (Section 3) with Berenstein-Zelevinsky’squantum A -mutation and Fomin-Zelevinsky’s A -mutation with coefficients. We focus on the case of principalcoefficients.Using the quantum family c X q and taking the classical limit q →
1, we prove in Section 5 that the family c X has a Poisson structure, as do its fibers. Finally, for the reader who wishes to see a more elementary proofof this result, in Appendix A we provide a direct proof of the Poisson structure of the c X family withoutpassing through quantization. Before going into the quantum setting, we recall the definition of cluster algebras and cluster varietiesfollowing [GHK15] and [GHKK18]. We start with the fixed data Γ: • a finite set I of directions with a subset of unfrozen directions I uf ; • a lattice N of rank | I | ; • a saturated sublattice N uf ⊆ N of rank | I uf | ; • a skew-symmetric bilinear form {· , ·} : N × N → Q ; • a sublattice N ◦ ⊆ N of finite index satisfying { N uf , N ◦ } ⊂ Z and { N, N uf ∩ N ◦ } ⊂ Z ; • a tuple of positive integers ( d i : i ∈ I ) with greatest common divisor 1; • M = Hom( N, Z ) and M ◦ = Hom( N ◦ , Z ).Then a seed is a tuple s = ( e i ; s ∈ N : i ∈ I ) such that { e i ; s : i ∈ I } is a basis of N , { e i ; s : i ∈ I uf } is a basisof N uf , and { d i e i ; s : i ∈ I } is a basis N ◦ . Define the matrix ˆ ǫ ij = { e i ; s , e j ; s } . The exchange matrix is definedas ǫ = ( ǫ ij ) , where ǫ := { e i ; s , e j ; s } d j . We denote the dual basis to s by (cid:8) e ∗ i ; s : i ∈ I (cid:9) , and we set f i ; s = d − i e ∗ i ; s . Observe that { f i ; s : i ∈ I } is abasis for M ◦ . For r ∈ R , define [ r ] + = max(0 , r ). Given seed data s and k ∈ I uf , the mutation s ′ of s is anew basis e i ; s ′ := ( e i ; s + [ ǫ ik ] + e k ; s for i = k, − e k ; s for i = k. (1)Note that { e i ; s ′ : i ∈ I uf } is still a basis for N uf and { d i e i ; s ′ : i ∈ I } is still a basis for N ◦ . The new basis { f i ; s ′ : i ∈ I } of M ◦ is obtained as the dual of { d i e i ; s ′ : i ∈ I } . We can now associate two tori X s = T M = Spec C [ N ] and A s = T N ◦ = Spec C [ M ◦ ] . (2)Write { X i : i ∈ I } for the coordinates on X s corresponding to the basis vectors { e i ; s : i ∈ I } , i.e. X i = X e i ; s ,and similarly write { A i : i ∈ I } for the coordinates on A s corresponding to the basis vectors { f i ; s : i ∈ I } ,3.e. A i = A f i ; s . The coordinates X i , A i are called cluster variables . These coordinates give identifications X s → ( C ∗ ) | I | , A s → ( C ∗ ) | I | . We define the birational maps (“ mutation maps ”) µ k : X s X s ′ , µ k : A s A s ′ . via pull-back of functions µ ∗ k ( X n ) = X n (1 + X e k ; s ) −{ n,e k ; s } d k for n ∈ N,µ ∗ k ( A m ) = A m (1 + A v k ; s ) −h d k e k ; s ,m i for m ∈ M ◦ . In terms of cluster variables, these mutation maps may be rewritten as µ ∗ k ( X i ; s ′ ) = X − k ; s for i = kX i ; s (cid:16) X − sgn( ǫ ik ) k ; s (cid:17) − ǫ ik for i = k (3)and µ ∗ k ( A i ; s ′ ) = ( A − k ; s (cid:16)Q j : ǫ kj > A ǫ kj j ; s + Q j : ǫ kj < A − ǫ kj j ; s (cid:17) for i = kA i ; s for i = k . (4)The A and X cluster varieties associated to Γ are the schemes A Γ := [ s A s / ∼ where tori are glued via the birational maps (4), and X Γ := [ s X s / ∼ where tori are glued via the birational maps (3). See [GHK15, Proposition 2.4] and the discussion followingit for further details. Remark . When there is no risk of confusion, we will generally drop the subscript Γ and simply write A and X .The A - and X - cluster algebras are the spaces of regular functions Γ( A , O A ) and Γ( X , O X ) on these clustervarieties. Remark . There is a notion of duality for the fixed data Γ and seed s . The Langlands dual fixed data Γ ∨ is defined by setting I ∨ := I, I ∨ uf := I uf , N ∨ = N ◦ , { · , · } ∨ := 1lcm ( d i : i ∈ I ) { · , · } , ( N ∨ ) ◦ := lcm ( d i : i ∈ I ) · N, and d ∨ j := lcm ( d i : i ∈ I ) d j . The
Langlands dual seed is s ∨ := ( d i e i ; s : i ∈ I ). Naturally, this Langlands dual data determines a newpair of cluster varieties: X Γ ∨ and A Γ ∨ are said to be the Fock-Goncharov duals of A Γ and X Γ . See [FG09,Section 1.2.10].For both A and X , the mutation relations may be generalized to allow coefficients in a semi-field. Thiswas treated for A in [FZ02] and for X in [BFMN18]. Importantly, the A and X mutations with coefficientsremain dual to one another in a precise sense: see [BFMN18, Section 3.3]. An instance of this constructionof particular interest– and the setting of the article– is the case of principal coefficients . We review clustervarieties with principal coefficients in the following subsection.4 .1.1 Cluster varieties with principal coefficients We first treat principal coefficients for A -varieties following [GHK15]. Let Γ be the fixed data and s the seeddata of a (coefficient-free ) pair of cluster varieties ( A , X ). Now consider the lattice and skew-symmetricbilinear form e N := N ⊕ M ◦ , { ( n , m ) , ( n , m ) } prin := { n , n } + h n , m i − h n , m i . Take e I = I ⊔ I with e I uf the inclusion of I uf into the first copy of I , e N uf = N uf ⊂ e N , e N ◦ = N ◦ ⊕ M , and e d i = d i for i in either copy of I . This defines new fixed data e Γ. Meanwhile, from the seed s we obtain a seed e s as (( e i ; s ,
0) : i ∈ I ) ∪ ((0 , f i ; s ) : i ∈ I ). This data defines an A -variety denoted A prin closely related to theoriginal A -variety. In fact, denoting e A (0 ,f i ; s ) by t i , A prin may be viewed as a family of deformations of A over Spec (cid:0) C (cid:2) t ± i : i ∈ I (cid:3)(cid:1) . Moreover a choice of seed s determines an extension of A prin to a family A prin , s over Spec ( C [ t i : i ∈ I ]). Explicitly, mutation of A -variables takes the form µ ∗ k (cid:16) e A i ; s ′ (cid:17) = ( e A − k ; s (cid:16) t [ c k ; s ] + Q j : ǫ kj > e A ǫ kj j ; s + t [ − c k ; s ] + Q j : ǫ kj < e A − ǫ kj j ; s (cid:17) if i = k e A i ; s if i = k . (5)Here c k ; s is a c -vector . If I and I are the first and second copies of I in e I , then c k ; s is the k th row of thesubmatrix e ǫ I × I of the exchange matrix e ǫ .Next, the X -mutation with principal coefficients defining the corresponding family of deformations X of X from [BFMN18] is given by µ ∗ k (cid:16) e X i ; s ′ (cid:17) = e X − i ; s for i = k e X i ; s (cid:16) t [sgn( ǫ ik ) c k ; s ] + + t [ − sgn( ǫ ik ) c k ; s ] + e X − sgn( ǫ ik ) k ; s (cid:17) − ǫ ik for i = k . (6)Note that X is not the scheme X prin of [GHKK18]. The difference stems from treatment of coefficients andfrozen variables. In the skew-symmetric case, X is dual to A prin as cluster varieties over Spec C (cid:2) t ± i : i ∈ I (cid:3) ,while X prin is dual to A prin as cluster varieties over C . Remark . As mentioned previously, the mutation relations with principal coefficients (5) and (6) areboth special case of more general mutation relations with coefficients in an arbitrary semifield P . The A mutation with coefficients in P was described by Fomin and Zelevinsky in [FZ02], while the X -mutationswith coefficients in P was given in [BFMN18]. For reference, these more general relations take the form: µ ∗ k (cid:16) e A i ; s ′ (cid:17) = ( e A − k ; s (cid:16) p + k Q j : ǫ kj > e A ǫ kj j ; s + p − k Q j : ǫ kj < e A − ǫ kj j ; s (cid:17) if i = k e A i ; s if i = k (7)and µ ∗ k (cid:16) e X i ; s ′ (cid:17) = e X − i ; s for i = k e X i ; s (cid:16) p J ǫ ik K k + p J − ǫ ik K k e X − sgn( ǫ ik ) k ; s (cid:17) − ǫ ik for i = k , (8)where p + := pp ⊕ , p − := 1 p ⊕ , and p J x K := p − if x < , x = 0 ,p + if x > . We distinguish frozen variables and coefficients, so I may have frozen directions. The distinction is not so important for A ,but it is significant for X and affects the notion of cluster duality. When the exchange matrix is only skew-symmetrizable, we also need to take the Langlands dual data to obtain the dualcluster variety with principal coefficients. .2 Scattering diagrams The geometry and combinatorics of cluster varieties and their associated cluster algebras can be encodedusing scattering diagrams, as formulated in [GHKK18]. In this section, we review the definition of scatteringdiagrams over skew-symmetric Lie algebras as in [KS14], [GHKK18], [CM19], [DM19]. We then use thismachinery to discuss quantum scattering diagrams in Section 2.4.Consider a finite rank lattice L with a skew-symmetric form {· , ·} . Let L ∨ be the dual lattice. Choose astrictly convex rational polyhedral cone C in L R = L ⊗ R . Let L ⊕ = C ∩ L and L + = L ⊕ \ { } . Now let g be an L + -graded Lie algebra: g = M n ∈ L + g n , [ g n , g n ] ⊂ g n + n . (9)Assume g is skew-symmetric with respect to {· , ·} , meaning [ g n , g n ] = 0 whenever { n , n } = 0. Choose alinear function d : L → Z such that d ( n ) > n ∈ L + . Then define g >k := M d ( n ) >k g n , which is a Liesubalgebra of g , and let g ≤ k = g / g >k . For each n ∈ L + , define g k n := M k ≥ g kn . We have corresponding Liegroups G k := exp g ≤ k , G := exp g = lim ←− G k , and G k n := exp( g k n ) ⊂ G . Definition 2.4. A wall in L ∨ R over g is a pair ( d , g d ) such that • g d ∈ G k n d for some primitive n d ∈ L + . • d ⊂ L ∨ R is a closed, convex (but not necessarily strictly convex), rational-polyhedral, codimension-oneaffine cone in L ∨ R , parallel to n ⊥ d . We call d the support of the wall.A scattering diagram D over g is a set of walls in L ∨ R over g such that for each k >
0, there are only finitelymany ( d , g d ) ∈ d with g d not 1 in G k . If ( d , g d ) and ( d , g d ) are two walls of d , and if codim L ∨ R ( d ∩ d ) = 1,then we require that [ g d , g d ] = 0. The singular locus of the scattering diagram is defined asSing( D ) = [ d ∈ D ∂ d ∪ [ d , d ∈ D dim d ∩ d = n − d ∩ d , Now consider a smooth immersion γ : [0 , → L ∨ R \ Sing( D ), with endpoints not contained in the support of D , that only crosses walls transversely. For each k > γ will cross only a finite number s k of walls, and welabel them by ( d i , g d i ), i = 1 , . . . , s k . Let t i be the time at which γ crosses the wall d i For this k , we define p kγ = g sgn( h n d s , − γ ′ ( t s ) i ) d s ◦ · · · ◦ g sgn( h n d , − γ ′ ( t ) i ) d , and taking the limit k → ∞ we define the path-ordered product p γ = lim k →∞ p kγ ∈ G .Two scattering diagram D and D ′ are said to be equivalent if p γ, D = p γ, D ′ for all paths γ for which bothare defined. A scattering diagram is consistent if p γ, D only depends on the endpoints of γ for any path γ for which p γ, D is defined.Versions of the following fundamental result have been proved in various contexts and degrees generality byvarious authors. Proposition 2.5 ([KS06], [GS11], [KS14], [GHKK18], [Bri17]) . Let g be a skew-symmetric L + -graded Liealgebra, and let D in be a finite scattering diagram whose walls are of the form ( n ⊥ i , g i ) for primitive n i ∈ L + .Then there is a unique-up-to-equivalence scattering diagram D such that D is consistent, D ⊃ D in . . Restricting to the cluster setting and following [GHKK18, Section 1], consider the fixed data Γ reviewed inSection 2.1. We need to use Γ and a choice of seed s = ( e i : i ∈ I ) to construct a graded Lie algebra as in(9).Naturally, the lattice N , which comes equipped with the skew-form {· , ·} , replaces the lattice L from thegeneral discussion. Now define N + s := ( X i ∈ I uf a i e i : a i ≥ , X i ∈ I uf a i > ) . (10)This plays the role of L + from the general discussion.Define the map p ∗ : N uf → M ◦ n
7→ { n, · } . In order to construct the cluster scattering diagram, [GHKK18] require p ∗ to be injective. Using theinjectivity assumption, we can choose a strictly convex top-dimensional cone σ ⊆ M ◦ R with associated monoid P := σ ∩ M ◦ such that p ∗ ( e i ) ∈ P \ { } for all i ∈ I uf . Denote by [C [ P ] the completion of C [ P ] with respectto the monomial ideal generated by P \ { } . Now consider the Lie algebra C [ P ] ⊗ Z N ◦ , with Lie bracketgiven by [ A m ⊗ n , A m ⊗ n ] = A m + m ⊗ ( h n , m i n − h n , m i n ) . Let it act on C [ P ] as follows: ( f ⊗ n )( A m ) = h n, m i f A m . Define g n to be the one-dimensional subspace of C [ P ] ⊗ Z N ◦ spanned by A p ∗ ( n ) ⊗ n . Then the Lie subalgebra g ⊂ C [ P ] ⊗ Z N ◦ is defined to be g := M n ∈ N + s g n . Observe that g is skew-symmetric with respect to {· , ·} . We can again choose a linear function d : N → Z such that d ( n ) > n ∈ N + s . We then define g >k , g ≤ k , g k n , and the corresponding Lie groups G k , G ,and G k n precisely as before.All that remains is to give an initial scattering diagram. For n ∈ N + s and f ∈ [C [ P ] of the form f = 1 + ∞ X k =1 c k A kp ∗ ( n ) , define p f ∈ G k n by p f ( A m ) = f h n ′ ,m i A m This is known as the injectivity assumption . While it may seem limiting at first glance, this always holds after introducingprincipal coefficients, which we reviewed in Section 2.1.1. [GHKK18] use A prin to draw conclusions about A (which is a fiberof A prin ) and X (which is a quotient of A prin by a torus action). n ′ is the generator of R ≥ · n ∩ N ◦ . Assuming the injectivity assumption is satisfied (so the scatteringdiagram may be defined), the initial walls of the cluster scattering diagram are n(cid:0) e ⊥ i , p f d i (cid:1) : i ∈ I uf , f d i = 1 + A p ∗ ( e i ) o . As the injectivity assumption is always satisfied after introducing principal coefficients, we can always workon the level of A prin to write: D A prin in , s := n(cid:0) ( e i , ⊥ , p f d i (cid:1) : i ∈ I uf , f d i = 1 + e A p ∗ (( e i , o . By Proposition 2 .
5, there exists a scattering diagram D A prin containing D A prin in , s . This is the cluster scatteringdiagram for A prin . [GHKK18, Construction 7.11] gives precisely how this is related to A and X , allowing therecovery of a scattering diagram for X as a slice of the A prin scattering diagram and a similar combinatorialstructure for A whose support is a quotient of the support of the A prin scattering diagram. An overviewappears in [CMN19, Section 2.2.1 and 2.2.2].The follow result about cluster scattering diagrams plays an important role in this paper. Theorem 2.6 ([GHKK18, Lemma 2.10]) . Let Γ be fixed data for which the injectivity assumption is satisfied,let D be the associated cluster scattering diagram, and let M ◦ R be the ambient space of the support of D . Then D induces a chamber decomposition of a full dimensional subset of M ◦ R , yielding a (usually infinite) simplicialgeneralized fan ∆ + whose maximal cones are in 1-1 correspondence with clusters. Fan structure and cluster varieties
The fan ∆ + is known as the cluster complex . If there are no frozen directions, the rays of ∆ + correspondprecisely to cluster variables. Each ray is generated by the g -vector of a cluster variable, a notion which wewill now review. Let p ∗ : N → M ◦ /N ⊥ uf be the lattice map defined by p ∗ ( n ) = { n, · }| N uf ∩ N ◦ , and denote the projection M ◦ → M ◦ /N ⊥ uf by π . Now let p ∗ : N → M ◦ be any lattice map satisfying • p ∗ | N uf = p ∗ ; • π ◦ p ∗ = p ∗ .The inclusion N ◦ ֒ → N ◦ ⊕ Mn ( n, p ∗ ( n ))induces a T N ◦ action on A prin . After fixing an initial seed s , every cluster variable A i ; s on A has a canonicalextension e A i ; s to A prin , and e A i ; s is an eigenfunction of this T N ◦ action. The g -vector of e A i ; s at s is theweight of e A i ; s under this T N ◦ action. See [GHKK18, Section 5] for further details.Even if the injectivity assumption is not satisfied for Γ, it is for e Γ and projecting from f M ◦ R to M ◦ R once againyields a simplicial generalized fan ∆ + ([GHKK18, Theorem 2.13]). This is still called the cluster complex . In general the “scattering functions” of this structure may not have the same form as scattering functions of a scatteringdiagram. The cones are convex, but not strictly convex if there are frozen directions.
8e obtain an honest simplicial fan (as opposed to generalized fan) by defining the maximal cone associatedto a cluster to be the R ≥ -span of the g -vectors of the variables (both frozen and unfrozen) of that cluster.In this way each cone is contained in a chamber of ∆ + , and the integral points of maximal cones correspondthe monomials in the variables of the corresponding cluster. We call the resulting fan Σ the g -vector fan .We can avoid certain redundancies in the atlas of A by indexing torus charts by maximal cones of Σ ratherthan seeds, as multiple seeds may yield the same unordered cluster coordinates. In this language, we have A Γ = [ G∈ Σ(max)
Spec ( C [( G ∩ M ◦ ) gp ]) / ∼ , where ( G ∩ M ◦ ) gp is the group completion of the monoid G ∩ M ◦ .While Σ is clearly closely related to A Γ , it also has a strong connection to X Γ ∨ . Observe that Σ is a fan in M ◦ R ,and M ◦ is the cocharacter lattice of cluster tori in X Γ ∨ . In fact, X -cluster variables on X Γ ∨ can be associatedto c -vectors (Section 2.1.1) in much the same way that A -variables on A Γ are associated to g -vectors.(See [BFMN18, Definition 5.10].) These c -vectors { c i ; s : i ∈ I } associated to a cluster { X i ; s ∨ : i ∈ I ∨ = I } generate the dual cone to G := span R ≥ { g i ; s : i ∈ I } . (See [NZ12, Theorem 1.2].) We can once again indexby maximal cones of Σ rather than seeds to avoid redundancies in the atlas of X Γ ∨ . In these terms, X Γ ∨ isthe scheme X Γ ∨ = [ G∈ Σ(max)
Spec (cid:0) C (cid:2) ( G ∨ ∩ N ◦ ) gp (cid:3)(cid:1) / ∼ , where G ∨ is the dual cone of G and ( G ∨ ∩ N ◦ ) gp is the group completion of the monoid G ∨ ∩ N ◦ . Observethat this description is highly reminiscent of the construction of toric varieties in terms of a fan. There aretwo key differences: the gluing maps and the group completion. The gluing maps are what put us in thecluster world to begin with, and if we were to replace the mutation maps with toric gluing maps the endresult would simply be a torus. This suggests a very natural partial compactification of X Γ ∨ , analogous toa toric variety partially compactifying a torus: b X Γ ∨ = [ G∈ Σ(max)
Spec ( C [ G ∨ ∩ N ◦ ]) / ∼ . The scheme b X Γ ∨ is known as the special completion of X Γ ∨ , and it was introduced by Fock and Goncharovin [FG16]. Remark . Note that Langlands duality is an involution, so if we were to start with Γ ∨ we would obtainthe special completion b X Γ . We will generally make use of this fact to avoid continually writing “Langlandsdual” and ∨ . Remark . From Section 3 on we will make use of the discussion in this section to index by maximal cones G of Σ rather than by seeds s . For instance, we will denote an X -cluster on X by n e X i ; G : i ∈ I o . Here we review the quantization of X by Fock and Goncharov ([FG09]), and the quantization of A by Beren-stein and Zelevinsky ([BZ05]). X q (respectively A q ) is a noncommutative q -deformation of X (respectively A q ), where tori are replaced by quantum tori (which are ∗ -algebras) and the mutation maps are ∗ -algebraisomorphisms of the noncommuative fraction fields of the quantum tori. X -varieties We again begin with fixed data Γ and a seed s (Section 2.1). Consider the noncommutative torus T N := C [ q ± d ] h X n : n ∈ N, X n · X n ′ = q { n,n ′ } X n + n ′ i . q { n ′ ,n } X n · X n ′ = q { n,n ′ } X n ′ · X n . (11)In particular, if we denote X i := X e i , then T N is a ∗ -algebra over C [ q ± d ], with the involutive antiautomor-phism ∗ : T N → T N defined by ∗ ( X n ) = X n , ∗ ( q ) = q − . (12) Quantum dilogarithm
To construct a non-commutative deformation of the space X , the quantum space X q , Fock and Goncharov[FG09] consider the quantum dilogarithm defined by Ψ q ( x ) = ∞ Y ℓ =1 (cid:0) q ℓ − x (cid:1) − . Write Li ( x ; q ) := X ℓ ≥ x ℓ ℓ ( q ℓ − q − ℓ ) . (13)One can check that Ψ q ( x ) = exp ( − Li ( − x ; q )) . (14)See [FG09, Section 3.2] for further details.For each seed s we will have a copy of T N , which we denote T N ; s . Since T N ; s is an Ore domain , we cantake its noncommutative fraction field T N ; s . The quantum mutation map µ qk , k ∈ I uf , is the isomorphism µ qk : T N ; s ′ → T N ; s given by conjugation by Ψ q /dk ( X k ; s ), µ qk := Ad Ψ q /dk ( X k ; s ) . Fock and Goncharov [FG09] showed that µ qk is a ∗ -algebras homomorphism, which follows from the factthat the equality Ψ q − ( x ) = Ψ q ( x ) − holds. They then decompose µ qk and express it in terms of clustervariables. That is, they write µ qk = µ ♯k ◦ µ ′ k , where µ ♯k := Ad Ψ q /dk ( X k ) : T N ; s → T N ; s , and µ ′ k : T N ; s ′ → T N ; s relates the coordinates associated to thebasis s ′ of N with the coordinates associated to the basis s . Explicitly, µ ′ k : X i ; s ′ = X e i ; s ′ X e i ; s ′ = X e i ; s +[ ǫ ik ] + e k ; s = q − ˆ ǫ ik [ ǫ ik ] + X e i ; s X [ ǫ ik ] + e k ; s = q − ˆ ǫ ik [ ǫ ik ] + X i ; s X [ ǫ ik ] + k ; s . One can compute µ ♯k more explicitly as X i ( X i (1 + q /d k X k )(1 + q /d k X k ) · · · (1 + q (2 | ǫ ik |− /d k X k ) ǫ ik ≤ ,X i (1 + q − /d k X k ) − (1 + q − /d k X k ) − · · · (1 + q (1 − | ǫ ik | ) /d k X k ) − ǫ ik ≥ . (15)Finally, in terms of cluster coordinates, we have µ qk ( X i ; s ′ ) = X − i ; s if i = kX i ; s (cid:16)Q | ǫ ik | ℓ =1 (cid:16) q (2 ℓ − /d k X − sgn ǫ ik k ; s (cid:17)(cid:17) − sgn( ǫ ik ) if i = k. (16) A domain A is a right Ore domain if for each nonzero elements x, y ∈ A , there exist r, s ∈ A such that xr = ys = 0. .3.2 Quantum A -cluster algebras Now we recall Berenstein and Zelevinsky’s treatment of quantum A -cluster algebras from [BZ05]. Weillustrate how one can translate between the two frameworks in Section 4.Let Λ be an | I | × | I | skew-symmetric integer matrix, and let e B be an | I | × | I uf | integer matrix. The pair (cid:16) Λ , e B (cid:17) is called compatible if e B T Λ = (cid:0) D (cid:1) , for a diagonal matrix D . Berenstein and Zelevinsky show([BZ05, Proposition 3.3]) that the pair (Λ , e B ) being compatible implies e B is of full rank and the principal | I uf | × | I uf | submatrix B is skew-symmetrizable by noting DB = e B T Λ e B . A compatible pair is the input forthe Berenstein-Zelevinsky quantum A -cluster algebra construction. We again consider a quantum torus ∗ -algebra. In Section 2.3.1, the cluster coordinates on T N ; s were expo-nentials of the elements of the basis s = { e i ; s : i ∈ I } of N , and the q -commutation relations were definedin terms of the skew form {· , ·} on N . In this case the cluster variables are exponentials of { f i ; s : i ∈ I } , abasis for M ◦ . We view Λ as a skew form on M ◦ and define T M ◦ as T M ◦ := C [ q ± ] h A m : m ∈ M ◦ , A m · A m ′ = q Λ ( m,m ′ ) A m + m ′ i . This is a ∗ -algebra over C [ q ± ], with ∗ ( A m ) = A m and ∗ ( q ) = q − . Again, we have a copy T M ◦ ; s for eachseed s , and we denote the noncommutative fraction field of T M ◦ ; s by T M ◦ ; s . We denote A f i ; s ∈ T M ◦ ; s by A i ; s . The quantum A -mutation defining A q is the ∗ -algebra isomorphism T M ◦ ; s ′ → T M ◦ ; s given by µ qk ( A i ; s ′ ) = ( A − f k ; s + P j : ǫkj> ǫ kj f j ; s + A − f k ; s − P j : ǫkj< ǫ kj f j ; s for i = kA i ; s for i = k . See [BZ05, Proposition 4.9].
In this section, we will outline how to construct the quantum version of scattering diagrams for X as in[GHKK18, arXiv version 1, Construction 1.31] and [Man15]. We have outlined how to obtain consistentscattering diagrams from skew-symmetric Lie algebras in Section 2.2. Thus we only need to merge thequantum algebras into the formulation. Gross-Hacking-Keel-Kontsevich define the following Lie bracket on T N : [ X, Y ] =
Y X − XY.
It is straightforward to check that this indeed satisfies the properties of a Lie bracket using (11). ˆ g willbe a Lie subalgebra of T N with this Lie bracket. To define it, Gross-Hacking-Keel-Kontsevich first take amultiplicative subset S of C [ q ± /d ], defined as the complement of the union of the two prime ideals ( q /d − q /d + 1). They then take C q to be the localization of C [ q ± /d ] by S : C q := S − C [ q ± /d ]. Then b g isdefined to be the free C q -submodule of T N with basis n b X n := X n q − q − (cid:12)(cid:12)(cid:12) n ∈ N + s o . (See (10) for the definitionof N + s .) This is in fact a Lie subalgebra as[ b X n , b X n ′ ] = q { n ′ ,n } − q { n,n ′ } q − q − b X n + n ′ , (17)and q { n ′ ,n } − q { n,n ′} q − q − ∈ C q . Observe that b g is a skew-symmetric (with respect to [ · , · ]) N + s -graded Lie algebra,and in order to apply Proposition 2 . Note in particular that while we can always quantize an X -variety, if the exchange matrix e B (which is ǫ tI uf × I in theFock-Goncharov formalism) is not full rank, then A does not admit a quantization by this procedure. P ⊂ N as in the classical cluster scattering diagrams construction. Then elements of b G = exp( b g ) act on \C [ q ± /d ][ P ] by conjugation, i.e. exp( g ) : X n exp( − g ) X n exp( g ). Recall the functionLi ( x ; q ) defined as in (13). Note that in particular Li ( − X e i ; q /d i ) ∈ lim ←− b g / b g >k . Indeed the conjugation by Ψ q /di can be expressed as Ψ q /di ( X e i ) − X e j Ψ q /di ( X e i )= ( (1 + q /d i X e i )(1 + q /d i X e i ) · · · (1 + q (2 { e j ,e i } d i − /d i X e i ) X e j { e j , e i } > q − /d i X e i ) − (1 + q − /d i X e i ) − · · · (1 + q (2 { e j ,e i } d i +1) /d i X e i ) − X e j { e j , e i } ≤ µ ♯k in Fockand Goncharov’s setting.Now consider the scattering diagram b D in , s := (cid:8)(cid:0) v ⊥ i , Ψ q /di ( X e i ) (cid:1) | i ∈ I uf (cid:9) where v i = p ∗ ( e i ). Then by Proposition 2 .
5, there is a unique (up to equivalence) consistent scatteringdiagram b D s containing b D in , s .For the A -cluster algebra, if the compatible pair exists, one can put Λ as the skew-symmetric form in theconstruction. Then one can similarly define the quantum A scattering diagram. Instead of repeating thewhole construction, we will demonstrate the quantum A scattering diagram as an example in the nextsubsection.We will show in Section 4 that the Berenstein-Zelevinsky quantization and the Fock-Goncharov quantizationare compatible via a p ∗ map. Then one can deduce the A and X scattering diagrams from the A prin scatteringdiagram as in the classical case. However, unlike the classical case, if a compatible pair does not exist forthe A -algebra, one cannot restrict the quantum A prin theta functions to obtain quantum A theta functions–which do not exist. The issue is that in this case the new “principal coefficient” variables that have beenintroduced will not commute with the original cluster variables. To recover the A -algebra, we would haveto specialize the principal coefficient variables to 1, at which point their failure to commute with the originalcluster variables would yield relations of the form A i = q c A i for some c = 0. While the construction of quantum scattering diagrams is very similar to the classical construction, thereis an important difference between the classical and quantum settings which we would like to highlighthere. In [Dav18], Davison proved the positivity conjecture for skew-symmetric quantum A -cluster algebras: every cluster variable is a Laurent polynomial in the monomials of any other cluster, with coefficients in Z ≥ [ q ± ]. He has since taken this further with Mandel. In [DM19] they show the strong positivity conjecture for quantum theta bases for skew-symmetric type quantum cluster algebras, meaning that the structureconstants for decomposing products of quantum theta functions in this setting are also Laurent polynomialsin the quantum parameter with positive integer coefficients.Here we give an example illustrating the failure of the quantum positivity conjecture for quantum thetafunctions outside of the skew-symmetric case. That is, we show a quantum theta function for a quantumcluster algebra of skew-symmetrizable type which, when expressed in terms of initial quantum cluster vari-ables, has a coefficient which is in Z [ q ± ] but not Z ≥ [ q ± ]. [CGM +
17] indicates that the rank 2 (classical) They must be treated as variables rather than coefficients if the A prin algebra is to have a compatible pair when A doesnot. A (2 , ǫ = (cid:18) −
32 0 (cid:19) , i.e. D = (cid:18) (cid:19) and ˆ ǫ = (cid:18) −
11 0 (cid:19) . In the formulation ofBerenstein-Zelevinsky, it corresponds to the compatible pair (Λ , B ), where Λ = (cid:18) − (cid:19) and B = (cid:18) − (cid:19) ,and hence D ′ = (cid:18) (cid:19) .Consider the seed { e = (1 , , e = (0 , } , and the corresponding basis { f = e ∗ , f = e ∗ } of M ◦ .For ease of comparison, we will adopt the notation of [LLRZ14] here, set q BZ = v − , and so consider thequantum torus with relation A A = v A A . [GHKK18, arXiv version 1] tells how to write down the initialscattering diagram for X q . To obtain the initial scattering diagram for A q , we can simply apply p ∗ : b D A in = (cid:8)(cid:0) e ⊥ , Ψ v (cid:0) A − (cid:1)(cid:1) , (cid:0) e ⊥ , Ψ v (cid:0) A (cid:1)(cid:1)(cid:9) . Observe that, for u = u f + u f , Ψ v (cid:0) A − (cid:1) − A u Ψ v (cid:0) A − (cid:1) = | u | Y ℓ =1 (cid:16) v sgn( u )3(2 ℓ − A − (cid:17) sgn( u ) A u (18)and Ψ v (cid:0) A (cid:1) − A u Ψ v (cid:0) A (cid:1) = | u | Y ℓ =1 (cid:16) v sgn( u )2(2 ℓ − A (cid:17) sgn( u ) A u . (19)It is straightforward to check that the initial wall functions reproduce the quantum mutation formulae forthe initial seed of this quantum cluster algebra.Now set L := p ∗ ( N ), set L + := p ∗ ( N + s ) \ { } , and let d : L → Z be given by h e − e , · i . Then d ( p ∗ ( e )) = d ( p ∗ ( e )) = 1. Now set P := (cid:16) span R ≥ { p ∗ ( e ) , p ∗ ( e ) } (cid:17) T M ◦ = span Z ≥ { f , − f } , and let J be themonomial ideal in C [ P ] generated by L + . Observe that b D A in is consistent for G . To see the failure ofquantum positivity in this example, we only need to compute the consistent scattering diagram for G ,denoted b D A .Consider the following loop γ : We could also work on the level of A prin , which is isomorphic to X prin . The injectivity assumption is satisfied in thisexample and we can obtain the scattering diagram for A q from the scattering diagram for A prin ,q by projection. := Ad − v ( A ) g := Ad − Ψ v ( A − ) γ .For b D A in , the wall crossing automorphism associated to γ is p γ = g ◦ g − ◦ g − ◦ g . To obtain b D A , we addwalls such that p γ is the identity in G . But for u = u f + u f , (cid:0) p γ (cid:1) − = g − ◦ g ◦ g ◦ g − is given by (cid:0) p γ (cid:1) − ( A u ) = (cid:18) sgn( u ) (cid:0) v − + 1 + v (cid:1) | u | X ℓ =1 v sgn( u )3(2 ℓ − + sgn( u ) (cid:0) v − + v (cid:1) | u | X ℓ =1 v sgn( u )2(2 ℓ − + sgn( u ) sgn( u ) (cid:0) v − v − (cid:1) | u | X ℓ =1 | u | X ℓ =1 v sgn( u )3(2 ℓ − u )2(2 ℓ − (cid:19) A f − f + · · · ! A u . (20)Note that there is only one term whose exponent vector m satisfies d ( m ) = 2. All terms encompassed bythe ellipses are higher order, i.e. have exponent vectors m ′ with d ( m ′ ) >
2. As a result, b D A has only onenew wall. Let g ∈ G denote the map A u (cid:18) sgn( u ) (cid:0) v − + 1 + v (cid:1) | u | X ℓ =1 v sgn( u )3(2 ℓ − + sgn( u ) (cid:0) v − + v (cid:1) | u | X ℓ =1 v sgn( u )2(2 ℓ − + sgn( u ) sgn( u ) (cid:0) v − v − (cid:1) | u | X ℓ =1 | u | X ℓ =1 v sgn( u )3(2 ℓ − u )2(2 ℓ − (cid:19) A f − f + · · · ! A u . The new wall is ( d , g ), where d := R ≥ · ( − f + 3 f ). The failure of quantum positivity in this examplestems from this wall function having coefficients in Z [ v ± ] rather than Z ≥ [ v ± ]. See Appendix B for this computation. := Ad − v ( A ) g := Ad − Ψ v ( A − ) g .Figure 1: The scattering diagram b D A .While the theta basis is indexed by g -vectors, the greedy basis is indexed by d -vectors. A piecewise linearmap sending the g -vector of a classical theta function to its d -vector (the theta function now being viewedas a greedy basis element) is given in [CGM + + A ( b, c ), define T : M ◦ R → M ◦ R m ( m if m ≥ m + (0 , cm ) if m ≤ . Then T ( g ( ϑ m )) = − d ( ϑ m ) by a similar argument as in [CGM +
17, Remark 4.5].In [LLRZ14], they considered the greedy basis element corresponding to the d -vector (3 , X [3 ,
4] = X p,q ≥ e ( p, q ) X (2 p − , q − . In particular, e (2 ,
1) = v − v − . This term e (2 ,
1) is the coefficient of X (1 , − , or in the Fock-Goncharovnotation we have adopted A f − f .Note that T ( − ,
5) = ( − , − d -vector (3 ,
4) corresponds to the g -vector ( − ,
5) in our setting.Now we compute the coefficient of A f − f in ϑ ( − , , expressed in terms of the initial cluster. We pick abasepoint Q in the positive quadrant and consider all possible broken lines whose initial decorating monomialis A − f +5 f , final decorating monomial is a scalar multiple of A f − f , and endpoint is Q . There is in factonly one: 15 g g (cid:0) v − − + v (cid:1) A − f + f A − f + f (cid:0) v − − v (cid:1) A − f − f (cid:0) v − − v (cid:1) A f − f Q .Figure 2: The broken line giving the A f − f summand of ϑ − f +5 f . While we have drawn this atop b D A , itis not difficult to see that this is the only broken line contributing to any order.We obtain precisely the same non-positive coefficient (cid:0) v − − v (cid:1) of A f − f for the quantum theta function ϑ − f +5 f as Lee-Li-Rupel-Zelevinsky do for the quantum greedy basis element X [3 , c X q In Section 2.3.1, we recall the quantization of a cluster Poisson variety X , which directly extends to thequantization for its special completion b X , by describing the quantum mutation map µ qk as a composition µ ♯k ◦ µ ′ k .To establish the quantum version of the c X family deforming b X , we introduce coefficients to both µ ♯k and µ ′ k . Consider the ring R := C [ t i : i ∈ I ]. We are going to work over the ∗ -ring R [ q ± d ], where d := lcm { d i : i ∈ I uf } and the involution map ∗ is defined as in (12).Geometrically, we want to associate a noncommutative affine scheme to each maximal cone G of the g -vectorfan and glue them via a quantum version of (6), or a version of (16) with coefficients. Algebraically, eachchamber G determines a ∗ -algebra A G over R [ q ± d ], a noncommutative polynomial ring given by A G := R h q ± d i h e X i ; G : i ∈ I, q − b ǫ ij e X i ; G e X j ; G = q − b ǫ ji e X j ; G e X i ; G i . The quantum mutation with coefficients will be a ∗ -algebra isomorphism of the noncommutative fractionfields of these ∗ -algebras. Thinking of the geometric interpretation, we will express the noncommutativefraction field of A G by K (cid:16) A | I | M ; G ,q ( R ) (cid:17) . We review the notion of a noncommutative fraction field in the following Subsection 3.1 and show that it can be appliedhere. .1 Quantum affine algebra We begin by reviewing some properties of the spectrum of a not-necessarily-commutative ring. For furtherdetails about noncommutative Noetherian rings, we refer the reader to [GW04] and [MR01].
Definition 3.1.
Let A be a ring. • A proper ideal P of A is prime if whenever I and J are two-sided ideals of R with IJ ⊆ P , either I ⊆ P or J ⊆ P . • The prime spectrum
Spec( A ) is the set of all prime ideals of A .The prime spectrum Spec( A ) is a topological space with the patch topology : the closed sets are the subsets X of Spec( A ) such that any prime ideal of A which is the intersection of some primes in X belongs to X (see [MR01, 4.6.14]).It is convenient think of A G as an iterated skew polynomial ring . We illustrate this concept in the caseof adjoining three noncommuting variables. The general case is simply an iteration of this process. See[GW04, Chapter 1] for a more complete treatment of iterated skew polynomial rings. Consider the R [ q ± d ]-automorphism α of R [ q ± d ][ e X G ] given by α (cid:16) e X G (cid:17) = q − b ǫ e X G . The ring R [ q ± d ][ e X G ][ e X G ; α ] is the noncommutative polynomial ring where e X G e X G = α (cid:16) e X G (cid:17) e X G .Now consider the R [ q ± d ]-automorphism α of R [ q ± d ][ e X G ][ e X G ; α ] given by α (cid:16) e X G (cid:17) = q − b ǫ e X G , and α (cid:16) e X G (cid:17) = q − b ǫ e X G . Then the ring R [ q ± d ][ e X G ][ e X G ; α ][ e X G ; α ] is the noncommutative polynomial ring satisfying e X G e X G = α (cid:16) e X G (cid:17) e X G , e X G e X G = α (cid:16) e X G (cid:17) e X G , and e X G e X G = α (cid:16) e X G (cid:17) e X G . With this description, we can assert that A G is a Noetherian domain. Proposition 3.2.
The ring A G is a Noetherian noncommutative ring.Proof. By induction on | I | . Note that R [ q ± d ] is a (commutative) Noetherian ring, so R [ q ± d ][ e X G ] also is a(commutative) Noetherian ring. The skew Hilbert basis theorem (see [GW04, Theorem 1.14]) implies that R [ q ± d ][ e X G ][ e X G ; α ] is a Noetherian ring. Assume the claim holds for all i < | I | .Now, since R [ q ± d ] h e X G , . . . , e X | I | ; G i / ∼ is equal to R [ q ± d ][ e X G ][ e X G ; α ] · · · [ e X | I |− G ; α | I |− ][ e X | I | ; G ; α | I | ],by the induction hypothesis R [ q ± d ][ e X G ][ e X G ; α ] · · · [ e X | I |− G ; α | I |− ] is a Noetherian ring, then by theskew Hilbert basis theorem we concluded that R [ q ± d ][ e X G ][ e X G ; α ] · · · [ e X | I |− G ; α | I |− ][ e X | I | ; G ; α | I | ] is aNoetherian ring.By [Gol58, Theorem 1], a ring without zero divisors which satisfies the ascending chain condition for rightideals has a right noncommutative fraction field. Combining with Proposition 3 .
2, an immediate corollaryis that we can take the noncommutative fraction field of A G . We denote the right noncommutative fractionfield of A G by K (cid:16) A | I | M ; G ,q ( R ) (cid:17) . This definition is equivalent to the usual one in the commutative case, see [GW04, Proposition 3.1]. emark . By the universal property of right rings of fractions, the right noncommutative fraction field of A G is canonically isomorphic to the right noncommutative fraction field of any localization of A G . This willallow us to glue quantum affine schemes by noncommutative “birational” maps µ qk, t ; G in Section 3.2.We can consider the multiplicatively closed set S consisting of the powers of the cluster variables n e X i ; G : i ∈ I o of A G . Observe that the cluster variables are all normal elements of A G , meaning e X i ; G A G = A G e X i ; G . Then,we construct the noncommutative localization of A G by S as T G = R [ q ± d ] h e X ± i ; G : i ∈ I i / ∼ with the samerelation as before: q − b ǫ ij e X i ; G e X j ; G = q − b ǫ ji e X j ; G e X i ; G . The ring T G is the quantum torus algebra over R [ q ± d ]. Proposition 3.4.
There is a 1-1 correspondence between the elements in
Spec ( A G ) that do not contain theelement e X i ; G for all i = 1 , . . . , | I | and the elements of Spec ( T G ) .Proof. Since A G is a Noetherian ring, the results follows by [MR01, Proposition 2.1.16].Following the arguments given in [BZ05], we describe the elements of Spec ( T G ) in terms of the prime idealsof a commutative ring. Proposition 3.5.
Let Z be the center of the quantum torus algebra T G . There is a 1-1 correspondencebetween the ideals of Z and the two-sided ideals of T G given by extension from Z to T G , with inverse givenby contraction. In particular, this gives a bijection between the sets Spec ( Z ) and Spec ( T G ) .Proof. Recall the fixed data Γ and let s be any seed associated to the maximal cone G . (See the discussionpreceding Remark 2 .
8. We fix s here as we will use an ordered basis of N in the proof.) Define the latticemap p ∗ : N → M ◦ n
7→ { n, · } . Now for any m ∈ p ∗ ( N ) consider the set T G ,m = n e X ∈ T G : e X i ; s e X e X − i ; s = q h e i ; s ,m i e X for i ∈ I o . (Note that the pairing h e i ; s , m i takes values in d Z .) For n = P i n i e i ; s ∈ N , denote the monomial Q i e X n i i ; s by e X n ; s . Note that T G ,m is an R [ q ± d ]-module with generating set n e X n ; s : − p ∗ ( n ) = m o . Indeed, if e X n ; s issuch that − p ∗ ( n ) = m , then e X i ; s e X n ; s e X − i ; G = q − P j ∈ I n j b ǫ ji e X n ; s e X i ; s e X − i ; s = q − h e i ; s ,p ∗ ( n ) i e X n ; s = q h e i ; s ,m i e X n ; s . Now, let e X ∈ T G ,m . Since e X = P n λ n e X n ; s and by hypothesis q h e i ; s ,m i e X = e X i ; s (cid:16)X λ n e X n ; s (cid:17) e X − i ; s = X λ n e X i ; s e X n ; s e X − i ; s = X λ n q h e i ; s , − p ∗ ( n ) i e X n ; s , A more general p ∗ map was defined in Section 2.2.1. h e i ; s , − p ∗ ( n ) i = h e i ; s , m i for each index in the sum.Consequently, T G is graded by p ∗ ( N ): T G = M m ∈ p ∗ ( N ) T G ,m . In addition, multiplication by e X n ; s gives anisomorphism between T G ,m and T G ,m − p ∗ ( n ) . The inverse is given by multiplication by e X − n ; s . Using this grading, it is immediate that Z = T G , and also that, if I is an ideal of Z , then the contractionin Z of the extension I T G = T G I is equal to I . That is, extension followed by contraction is the identity onideals of Z .Now, let J be a two-sided ideal of T G . Clearly, ( J ∩ Z ) T G ⊆ J . Next, since J is two-sided, J is p ∗ ( N )-gradedalso, namely J = M m ∈ p ∗ ( N ) J m where J m = J ∩ T G ,m . The isomorphism given by multiplication by e X n ; s restricts to an isomorphism J m ∼ → J m − p ∗ ( n ) . In particular, when m = 0 it gives an isomorphism J ∩ Z = J ∼ → J − p ∗ ( n ) . Since e X n ; s ∈ T G for all n ∈ N , this establishes that J = M m ∈ p ∗ ( N ) J m ⊆ ( J ∩ Z ) T G . So, J = ( J ∩ Z ) T G and contraction followed by extension is the identity on two-sided ideals of T G .For the next claim, using the fact that the contraction and extension give the bijection between the sets oftwo-sided prime ideals of T G and the prime ideals of Z , it is enough to prove the following statements: • If P ∈ Spec ( T G ), then P ∩ Z ∈ Spec ( Z ), and • If Q ∈ Spec ( Z ), then Q T G ∈ Spec ( T G ).For the first, let P ∈ Spec ( T G ). Consider ideals I and I of Z such that I I ⊆ P ∩ Z . Then, using the factthat contraction followed by extension is the identity,( I T G ) ( I T G ) = ( I I ) T G ⊆ ( P ∩ Z ) T G = P. The hypothesis P prime implies that I T G ⊆ P or I T G ⊆ P . Hence,( I T G ) ∩ Z ⊆ P ∩ Z or ( I T G ) ∩ Z ⊆ P ∩ Z and since extension followed by contraction is the identity, we conclude that I ⊆ P ∩ Z or I ⊆ P ∩ Z . So, P ∩ Z is a prime ideal of Z .For the second one, let Q ∈ Spec ( Z ). Consider two-sided ideals J and J of T G such that J J ⊆ Q T G .Since extension followed by contraction is the identity,( J ∩ Z ) ( J ∩ Z ) ⊆ ( J J ) ∩ Z ⊆ ( Q T G ) ∩ Z = Q. From the hypothesis Q prime it follows that( J ∩ Z ) ⊆ Q or ( J ∩ Z ) ⊆ Q. Finally, since contraction followed by extension is the identity, we conclude that J ⊆ Q T G or J ⊆ Q T G .Hence, Q T G is a prime ideal of T G . 19 .2 Quantum mutation of the c X family We now describe explicitly the quantum mutation map with coefficients µ qk, t ; G . It would be analogous to[FG09, Section 3.3] but including coefficients. For simplicity, we denote q k := q /d k . First, we consider thequantum dilogarithm with coefficients t . Ψ q k , t (cid:16) e X k ; G (cid:17) := Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! (21)We then define µ ♯k, t ; G to be the automorphism of K (cid:16) A | I | M ; G ,q ( R ) (cid:17) given by conjugation by Ψ q k , t (cid:16) e X k ; G (cid:17) : µ ♯k, t ; G ( x ) := Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! x Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! − . (22)Next, up to a certain scale factor, µ ′ k, t ; G : K (cid:16) A | I | M ; G ′ ,q ( R ) (cid:17) → K (cid:16) A | I | M ; G ,q ( R ) (cid:17) just re-expresses the coordi-nates n e X i ; G ′ = e X e i ; G′ : i ∈ I o obtained by exponentiating the basis elements of s ′ in terms of the coordinates n e X i ; G = e X e i ; G : i ∈ I o obtained by exponentiating the basis elements of s . Explicitly, we define µ ′ k, t ; G (cid:16) e X v G ′ (cid:17) = t −{ v,e k ; G } d k [ − c k ; G ] + e X v G , which in terms of the two sets of cluster coordinates has form: µ ′ k, t ; G (cid:16) e X i ; G ′ (cid:17) = e X − i ; G if i = k, t − ǫ ik [ − c k ; G ] + q − b ǫ ik [ ǫ ik ] + e X i ; G e X [ ǫ ik ] + k ; G if i = k. (23) Lemma 3.6.
The automorphism µ ♯k is given in cluster coordinates by µ ♯k, t ; G (cid:16) e X i ; G (cid:17) = e X i ; G | ǫ ik | Y ℓ =1 t [ c k ; G ] + t [ − c k ; G ] + q ℓ − k e X k ; G ! if ǫ ik ≤ , e X i ; G ǫ ik Y ℓ =1 t [ c k ; G ] + t [ − c k ; G ] + q − ℓk e X k ; G !! − if ǫ ik ≥ . (24) Proof.
This is [FG09, Lemma 3.4] with X k replaced by t [ c k ; G ] + t [ − c k ; G ] + e X k ; G . We reproduce the argument here forthe reader’s convenience. Recall that q − b ǫ ki e X k ; G e X i ; G = q − b ǫ ik e X i ; G e X k ; G , so e X k ; G e X i ; G = q − b ǫ ik e X i ; G e X k ; G = q − ǫ ik k e X i ; G e X k ; G . Then writing Ψ q k ( x ) as ∞ X ℓ =0 c ℓ x ℓ , we have Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! e X i ; G = ∞ X ℓ =0 c ℓ t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! ℓ e X i ; G = ∞ X ℓ =0 c ℓ e X i ; G q − ǫ ik k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! ℓ = e X i ; G Ψ q k q − ǫ ik k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! . µ ♯k, t ; G (cid:16) e X i ; G (cid:17) = Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! e X i ; G Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! − = e X i ; G Ψ q k q − ǫ ik k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! − . (25)Next, we use the quantum dilogarithm difference relation [FG09, Equation 52]: Ψ q (cid:0) q x (cid:1) = (1 + qx ) Ψ q ( x ) , Ψ q (cid:0) q − x (cid:1) = (cid:0) q − x (cid:1) − Ψ q ( x ) . (26)Combining (26) and (25) yields the claim.We now define the quantum X -mutation with coefficients µ qk, t ; G : K (cid:16) A | I | M ; G ′ ,q ( R ) (cid:17) → K (cid:16) A | I | M ; G ,q ( R ) (cid:17) to bethe composition µ qk, t ; G := µ ♯k, t ; G ◦ µ ′ k, t ; G . Proposition 3.7.
The quantum X -mutation with coefficients µ qk, t ; G : K (cid:16) A | I | M ; G ′ ,q ( R ) (cid:17) → K (cid:16) A | I | M ; G ,q ( R ) (cid:17) isgiven in cluster coordinates by µ qk, t ; G (cid:16) e X i ; G ′ (cid:17) = e X − i ; G if i = k, e X i ; G | ǫ ik | Y ℓ =1 (cid:16) t [sgn ( ǫ ik ) c k ; G ] + + t [ − sgn ( ǫ ik ) c k ; G ] + q ℓ − k e X − sgn ( ǫ ik ) k ; G (cid:17) − sgn ( ǫ ik ) if i = k. (27) Proof.
We use Lemma 3 . µ qk, t ; G := µ ♯k, t ; G ◦ µ ′ k, t ; G . First, if i = k theresult is immediate. Next, if i = k but ǫ ik = 0, (23) reduces to µ ′ k, t ; G (cid:16) e X i ; G ′ (cid:17) = e X i ; G and the product in (24)is empty. So in this case µ qk, t ; G (cid:16) e X i ; G ′ (cid:17) = e X i ; G as claimed. Two cases remain: Case 1 : ǫ ik < µ ♯k, t ; G (cid:16) t − ǫ ik [ − c k ; G ] + q − b ǫ ik [ ǫ ik ] + e X i ; G e X [ ǫ ik ] + k ; G (cid:17) = t − ǫ ik [ − c k ; G ] + µ ♯k, t ; G (cid:16) e X i ; G (cid:17) = t | ǫ ik | [ − c k ; G ] + e X i ; G | ǫ ik | Y ℓ =1 t [ c k ; G ] + t [ − c k ; G ] + q ℓ − k e X k ; G ! = e X i ; G | ǫ ik | Y ℓ =1 (cid:16) t [ − c k ; G ] + + t [ c k ; G ] + q ℓ − k e X k ; G (cid:17) . ase 2 : ǫ ik >
0. Recall that by definition ǫ ik = { e i , e k } d k , q k = q /d k and b ǫ ik = { e i , e k } . µ ♯k, t ; G (cid:16) t − ǫ ik [ − c k ; G ] + q − b ǫ ik [ ǫ ik ] + e X i ; G e X [ ǫ ik ] + k ; G (cid:17) = t − ǫ ik [ − c k ; G ] + q − b ǫ ik ǫ ik µ ♯k, t ; G (cid:16) e X i ; G e X ǫ ik k ; G (cid:17) = t − ǫ ik [ − c k ; G ] + q − b ǫ ik ǫ ik e X i ; G e X ǫ ik k ; G ǫ ik Y ℓ =1 t [ c k ; G ] + t [ − c k ; G ] + q − ℓk e X k ; G !! − = e X i ; G ǫ ik Y ℓ =1 q b ǫ ik e X − k ; G (cid:16) t [ − c k ; G ] + + t [ c k ; G ] + q − ℓk e X k ; G (cid:17)! − = e X i ; G ǫ ik Y ℓ =1 q ǫ ik k q − ℓk (cid:16) t [ − c k ; G ] + q ℓ − k e X − k ; G + t [ c k ; G ] + (cid:17)! − = e X i ; G q ǫ ik k q − ǫ ik k ǫ ik Y ℓ =1 (cid:16) t [ − c k ; G ] + q ℓ − k e X − k ; G + t [ c k ; G ] + (cid:17)! − = e X i ; G ǫ ik Y ℓ =1 (cid:16) t [ c k ; G ] + + t [ − c k ; G ] + q ℓ − k e X − k ; G (cid:17)! − . This proves the claim.
Corollary 3.8.
Setting t = we recover the coefficient free quantum mutation formula (16) . Setting q = 1 ,we recover the degeneration formula (6) . Proposition 3.9.
The quantum X -mutation with coefficients µ qk ; t ; G is a ∗ -homomorphism.Proof. Since by construction µ qk ; t ; G = µ ♯k ; t ; G ◦ µ ′ k ; t ; G , it is enough to prove that the homomorphisms µ ♯k ; t ; G and µ ′ k ; t ; G are ∗ -homomorphisms.Using the fact that the quantum dilogarithm satisfies the equality Ψ q − k ( x ) = Ψ q k ( x ) − [FG09, Equation 53],it follows that ∗ (cid:16) µ ♯k ; t ; G (cid:16) e X i ; G (cid:17)(cid:17) = Ψ q − k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! − e X i ; G Ψ q − k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! = Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! e X i ; G Ψ q k t [ c k ; G ] + t [ − c k ; G ] + e X k ; G ! − = µ ♯k ; t ; G (cid:16) e X i ; G (cid:17) . For µ ′ k ; t ; G , the statement it is clear if i = k or i = k and ǫ ik = 0. Now, assume that i = k and ǫ ik = 0. Weconsider two cases: Case 1: ǫ ik < ∗ (cid:16) µ ′ k ; t ; G (cid:16) e X i ; G (cid:17)(cid:17) = ∗ (cid:16) t − ǫ ik [ − c k ; G ] + e X i ; G (cid:17) = t − ǫ ik [ − c k ; G ] + e X i ; G = µ ′ k ; t ; G (cid:16) e X i ; G (cid:17) . ase 2: ǫ ik > . Recall that e X k ; G X i ; G = q − b ǫ ik e X i ; G e X k ; G . ∗ (cid:16) µ ′ k ; t ; G (cid:16) e X i ; G (cid:17)(cid:17) = ∗ (cid:16) t − ǫ ik [ − c k ; G ] + q − b ǫ ik ǫ ik e X i ; G e X ǫ ik k ; G (cid:17) = t − ǫ ik [ − c k ; G ] + q b ǫ ik ǫ ik e X ǫ ik k ; G e X i ; G = t − ǫ ik [ − c k ; G ] + q b ǫ ik ǫ ik q − b ǫ ik ǫ ik e X i ; G e X ǫ ik k ; G = t − ǫ ik [ − c k ; G ] + q − b ǫ ik ǫ ik e X i ; G e X ǫ ik k ; G = µ ′ k ; t ; G (cid:16) e X i ; G (cid:17) . This completes the proof.
Definition 3.10.
The scheme c X G ,q over R is obtained by gluing the affine patches A | I | M ; G ,q ( R ) := Spec (cid:16) R h q ± d , e X G , . . . , e X | I | ; G i(cid:17) via the mutations given in (27), where G is the maximal cone of ∆ + associated to a choice of initial seed s and G is any maximal cone of ∆ + . Similarly, we obtained the scheme X G ,q by gluing the affine patchesSpec (cid:16) R h q ± d , e X ± G , . . . , e X ±| I | ; G i(cid:17) . Once we have fixed an initial seed s or cone G and there is no risk ofconfusion, we will often drop the subscript from our notation, writing simply c X q or X q . Remark . In [BFMN18], the family c X is viewed as a toric degeneration of the special completion b X of X to the toric variety defined by the underlying fan of the cluster complex in the A scattering diagram.Note that here we obtain a quantum version of this toric degeneration, where characters only q -commute. Example 3.12.
In this example, we compare the different cluster varieties associated to the A quiver. Thismay be compared to [BFMN18, Table 2] and [FZ07, Tables 1-4], where to make the comparison set ε s = B Ts .In Table 1, we compare the usual X -variety with the corresponding quantum variety X q obtained using thequantum mutation formula in [FG09] and in Table 2, we compare the family c X with the quantum family c X q using the mutation formula (27). X X q s ε s X s X s X s X s (cid:0) −
11 0 (cid:1) X X X X l µ (cid:0) − (cid:1) X (1 + X ) X X (1 + qX ) X − l µ (cid:0) −
11 0 (cid:1) X (1+ X ) X X + X +1 X (1 + qX ) − X − (cid:0) X − X (cid:1) − (cid:0) X − + q (1 + qX ) (cid:1) l µ (cid:0) − (cid:1) X +1 X X X X X + X +1 X − (cid:0) q − X − (cid:1) (cid:0) X − + q (1 + qX ) (cid:1) − (cid:0) X − X (cid:1) l µ (cid:0) −
11 0 (cid:1) X X X +1 1 X (cid:0) q − X − (cid:1) − X X − l µ (cid:0) − (cid:1) X X X X Table 1: Comparison between the usual cluster variety X and the quantum cluster variety X q in type A .23 X c X q s ε s C s e X s e X s e X s e X s (cid:0) −
11 0 (cid:1) ( ) X X X X l µ (cid:0) − (cid:1) (cid:0) − (cid:1) X (1 + t X ) X X (1 + t qX ) X − l µ (cid:0) −
11 0 (cid:1) (cid:0) − − (cid:1) X (1+ t X ) t t X X + t X +1 X (1 + t qX ) − X − (cid:0) X − X (cid:1) − (cid:0) X − + t q (1 + t qX ) (cid:1) l µ (cid:0) − (cid:1) (cid:0) − − − (cid:1) t X +1 X X X t t X X + t X +1 X − (cid:0) t + q − X − (cid:1) ( X − + t q (1 + t qX )) − ( X − X ) l µ (cid:0) −
11 0 (cid:1) (cid:0) −
11 0 (cid:1) X X t X +1 1 X (cid:0) t + q − X − (cid:1) − X X − l µ (cid:0) − (cid:1) ( ) X X X X Table 2: Comparison between the family c X and the quantum family c X q in type A .Here C s is the matrix of c -vectors for the cluster. Relation to Berenstein-Zelevinsky quantization of A prinIn this section we relate quantum X -mutation with coefficients (Equation (27)) with Berenstein-Zelevinsky’squantum A -mutation and Fomin-Zelevinsky’s A -mutation with coefficients. We are particularly interestedin the case of principal coefficients, so this will be our focus. The first task is simply to translate between the Berenstein-Zelevinsky framework and the Fock-Goncharovframework (employed in this paper). Let us recap the necessary notions in Section 2.3.2. Let (cid:16) Λ , e B (cid:17) bea compatible pair, i.e. e B T Λ = (cid:0) D ′ (cid:1) , where D ′ is the skew-symmetrizing matrix. Then D ′ B is skew-symmetric. Note that all matrices in this setup are integral.First note that ǫ I uf × I = e B T when comparing [FG09] to [BZ05]. Rather than multiplying ǫ by a diagonalmatrix to obtain an integral skew symmetric matrix, Fock and Goncharov multiply a rational skew symmetricmatrix b ǫ by a diagonal integral matrix D to obtain ǫ : ǫ = b ǫD. Then ǫ I uf × I uf = b ǫ I uf × I uf D I uf × I uf . We have that ( D ′ B ) T = ǫ I uf × I uf D ′ = b ǫ I uf × I uf D I uf × I uf D ′ is skew symmetric, and we can relate the two setups by taking D ′ to be αD − I uf × I uf for some α ∈ Z with α b ǫ I uf × I uf integral. Since ǫ I uf × I uf is integral and ǫ I uf × I uf = b ǫ I uf × I uf D I uf × I uf , it follows that d b ǫ I uf × I uf is integralas well, where d = lcm ( d i : i ∈ I uf ). We can take D ′ = dD − I uf × I uf – i.e. we can take D ′ to be the Langlandsdual of the unfrozen part of D to give a dictionary between the Berenstein-Zelevinsky and Fock-Goncharovprescriptions. Before relating quantum X -mutation with coefficients to quantum A -mutation with coefficients, we need toclarify a potential point of confusion. As in [BFMN18]: We distinguish between frozen variables and coefficients.
Geometrically, a coefficient is a parameter on the base of a family while a frozen variable is a coordinate onthe fibers of the family. Coefficients are in the base ring while frozen variables are in an algebra over thebase ring. Classically, A -mutation treats frozen variables and coefficients in exactly the same way, so theseconcepts are generally identified in the literature. However, X -mutation is genuinely affected the distinction,and in turn so is the notion of a dual cluster variety. For example, if we have a cluster variety A of dimension n over C , then A prin can be viewed as either a 2 n -dimensional A -cluster variety over C or an n -dimensional A -cluster variety over C [ t , . . . , t n ]. In the former case, the dual is a 2 n -dimensional X -cluster variety over C ( X prin in [GHKK18]), while in the latter case the dual is an n -dimensional X -cluster variety over C [ t , . . . , t n ]( X in [BFMN18]). The mutation formulas for X prin and X differ. We will however allow arbitrary frozen variables. We discuss the distinction we make between coefficients and frozenvariables later in this section. Note that this particular appearance of Langlands duality is simply to translate between differing conventions. The A - and X -varieties discussed in this section are associated to the same fixed data Γ, rather than Langlands dual fixed data Γ and Γ ∨ .It is important to keep this in mind in Proposition 4 .
25n the quantum setting, the distinction can also affect A -mutation. The reason deals with the relationbetween lattices and cluster variables. Classically, it is convenient to describe mutation with coefficients byextending the lattice and skew form to incorporate the coefficients. On the other hand, we could just as wellkeep the original lattice and skew form when adding coefficients and still obtain the same mutation formulawithout referencing any extended lattice, which is essentially how mutation with coefficients in a semifieldwas originally framed in [FZ02], [FZ07]. By contrast, in the quantum setting existence of a compatible pairimplies that ǫ I uf × I is full-rank. If the submatrix ǫ I uf × I uf is not full-rank, then there will be some frozen A -variable that does not commute with a mutable A -variable– Λ( f i , f j ) = 0 for some mutable index i andfrozen index j . So, if we extend the lattice to incorporate coefficients as frozen variables , we can obtaincoefficients that do not commute with variables. If, on the other hand, we keep the original lattice anddistinguish between frozen variables and coefficients, by construction the coefficients will always commutewith cluster variables while frozen variables may not. The rank of the lattice here is the number of variables(both mutable and frozen) in each cluster.With this in mind, we modify the quantum mutation formula of [BZ05] to include coefficients in a semifield P as in [FZ02], [FZ07], without altering the lattice and skew form of the coefficient-free case. As usual, if i = k , µ qk, t ; G (cid:16) e A i ; G ′ (cid:17) = e A i ; G . For i = k , we combine [BZ05, Proposition 4.9] and [FZ07, Equation 2.8] to set µ qk, p ; G (cid:16) e A i ; G ′ (cid:17) = p + k e A − f k ; G + P j : ǫkj> ǫ kj f j ; G + p − k e A − f k ; G − P j : ǫkj< ǫ kj f j ; G . (28)In the special case of principal coefficients at G , p + k = t [ c k ; G ] + , p − k = t [ − c k ; G ] + , and (28) becomes µ qk, t ; G (cid:16) e A i ; G ′ (cid:17) = t [ c k ; G ] + e A − f k ; G + P j : ǫkj> ǫ kj f j ; G + t [ − c k ; G ] + e A − f k ; G − P j : ǫkj< ǫ kj f j ; G . (29)Next, recall the lattice maps p ∗ : N → M ◦ reviewed in Section 2.2.1. We show that the two quantizationswith coefficients are compatible in the following sense: Proposition 4.1.
After making the identification q d FG = q − BZ , a p ∗ -map on the level of lattices induces a ∗ -algebra homomorphism from the quantum X -torus algebra over R q FG := C [ t i : i ∈ I ] h q ± d FG i to the quantum A -torus algebra over R q BZ := C [ t i : i ∈ I ] h q ± BZ i that commutes with mutation: p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = µ qk, t ; G (cid:16) p ∗ (cid:16) e X i ; G ′ (cid:17)(cid:17) . That is, there is map of quantum cluster varieties p : A prin ,q → X q from the Berenstein-Zelevinsky quantum A -variety with principal coefficients to the Fock-Goncharov quantum X -variety with principal coefficients inthe sense of [BFMN18].Proof. First, the identification of quantum parameters q d FG = q − BZ comes from the following computation.In the quantum X -torus algebra e X e i ; G e X e j ; G = q ǫ ij d − j FG e X e i ; G + e j ; G , and in the quantum A -torus algebra e A p ∗ ( e i ; G ) e A p ∗ ( e j ; G ) = q Λ( p ∗ ( e i ; G ) ,p ∗ ( e k ; G ))BZ e A p ∗ ( e i ; G + e j ; G ) = q − ǫ ij d ∨ j BZ e A p ∗ ( e i ; G + e j ; G ) . In light of this, we want to identify q dj FG with q − d ∨ j BZ for all j , which is accomplished by setting q d FG = q − BZ . This identification can be incorporated into the definition of the ∗ -algebra homomorphism, writing p ∗ (cid:16) q d (cid:17) = q − . i = k , p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = p ∗ (cid:16) e X − k ; G (cid:17) = e A p ∗ ( − e k ; G ) . Similarly, µ qk, t ; G (cid:16) p ∗ (cid:16) e X i ; G ′ (cid:17)(cid:17) = µ qk, t ; G (cid:16) e A p ∗ ( e k ; G′ ) (cid:17) = e A p ∗ ( − e k ; G ) . So it holds for i = k .Assume i = k . Then p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:18) t [sgn ( ǫ ik ) c k ; G ] + + t [ − sgn ( ǫ ik ) c k ; G ] + q ℓ − dk FG e A p ∗ ( − sgn ( ǫ ik ) e k ; G ) (cid:19) − sgn ( ǫ ik ) (30)and µ qk, t ; G (cid:16) p ∗ (cid:16) e X i ; G ′ (cid:17)(cid:17) = µ qk, t ; G (cid:16) e A p ∗ ( e i ; G′ ) (cid:17) = µ qk, t ; G (cid:16) e A P j ǫ ′ ij f j ; G′ (cid:17) = µ qk, t ; G (cid:16) e A ǫ ′ ik f k ; G′ + P j = k ǫ ′ ij f j ; G′ (cid:17) = q − ǫ ′ ik Λ ( p ∗ ( e i ; G′ ) ,f k ; G′ ) BZ e A P j = k ǫ ′ ij f j ; G′ (cid:16) t [ c k ; G ] + e A − f k ; G + P j : ǫkj> ǫ kj f j ; G + t [ − c k ; G ] + e A − f k ; G − P j : ǫkj< ǫ kj f j ; G (cid:17) ǫ ′ ik = e A p ∗ ( e i ; G′ ) e A − ǫ ′ ik f k ; G′ (cid:16) t [ c k ; G ] + e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + e A f k ; G′ (cid:17) ǫ ′ ik . (31)The statement follows immediately if ǫ ik = 0. Now take ǫ ik <
0. Then (31) becomes µ qk, t ; G (cid:16) p ∗ (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:18) t [ c k ; G ] + q ( | ǫ ik |− ℓ )Λ ( − f k ; G′ ,p ∗ ( e k ; G ) ) BZ e A − f k ; G′ e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + (cid:19) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:16) t [ c k ; G ] + q − ( | ǫ ik |− ℓ ) d ∨ k BZ e A − f k ; G′ e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + (cid:17) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:18) t [ c k ; G ] + q − ( | ǫ ik |− ℓ + ) d ∨ k BZ e A p ∗ ( e k ; G ) + t [ − c k ; G ] + (cid:19) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:16) t [ c k ; G ] + q − ℓ − d ∨ k BZ e A p ∗ ( e k ; G ) + t [ − c k ; G ] + (cid:17) . (32)In the final equality, we have used the fact that all term to the right of the product symbol commute withone another to reorder the product.Meanwhile, (30) becomes p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:18) t [ − c k ; G ] + + t [ c k ; G ] + q ℓ − dk FG e A p ∗ ( e k ; G ) (cid:19) . (33)27fter making the substitution q d FG = q − BZ , we find p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G ) | ǫ ik | Y ℓ =1 (cid:16) t [ − c k ; G ] + + t [ c k ; G ] + q − ℓ − d ∨ k BZ e A p ∗ ( e k ; G ) (cid:17) , in agreement with (32).Next, take ǫ ik >
0. Then (31) becomes µ qk, t ; G (cid:16) p ∗ (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G′ ) (cid:16) e A − ǫ ik f k ; G′ (cid:17) − ǫ ik Y ℓ =1 (cid:16) t [ c k ; G ] + e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + e A f k ; G′ (cid:17) − = e A p ∗ ( e i ; G′ ) (cid:16)(cid:16) t [ c k ; G ] + e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + e A f k ; G′ (cid:17) e A − ǫ ik f k ; G′ (cid:17) − ǫ ik − Y ℓ =1 (cid:16) t [ c k ; G ] + e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + e A f k ; G′ (cid:17) − = e A p ∗ ( e i ; G′ ) (cid:18) e A − ( ǫ ik − f k ; G′ (cid:18) t [ c k ; G ] + q Λ ( p ∗ ( e k ; G ) , − ( ǫ ik − ) f k ; G′ ) BZ e A p ∗ ( e k ; G ) + t [ − c k ; G ] + (cid:19)(cid:19) − ǫ ik − Y ℓ =1 (cid:16) t [ c k ; G ] + e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + e A f k ; G′ (cid:17) − = e A p ∗ ( e i ; G′ ) (cid:18) t [ c k ; G ] + q ( ǫ ik − ) d ∨ k BZ e A p ∗ ( e k ; G ) + t [ − c k ; G ] + (cid:19) − (cid:16) e A − ( ǫ ik − f k ; G′ (cid:17) − ǫ ik − Y ℓ =1 (cid:16) t [ c k ; G ] + e A p ∗ ( e k ; G )+ f k ; G′ + t [ − c k ; G ] + e A f k ; G′ (cid:17) − = e A p ∗ ( e i ; G + ǫ ik e k ; G ) ǫ ik Y ℓ =1 (cid:18) t [ c k ; G ] + q ( ǫ ik − ( ℓ − )) d ∨ k BZ e A p ∗ ( e k ; G ) + t [ − c k ; G ] + (cid:19) − = q ǫ ik d ∨ k BZ e A p ∗ ( e i ; G ) e A ǫ ik p ∗ ( e k ; G ) ǫ ik Y ℓ =1 (cid:16) t [ c k ; G ] + q ℓ − d ∨ k BZ e A p ∗ ( e k ; G ) + t [ − c k ; G ] + (cid:17) − = e A p ∗ ( e i ; G ) ǫ ik Y ℓ =1 (cid:16) t [ c k ; G ] + + t [ − c k ; G ] + q − ℓ − d ∨ k BZ e A p ∗ ( − e k ; G ) (cid:17) − . (34)To write the final equality, we use the identity r X ℓ =1 (2 ℓ −
1) = r .Meanwhile, (30) becomes p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G ) ǫ ik Y ℓ =1 (cid:18) t [ c k ; G ] + + t [ − c k ; G ] + q ℓ − dk FG e A p ∗ ( − e k ; G ) (cid:19)! − = e A p ∗ ( e i ; G ) ǫ ik Y ℓ =1 (cid:18) t [ c k ; G ] + + t [ − c k ; G ] + q ℓ − dk FG e A p ∗ ( − e k ; G ) (cid:19) − . (35)After making the substitution q d FG = q − BZ , we find p ∗ (cid:16) µ qk, t ; G (cid:16) e X i ; G ′ (cid:17)(cid:17) = e A p ∗ ( e i ; G ) ǫ ik Y ℓ =1 (cid:16) t [ c k ; G ] + + t [ − c k ; G ] + q − ℓ − d ∨ k BZ e A p ∗ ( − e k ; G ) (cid:17) − , in agreement with (34). This completes the proof. 28 emark . To compare the Berenstein-Zelevinsky and Fock-Goncharov quantizations in the coefficient-freecase rather than the principal coefficient case, we can simply take the coefficients t i to 1 in Proposition 4 . The Poisson structure on X can be realized as the semi-classical limit of X q ’s quantum structure, as describedin [FG09]. Let us summarize as follows:Recall from Section 2.3.1 that X q is a union of non-commutative tori T N ; s , glued via birational quantummutation maps µ qk : T N ; s ′ → T N ; s , where T N ; s is the noncommutative fraction field of T N ; s (Equation (16)).On each T N ; s we have the relation q −{ v ,v } X v X v = X v + v . This induces a Poisson structure on the corresponding X -torus T N ; s by taking the semi-classical limit shownbelow. { X v , X v } := lim q → X v X v − X v X v q −
1= lim q → q { v ,v } − q { v ,v } q − X v + v =2 { v , v } X v + v . Since µ qk : T N ; s ′ → T N ; s is a ∗ -algebra homomorphism that restricts to the identity on C h q ± d i , we have µ qk (cid:18) X v X v − X v X v q − (cid:19) = µ qk ( X v ) µ qk ∗ ( X v ) − µ qk ( X v ) µ qk ∗ ( X v ) q − . Since the usual mutation formula can be recovered by taking q →
1, we have µ k ( { X v , X v } s ′ ) := µ k (cid:18) lim q → X v X v − X v X v q − (cid:19) = lim q → µ qk ∗ (cid:18) X v X v − X v X v q − (cid:19) = lim q → µ qk ( X v ) µ qk ( X v ) − µ qk ( X v ) µ qk ( X v ) q −
1= : { µ k ( X v ) , µ k ( X v ) } s . Thus, the Poisson structures on tori T N ; s patch together giving a global Poisson structure on X .In this section we extend the above argument to c X and its fibers b X t . We start by giving the key definitions.In the following definitions, A is a commutative ring with 1. Definition 5.1.
Let F be a sheaf of A -algebras on X . We say that F is an A -Poisson sheaf if for eachopen set U ⊂ X , we can endow F ( U ) with an A -bilinear Poisson bracket {· , ·} U , with the Poisson bracketssatisfying the following compatibility property:If V ⊂ U and f, g ∈ F ( U ), then { f | V , g | V } V = { f, g } U . Definition 5.2.
Let X be a scheme over A . We say X is a Poisson scheme over A if O X is an A -Poissonsheaf. 29 roposition 5.3. c X is a Poisson scheme over R , with the Poisson structure inherited from the Poissonstructure on affine patches A | I | M ; G ,q ( R ) := Spec (cid:16) R h e X i ; G : i ∈ I i(cid:17) : n e X v G , e X v G o G := 2 { v , v } e X v + v G . Proof.
By definition, c X q is a union of non-commutative affine R -schemes A | I | M ; G ,q , each corresponding to aring A G = R h q ± d , ˜ X G , . . . , ˜ X | I | ; G i where the variables satisfy the q -commutation relations q − ˆ ǫ ij ˜ X i ; G ˜ X i ; G = q − ˆ ǫ ji ˜ X j ; G ˜ X i ; G . These patches are glued via the mutations given in (27). By Proposition 3 .
9, these mutation maps are ∗ -algebra homomorphisms, and by construction they restrict to the identity on R h q ± d i . We have preciselythe same semi-classical limit as the coefficient free case:lim q → e X v G e X v G − e X v G e X v G q −
1= lim q → q { v ,v } − q { v ,v } q − e X v + v ; G =2 { v , v } e X v + v ; G . Next, by Corollary 3 . q → µ k, t (cid:18)n e X v G ′ , e X v G ′ o G ′ (cid:19) = n µ k, t (cid:16) e X v G ′ (cid:17) , µ k, t (cid:16) e X v G ′ (cid:17)o G . The R -Poisson structures on affine patches glue together giving a global R -Poisson structure on c X .An immediate corollary of Proposition 5 . Corollary 5.4.
Let t be a closed point of Spec ( R ) . Then b X t is a Poisson scheme over C . Appendices
A Direct calculation of the Poisson structure
It can of course be shown that c X is an R -Poisson scheme without passing through its quantization. As somereaders may appreciate having a low-tech, direct proof of Proposition 5 . Proof.
Let µ k : A | I | M ; G ( R ) A | I | M ; G ′ ( R ) be a mutation gluing neighboring affine patches in c X . To rephrase30he statement of the proposition, we have the following commutative diagram: K (cid:16) A | I | M ; G ( R ) (cid:17) × K (cid:16) A | I | M ; G ( R ) (cid:17) K (cid:16) A | I | M ; G ′ ( R ) (cid:17) × K (cid:16) A | I | M ; G ′ ( R ) (cid:17) K (cid:16) A | I | M ; G ( R ) (cid:17) K (cid:16) A | I | M ; G ′ ( R ) (cid:17) {· , ·} G ( µ ∗ k ,µ ∗ k ) {· , ·} G′ µ ∗ k . (36)We claim first that this diagram commutes when we restrict to cluster variables e X i ; G ′ . Assuming this claim,we would have n µ ∗ k (cid:16) e X i ; G ′ (cid:17) , µ ∗ k (cid:16) e X j ; G ′ · · · e X j r ; G ′ (cid:17)o G = n µ ∗ k (cid:16) e X i ; G ′ (cid:17) , µ ∗ k (cid:16) e X j ; G ′ (cid:17)o G µ ∗ k (cid:16) e X j ; G ′ (cid:17) · · · µ ∗ k (cid:16) e X j r ; G ′ (cid:17) + · · · + n µ ∗ k (cid:16) e X i ; G ′ (cid:17) , µ ∗ k (cid:16) e X j r ; G ′ (cid:17)o G µ ∗ k (cid:16) e X j ; G ′ (cid:17) · · · µ ∗ k (cid:16) e X j r − ; G ′ (cid:17) = µ ∗ k (cid:18)n e X i ; G ′ , e X j ; G ′ o G ′ (cid:19) µ ∗ k (cid:16) e X j ; G ′ (cid:17) · · · µ ∗ k (cid:16) e X j r ; G ′ (cid:17) + · · · + µ ∗ k (cid:18)n e X i ; G ′ , e X j r ; G ′ o G ′ (cid:19) µ ∗ k (cid:16) e X j ; G ′ (cid:17) · · · µ ∗ k (cid:16) e X j r − ; G ′ (cid:17) = µ ∗ k (cid:18)n e X i ; G ′ , e X j ; G ′ · · · e X j r ; G ′ o G ′ (cid:19) . We extend to monomials in both arguments similarly, and use linearity of µ ∗ k and bilinearity of Poissonbrackets to extend to polynomials in each argument. Now suppose we have a pair of rational functions. (cid:26) µ ∗ k (cid:18) f g (cid:19) , µ ∗ k (cid:18) f g (cid:19)(cid:27) G = µ ∗ k (cid:18) g g (cid:19) { µ ∗ k ( f ) , µ ∗ k ( f ) } G + µ ∗ k (cid:18) f g (cid:19) (cid:26) µ ∗ k ( f ) , µ ∗ k (cid:18) g (cid:19)(cid:27) G + µ ∗ k (cid:18) f g (cid:19) (cid:26) µ ∗ k (cid:18) g (cid:19) , µ ∗ k ( f ) (cid:27) G + µ ∗ k ( f f ) (cid:26) µ ∗ k (cid:18) g (cid:19) , µ ∗ k (cid:18) g (cid:19)(cid:27) G To simplify we note that { f, } = 0, so0 = (cid:26) f, gg (cid:27) = 1 g { f, g } + g (cid:26) f, g (cid:27) and (cid:26) f, g (cid:27) = − g { f, g } . So, (cid:26) µ ∗ k (cid:18) f g (cid:19) , µ ∗ k (cid:18) f g (cid:19)(cid:27) G = µ ∗ k (cid:18) g g (cid:19) { µ ∗ k ( f ) , µ ∗ k ( f ) } G + µ ∗ k (cid:18) f g g (cid:19) { µ ∗ k ( f ) , µ ∗ k ( g ) } G + µ ∗ k (cid:18) f g g (cid:19) { µ ∗ k ( g ) , µ ∗ k ( f ) } G + µ ∗ k (cid:18) f f g g (cid:19) { µ ∗ k ( g ) , µ ∗ k ( g ) } G = µ ∗ k (cid:18) g g (cid:19) µ ∗ k (cid:0) { f , f } G ′ (cid:1) + µ ∗ k (cid:18) f g g (cid:19) µ ∗ k (cid:0) { f , g } G ′ (cid:1) + µ ∗ k (cid:18) f g g (cid:19) µ ∗ k (cid:0) { g , f } G ′ (cid:1) + µ ∗ k (cid:18) f f g g (cid:19) µ ∗ k (cid:0) { g , g } G ′ (cid:1) = µ ∗ k (cid:18)(cid:26) f g , f g (cid:27) G ′ (cid:19) . This shows that the claim that the diagram commutes for cluster variables implies the proposition. We nowtackle this claim. 31enote the top path {· , ·} G ◦ ( µ ∗ k , µ ∗ k ) by p and the bottom path µ ∗ k ◦ {· , ·} G ′ by y . Both p ( f, f ) and y ( f, f ) are clearly 0, so assume from now on that the arguments are distinct.Note that {· , ·} G induces the Poisson bivector field π G = X i,j { e i ; G , e j ; G } e X i ; G e X j ; G ∂ e X i ; G ∧ ∂ e X j ; G and { f, g } G = π G ( df ∧ dg ). Case 1:
Check p (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) = y (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) p (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) = π G X l,m ∂µ ∗ k (cid:16) e X i ; G ′ (cid:17) ∂ e X l ; G ∂µ ∗ k (cid:16) e X k ; G ′ (cid:17) ∂ e X m ; G d e X l ; G ∧ d e X m ; G We compute the partial derivatives. If ǫ ik = 0, ∂µ ∗ k ( e X i ; G′ ) ∂ e X l ; G is given by ∂∂ e X l ; G (cid:18) e X i ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik (cid:19) = δ il (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik + δ kl sgn ( ǫ ik ) ǫ ik t [ − sgn( ǫ ik ) c k ; G ] + e X i ; G e X − sgn( ǫ ik ) − k ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik − . If ǫ ik = 0, instead we simply get ∂µ ∗ k (cid:16) e X i ; G ′ (cid:17) ∂ e X l ; G = δ il . Meanwhile ∂µ ∗ k (cid:16) e X k ; G ′ (cid:17) ∂ e X m ; G = ∂ e X − k ; G ∂ e X m ; G = − δ km e X − k ; G . Subcase 1.a: ǫ ik = 0Using d e X k ; G ∧ d e X k ; G = 0 we have p (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) = π G (cid:18) − e X − k ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik d e X i ; G ∧ d e X k ; G (cid:19) = − { e i ; G , e k ; G } e X i ; G e X k ; G e X − k ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik = { e i ; G ′ , e k ; G ′ } e X i ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik e X − k ; G = y (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) Subcase 1.b: ǫ ik = 0Then { e i ; G , e k ; G } = { e i ; G ′ , e k ; G ′ } = 0, and p (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) = y (cid:16) e X i ; G ′ , e X k ; G ′ (cid:17) = 0.32 ase 2: Check p (cid:16) e X i ; G ′ , e X j ; G ′ (cid:17) = y (cid:16) e X i ; G ′ , e X j ; G ′ (cid:17) , i, j = k p (cid:16) e X i ; G ′ , e X j ; G ′ (cid:17) = π G X l,m ∂µ ∗ k (cid:16) e X i ; G ′ (cid:17) ∂ e X l ; G ∂µ ∗ k (cid:16) e X j ; G ′ (cid:17) ∂ e X m ; G d e X l ; G ∧ d e X m ; G We computed the necessary partial derivatives while addressing the previous case.
Subcase 2.a: ǫ ik , ǫ jk = 0Then p (cid:16) e X i ; G ′ , e X j ; G ′ (cid:17) is given by { e i ; G , e j ; G } e X i ; G e X j ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik × (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − ǫ jk + { e i ; G , e k ; G } e X i ; G e X j ; G e X − sgn( ǫ jk ) k ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik × | ǫ jk | t [ − sgn( ǫ jk ) c k ; G ] + (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − ǫ jk − + { e k ; G , e j ; G } e X i ; G e X j ; G e X − sgn( ǫ ik ) k ; G | ǫ ik | t [ − sgn( ǫ ik ) c k ; G ] + (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik − × (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − ǫ jk This simplifies as follows. e X i ; G (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − ǫ ik e X j ; G (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − ǫ jk × (cid:18) { e i ; G , e j ; G } + { e i ; G , e k ; G } | ǫ jk | e X − sgn( ǫ jk ) k ; G t [ − sgn( ǫ jk ) c k ; G ] + (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − + { e k ; G , e j ; G } | ǫ ik | e X − sgn( ǫ ik ) k ; G t [ − sgn( ǫ ik ) c k ; G ] + (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − (cid:19) The top line of this expression is just µ ∗ k (cid:16) e X i ; G ′ (cid:17) µ ∗ k (cid:16) e X j ; G ′ (cid:17) . So, we would like to show that the three-lineexpression in parenthesis reduces to { e i ; G ′ , e j ; G ′ } = n e i ; G + [ ǫ ik ] + e k ; G , e j ; G + [ ǫ jk ] + e k ; G o = { e i ; G , e j ; G } + { e i ; G , e k ; G } [ ǫ jk ] + + { e k ; G , e j ; G } [ ǫ ik ] + . That is, we would like to see that { e i ; G , e k ; G } | ǫ jk | e X − sgn( ǫ jk ) k ; G t [ − sgn( ǫ jk ) c k ; G ] + (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − + { e k ; G , e j ; G } | ǫ ik | e X − sgn( ǫ ik ) k ; G t [ − sgn( ǫ ik ) c k ; G ] + (cid:16) t [sgn( ǫ ik ) c k ; G ] + + t [ − sgn( ǫ ik ) c k ; G ] + e X − sgn( ǫ ik ) k ; G (cid:17) − is just an overly complicated way to write { e i ; G , e k ; G } [ ǫ jk ] + + { e k ; G , e j ; G } [ ǫ ik ] + .33f sgn ( ǫ ik ) = sgn ( ǫ jk ) =: σ , we have the following simplification. e X − σk ; G t [ − σ c k ; G ] + (cid:16) t [ σ c k ; G ] + + t [ − σ c k ; G ] + e X − σk ; G (cid:17) − ( { e i ; G , e k ; G } | ǫ jk | + { e k ; G , e j ; G } | ǫ ik | )= e X − σk ; G t [ − σ c k ; G ] + (cid:16) t [ σ c k ; G ] + + t [ − σ c k ; G ] + e X − σk ; G (cid:17) − ( { e i ; G , e k ; G } |{ e j ; G , e k ; G } d k | + { e k ; G , e j ; G } |{ e i ; G , e k ; G } d k | )= σd k e X − σk ; G t [ − σ c k ; G ] + (cid:16) t [ σ c k ; G ] + + t [ − σ c k ; G ] + e X − σk ; G (cid:17) − ( { e i ; G , e k ; G } { e j ; G , e k ; G } − { e j ; G , e k ; G } { e i ; G , e k ; G } )=0Likewise, { e i ; G , e k ; G } [ ǫ jk ] + + { e k ; G , e j ; G } [ ǫ ik ] + = { e i ; G , e k ; G } [ { e j ; G , e k ; G } d k ] + + { e k ; G , e j ; G } [ { e i ; G , e k ; G } d k ] + = d k { e i ; G , e k ; G } { e j ; G , e k ; G } (cid:0) [ σ ] + − [ σ ] + (cid:1) = 0 . On the other hand, if sgn ( ǫ ik ) = − sgn ( ǫ jk ) =: σ , we have { e i ; G , e k ; G } | ǫ jk | e X σk ; G t [ σ c k ; G ] + (cid:16) t [ − σ c k ; G ] + + t [ σ c k ; G ] + e X σk ; G (cid:17) − + { e k ; G , e j ; G } | ǫ ik | e X − σk ; G t [ − σ c k ; G ] + (cid:16) t [ σ c k ; G ] + + t [ − σ c k ; G ] + e X − σk ; G (cid:17) − = (cid:18) − σd k { e i ; G , e k ; G } { e j ; G , e k ; G } e X σk ; G t [ σ c k ; G ] + (cid:16) t [ σ c k ; G ] + + t [ − σ c k ; G ] + e X − σk ; G (cid:17) + σd k { e k ; G , e j ; G } { e i ; G , e k ; G } e X − σk ; G t [ − σ c k ; G ] + (cid:16) t [ − σ c k ; G ] + + t [ σ c k ; G ] + e X σk ; G (cid:17) (cid:19) × (cid:16) t [ σ c k ; G ] + + t [ − σ c k ; G ] + e X − σk ; G (cid:17) − (cid:16) t [ − σ c k ; G ] + + t [ σ c k ; G ] + e X σk ; G (cid:17) − = − σd k { e i ; G , e k ; G } { e j ; G , e k ; G } t [ − σ c k ; G ] + t [ − σ c k ; G ] + + t − σ c k ; G ] + e X − σk ; G + t σ c k ; G ] + e X σk ; G t [ − σ c k ; G ] + t [ − σ c k ; G ] + + t − σ c k ; G ] + e X − σk ; G + t σ c k ; G ] + e X σk ; G = − σd k { e i ; G , e k ; G } { e j ; G , e k ; G } Similarly, { e i ; G , e k ; G } [ ǫ jk ] + + { e k ; G , e j ; G } [ ǫ ik ] + = { e i ; G , e k ; G } [ { e j ; G , e k ; G } d k ] + + { e k ; G , e j ; G } [ { e i ; G , e k ; G } d k ] + = d k { e i ; G , e k ; G } { e j ; G , e k ; G } (cid:0) [ − σ ] + − [ σ ] + (cid:1) = − σd k { e i ; G , e k ; G } { e j ; G , e k ; G } . Subcase 2.b: ǫ ik = 0 or ǫ jk = 0If both are 0, the result is immediate. Assume one is non-zero, say ǫ jk = 0. p (cid:16) e X i ; G ′ , e X j ; G ′ (cid:17) = { e i ; G , e j ; G } e X i ; G e X j ; G (cid:16) t [sgn( ǫ jk ) c k ; G ] + + t [ − sgn( ǫ jk ) c k ; G ] + e X − sgn( ǫ jk ) k ; G (cid:17) − ǫ jk + { e i ; G , e k ; G } e X i ; G e X k ; G ( · · · )= { e i ; G , e j ; G } µ ∗ k (cid:16) e X i ; G ′ (cid:17) µ ∗ k (cid:16) e X j ; G ′ (cid:17) (since ǫ ik = 0)= { e i ; G ′ , e j ; G ′ } µ ∗ k (cid:16) e X i ; G ′ (cid:17) µ ∗ k (cid:16) e X j ; G ′ (cid:17) = y (cid:16) e X i ; G ′ , e X j ; G ′ (cid:17) This establishes the proposition. 34
Computation of (cid:0) p γ (cid:1) − (cid:0) p γ (cid:1) − = g − ◦ g ◦ g ◦ g − to order 2, meaning we keep terms whose exponent vectors m satisfy d ( m ) ≤
2. We used this in Section 2.4.1 to compute b D A .First note that for u = u f + u f , g ( A u ) = | u | Y ℓ =1 (cid:16) v sgn( u )2(2 ℓ − A (cid:17) sgn( u ) A u = | u | Y ℓ =1 (cid:16) u ) v sgn( u )2(2 ℓ − A + [ − sgn( u )] + v sgn( u )4(2 ℓ − A + · · · (cid:17) A u = u ) | u | X ℓ =1 v sgn( u )2(2 ℓ − A + (cid:18) X ≤ a
The authors would like to thank Mark Gross, Dylan Rupel and Elmar Wagner for helpful discussions. M.Cheung is supported by NSF grant DMS-1854512, and AMS Simons Travel Grants. J.B. Fr´ıas-Medina issupported by Programa de Becas Posdoctorales 2019, DGAPA, UNAM. T. Magee is supported by EPSRCgrant EP/P021913/1.
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Man-Wai CheungDepartment of Mathematics, One Oxford Street Cambridge, Harvard University, MA 02138,USA e-mail: [email protected]
Juan Bosco Fr´ıas-MedinaCentro de Ciencias Matem´aticas, UNAM Campus Morelia, C.P. 58089, Michoac´an, Mexico e-mail: [email protected]
Timothy MageeSchool of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK e-mail:e-mail: