Twist automorphisms and Poisson structures

We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their compatibility with Poisson structures and quantization. The twist automorphisms always permute well-behaved bases for cluster algebras. We explicitly construct (quantum) twist automorphisms of Donaldson-Thomas type and for principal coefficients.


Cluster algebras
The theory of cluster algebras were introduced by Fomin and Zelevinsky [13] as a combinatorial framework to study the dual canonical bases of quantum groups [28,34,35].In this theory, one has the cluster A-variety A (also called the cluster K 2 -variety).It is a scheme equipped with a cluster structure: A is the union of many tori which are glued by birational maps called mutations [24].Let U A denote the "function ring" of A (by which we mean the ring of global sections of its structure sheaf; it is the coordinate ring when A is affine).Then U A is called the upper cluster algebra (or upper cluster A-algebra).Under a mild assumption (full rank assumption), one can endow A with a Poisson structure [19,20].Correspondingly, the upper cluster algebra U A becomes a Poisson algebra, which can be naturally quantized [5].
One also has the cluster X-variety X (also called the cluster Poisson variety).It is a scheme equipped with the same cluster structure: X is the union of many dual tori which are glued by mutations.It has a canonical Poisson structure.Let U X denote its function ring, which is called the cluster Poisson algebra (or upper cluster X-algebra).Then U X is a Poisson algebra, and one can quantize it naturally.
Fock and Goncharov [9,10] found that the cluster varieties A and X naturally arise in the study of the (higher) Teichmüller theory of a surface.They further conjectured that the upper cluster algebra U A should possess a basis naturally parametrized by the tropical points of the cluster X-variety associated to the Langlands dual cluster structure and, conversely, the cluster Poisson algebra U X should possess a basis naturally parametrized by the tropical points of the cluster A-variety associated to the Langlands dual cluster structure.(By [25], Fock-Goncharov conjecture is true for many cases, but might not be true in general.) In view of Fock-Goncharov conjecture, it is important to understand the upper cluster Aalgebras, upper cluster X-algebras and their bases.
On the one hand, many well-known cluster A-varieties (strictly speaking, the sets of their rational points) are smooth manifolds.Examples include the unipotent cells N w − [15], double Bruhat cells [3], and top dimensional cells of the Grassmannians Gr(k, n), k ≤ n ∈ N [50].On the other hand, there exists few literature on cluster X-varieties.[26,49,51] embedded the Drinfeld double quantum groups of Dynkin types to quantized U X .[48] embedded a subalgebra of a K-theoretic Coulomb branch to quantized U X .
The bases for (quantized) U A have been extensively studied and they have been related to representation theory and geometry (see the survey [46]).Not much is known for the bases of (quantized) U X (see [1,9,25] for some results).
Twist automorphisms [2,4] introduced an automorphism η w on the unipotent cell N w − .Its gives rise to an automorphism η w on the coordinate ring C[N w − ]. η w was called a twist automorphism and has been studied via cluster algebras [16].A quantum unipotent cell A q [N w − ] is a quantum analog of the coordinate ring C[N w − ], which is defined for all Kac-Moody types.In a joint work [33], the first author introduced the quantum twist automorphism η w on A q [N w − ] and further showed that the dual canonical basis B * of A q [N w − ] is permuted by η w .By [15,17,21,23], A q [N w − ] is a (quantum) upper cluster algebra U A .By a motivational conjecture of Fomin and Zelevinsky [13] and its natural generalization [32], the dual canonical basis of A q [N w − ] should contain all cluster monomials, i.e., the monomials of coordinate functions in some toric chart which are globally regular on A. With the help of the existence of the twist automorphism η w and the fact that it permutes the dual canonical basis B * , the second author gave a proof of the generalized conjecture for all cases [47].(See [27,37,44] for other approaches and the corresponding results.) In view of the successful application of the twist automorphism η w to the (quantum) upper cluster algebra A q [N w − ], it is natural to ask for a twist automorphism in general cases.

Preliminaries
Let I denote a finite set of vertices and I = I uf ⊔ I f a partition of I. Choose skew-symmetrizers d i ∈ Z >0 , i ∈ I.The vertices in I uf and I f are called unfrozen and frozen, respectively.A seed t is a collection ((A i ) i∈I , (X i ) i∈I , B) whose matrix B = (b ij ) i,j∈I is Z-valued such that 1 d i b ij = − 1 d j b ji , and A i , X i are indeterminates called cluster A-variables and cluster X-variables, respectively.
Choose a base ring k = Z or Z v ± 1 d ′ for the classical cases or the quantum cases, respectively, where d ′ is a sufficiently divisible positive integer.We associate to t two Laurent polynomial rings LP A = k A ± i i∈I , LP X = k X ± i i∈I .Then, if we take k = C, LP • is the coordinate ring of the corresponding torus T • := Spec LP • , where • stands for A or X.We denote the Laurent monomials by A m for any m = (m i ) i∈I ∈ Z I and, similarly, X n for any n = (n i ) i∈I ∈ Z I .
Unless otherwise specified, we will use a v-twisted product on LP • (see Section 2.3).We will set v = 1 if we work at the classical level.Let F • denote the skew-field of fractions of LP • .A monomial map between LP • is an algebra homomorphism sending Laurent monomials to Laurent monomials.We have a canonical a monomial map p * : LP X → LP A .Following [43,44], we consider the pointed elements, i.e., the elements of the form , where F m and F n are polynomials in indeterminates Z k , k ∈ I uf , with constant term 1 (called F -polynomials).Here, Z k → p * X k and Z k → X k denote the evaluation of Z k , respectively, and • denotes the commutative product.
For any unfrozen vertex k ∈ I uf , we have an algorithm called mutation which generates a new seed µ k t = ((A i (µ k t)) i∈I , (X i (µ k t)) i∈I , B(µ k t)).
Let ∆ + t denote the set of seeds t ′ = µt obtained from t by any finite sequence of mutations µ, where µ takes the form µ ks • • • µ k 2 µ k 1 and starts from the seed t.Note that the mutation sequence µ depends on t, but we omit the symbol t as in standard literature.Recall that the cluster •-variety is the union ) such that the tori (local charts) are glued by mutation birational maps (µ • ) * (coordinate change).The upper cluster •-algebra U • is defined to be its function ring, which turns out to be U ) by choosing any initial seed t (local chart).It is known that, up to identification by mutation maps, these objects are independent of the choice of the initial seed t.So we can simply write U • .

Let
) denote two similar seeds, i.e., there exists a permutation σ on We can relabel the vertices I uf when working with t ′ , so that we can assume σ = Id from now on.
Let pr I uf denote the natural projection from Z I to Z I uf .Following [43,44], two pointed elements S A m ∈ LP A (t) and S A m ′ ∈ LP A (t ′ ) are said to be similar if pr I uf m = pr I uf m ′ and they have the same F -polynomial.Similarly, we define two pointed elements S X n ∈ LP X (t) and S X n ′ ∈ LP X (t ′ ) to be similar if pr I uf B(t)n = pr I uf B(t ′ )n ′ and they have the same F -polynomial.Following [43,44], a variation map (or a correction map, a coefficient twist map) var • t is a map sending pointed elements in LP • (t) to similar pointed elements in LP • (t ′ ), where • stands for A or X, see Remarks 3.5 and 4.5.For the purpose of this paper, we require that var • t is an algebra homomorphism from LP • (t) to LP • (t ′ ).
We define a twist endomorphism tw • t on U • (t) to be the composition µ * var • t of the mutation map µ * with a variation map var • t .In the classical case, it is called Poisson if it preserves the Poisson structure.We can show that the construction is independent of the choice of the initial seed, so we can simply say that tw • acts on U • (Propositions 3.14 and 4.12).Note that different choices of variation maps still give different twist endomorphisms.
Let Z Q denote the multiplicative group generated by the roots of coefficients (frozen cluster The notion of the twist endomorphism tw A can be naturally generalized for U A Z Q .In the classical case, the existence of the twist endomorphisms tw A is given by Theorem 3.19 see Remark 4.21 for the existence of tw X : Assume that the full rank assumption holds.Then the twist endomorphisms tw A on U A Z Q exist, and they are in bijection with the solutions of an inhomogeneous linear equation system. Let var X t ′ denote a monomial map from LP X (t ′ ) to LP X (t) and var A t the corresponding monomial map from LP A (t) Z Q to LP A (t ′ ) Z Q constructed using the pullback (see Theorem 4.19).Let tw • denote their compositions with the mutation maps, respectively.Then tw • are related by Theorem 4.19: tw X is a twist endomorphism (resp.twist automorphism) on U X if and only if tw A is a twist endomorphism (resp.twist automorphism1 ) on U A Z Q .Assume that tw X is a twist automorphism on U X .Then it is Poisson if and only if tw A and tw X −1 commute with the natural homomorphism p * : U X → U A , i.e., p * tw X −1 = tw A p * .Note that a variation map var X t ′ is Poisson if and only if the corresponding linear map satisfies the quadratic equation in Lemma 4.10.
We also explicitly construct Poisson (or quantum) twist automorphisms tw • on U • in the following cases: The case t ′ = t [1] (see Definition 2.19 and Theorem 5.2): the corresponding twist automorphisms are said to be of Donaldson-Thomas type (DT-type for short).The original twist automorphism η w is of this type [47].
The case when there is a seed of principal coefficients (see Section 2.10 and Theorem 5.4).
We prove general results that well-behaved bases for U • satisfying Assumptions 6.1 or 6.3 are permuted by twist automorphisms tw • (see Theorems 6.2 and 6.4).In addition, we proposed a method for constructing bases of U X in Theorem 6.6.
Recall that we can naturally quantize U • if a compatible Poisson structure is given.Then we can lift a Poisson twist endomorphism to a quantum twist endomorphism.Conversely, the classical limit of a quantum twist endomorphism is a Poisson twist endomorphism.See Remarks 3.7 and 4.7.
Remark 1.1 (a comparison with the previous literature).The second author introduced the correction technique to compare similar pointed elements for similar seeds t and t ′ in [43].In order to facilitate the comparison, in [44], he introduced a variation map var A t sending pointed elements in LP A (t) to similar pointed elements in LP A (t ′ ).The variation map var A : LP A (t) Z Q → LP A (t ′ ) Z Q considered in this paper is defined in the same spirit, but chosen slightly differently, such that it becomes an algebra homomorphism.The variation map var X : LP X (t ′ ) → LP X (t) has not been considered before.
In the classical case k = Z, it is easy to see that a variation map var A : LP A (t) → LP A (t ′ ) in this paper is the same as a quasi-homomorphism introduced by Fraser [14].When one identifies F A (t) and F A (t ′ ) using the mutation map µ * , a twist endomorphism tw A on U A becomes the same as a quasi-homomorphism for a normalized seed pattern in the sense of [14].[6] studied the group of the twist automorphisms on U A for principal coefficient cases and several other special cases.
To the best of the authors' knowledge, the Poisson or quantum twist endomorphisms as well as the twist endomorphisms tw X have not been introduced in the previous literature (although specific examples have risen from Lie theory and from higher Teichmüller theory, see Section 7).

Remark 1.2 (morphisms for schemes). Consider the classical cases.
On the one hand, our twist endomorphism tw • provides an endomorphism for the affine scheme Spec U • .
On the other hand, a cluster variety might not be affine, and we do not know if tw • provide an endomorphism for it.While the variation map is always a scheme morphism, the mutations might be not.
See [24] for the comparison between the schemes.

Contents
We provide preliminaries for the theory of cluster algebras in Section 2.
In Sections 3 and 4, we introduce (Poisson, quantum) twist endomorphisms for upper cluster A-algebras and upper cluster X-algebras, respectively.We discuss their existence.
In Section 5, we explicitly construct (quantum) twist automorphisms for two special cases: the Donaldson-Thomas type and the principal coefficients.
In Section 6, we prove that a basis with nice properties is permuted by a twist automorphism.We also construct bases for cluster Poisson algebras.
In Section 7, we give some explicit examples for Poisson twist automorphisms.

Convention
We fix a finite set of vertices I together with a partition I = I uf ⊔ I f .The elements in I uf and I f are said to be unfrozen and frozen, respectively.We also fix skew-symmetrizers d i ∈ Z >0 for i ∈ I. Let D denote the diagonal matrix whose diagonal entries are 1 d i , i ∈ I.We choose a base ring k of characteristic 0 and a unit v ∈ k.For classical cluster algebras, we choose k = Z and v = 1.For quantum cluster algebras, we choose where v is an indeterminate and d ′ ∈ N >0 is sufficiently divisible.The ring multiplication for cluster algebras will be the v-twisted product * .We will also use the commutative product •.
An I × I matrix H is said to be skew-symmetrizable by D if DH is skew-symmetric.
For any I × I-matrix H and any J 1 , J 2 ⊂ I, let H J 1 ,J 2 denote the J 1 × J 2 -submatrix of H. Then we can denote H as a block matrix: Let col i H and row i H denote the i-th column and the i-th row of H, respectively.For any permutation σ of I (resp. of I uf ), let P σ denote the I × I-matrix (resp.the I uf × I uf -matrix) such that col i P σ is the σi-th unit vector.Then we have col k (HP σ ) = col σk H as column vectors and row i (P σ H) = row σ −1 i H as row vectors.We work with column vectors unless otherwise specified.
Assume that σ is a permutation on For a Z-lattice L, we denote Let pr I uf denote the natural projection from Z I to Z I uf .

Seeds and tori
Definition 2.1 (seeds).A seed t is a collection ((A i ) i∈I , (X i ) i∈I , B), where A i and X i are indeterminates called cluster A-variables and cluster X-variables, respectively, and B = (b ij ) i,j∈I is a Z-valued matrix such that DB is skew-symmetric.
Define Laurent polynomial rings LP A = k A ± i i∈I and LP X = k X ± i i∈I .For any vector m, n ∈ Z I , denote the Laurent monomials A m = A m i i and X n = X n i i .We use • to denote their commutative multiplication, which is often omitted for simplicity.
We associate to t a lattice N = Z I with the natural basis e i , i ∈ I, whose elements n = n i e i = (n i ) ∈ Z I are viewed as the Laurent degrees for X n ∈ LP X .Let M denote its dual lattice with the dual basis e * i .Let ⟨ , ⟩ denote the natural pairing between N Q and M Q .Define f i = 1 d i e * i and the sublattice M • = ⊕ i∈I Zf i ⊂ M Q .We identify M • with Z I such that f i become the unit vectors.View its elements m = m i f i = (m i ) ∈ Z I as the Laurent degrees for A m ∈ LP A .
Let e and f denote the I-labeled bases (e i ) i∈I and (f i ) i∈I , respectively.We can view them as matrices whose columns are the basis elements.Define the linear map p * : N → M • such that p * e j = i b ij f i , ∀j.It has the following matrix presentation: In the classical case, extend k to a field containing Z. Denote the affine schemes T A = Spec LP A and T X = Spec LP X .We will call T A and T X the tori associated to t, since T A (k) and T X (k) coincide with the split algebraic torus (k × ) I .

Poisson structures
Recall that d is the least common multiplier of all d i .We can associate to N a 1 d Z-valued canonical skew-symmetric bilinear form −ω such that ω(e i , e j ) = 1 d j b ji , ∀i, j.The corresponding canonical Poisson structure on T X is given by (2.1) Let W denote the Q-valued matrix (ω(e i , e j )) i,j .Recall that the diagonal entries of D are Remark 2.2.For any i ∈ I, we have p * e i ( ) = ω(e i , ).
Remark 2.3.Following [19,20], the Poisson structure of the form (2.1) is usually called logcanonical in the sense that the following holds (for k = R): We often impose the following assumption.Denote the matrix B := (b ik ) i∈I,k∈I uf .Assumption 2.4 (full rank assumption).We assume that the linear map p * restricts to an injective map on N uf .Equivalently, the matrix B is of full rank.

Definition 2.5 ([5]
).By a (compatible) Poisson structure on T A , we mean a collection of strictly positive numbers The corresponding Poisson bracket on T A is defined as The bilinear form λ is represented by the Λ-matrix Λ = (Λ ij ) i,j∈I := (λ(f i , f j )) i,j∈I .
Note that the existence of a compatible Poisson structure implies the full rank assumption (see Assumption 2.4).Conversely, by [19,20], when the full rank assumption is satisfied, we can always choose a (not necessarily unique) compatible Poisson structure.
From now on, we will make the choice such that d ′ k = 1 d k , ∀k ∈ I uf (see [5,Proposition 3.3]).Correspondingly, λ is 1 d ′ Z-valued for some sufficiently divisible d ′ ∈ N >0 .We choose d ′ such that d| d ′ .
Note that the bilinear form λ on M • (t) naturally induces a bilinear form λ on N uf such that λ(n, n ′ ) := λ(p * n, p * n ′ ).
Definition 2.6 (connected matrix).An I ×I matrix B is said to be connected, if for any i, j ∈ I, there exists finitely many vertices i s ∈ I, 0 ≤ s ≤ l, such that i 0 = i, i l = j, and b isi s+1 ̸ = 0 for any 0 ≤ s ≤ l − 1.
It is straightforward to check the following result.
Lemma 2.7.Assume that B is a connected I × I matrix.If DB and D ′ B are both skewsymmetric for some invertible diagonal matrices D, D ′ , then there exists some 0

Quantum torus algebras
Recall that we have chosen d| d ′ and ω on N is 1 d Z-valued.Using the canonical Poisson structure on T X , we equip LP X with an extra multiplication * , called the v-twisted product, such that Then LP X is called a quantum torus algebra for the quantum case k The canonical Poisson bracket (2.1) can be recovered from the twisted product by If there exists a 1 d ′ Z-valued compatible Poisson structure λ on T A (see Definition 2.5), we can equip LP A with an extra multiplication * , called the v-twisted product, such that Then LP A is called a quantum torus algebra for the quantum case k = Z v ± 1 d ′ .As before, the compatible Poisson structure can be recovered from the twisted product by For the classical case k = Z, we have v = 1 and we simply define the v-twisted product * of LP • to be the commutative products •.From now on, we always view LP • as a k-algebra whose multiplication is the v-twisted product. Let be two quantum torus algebras as above, viewed as k-modules.Here L and L ′ denote the lattices of Laurent degrees, respectively, and χ can denote X or A. By a monomial map Φ from LP to LP ′ we mean a k-linear map such that, there exists some linear map Ψ : L → L ′ satisfying Φ(χ u ) = χ Ψ (u) ∀u ∈ L. In this case, Φ is called the monomial map associated to Ψ and Ψ the linear map associated to Φ.
In particular, the linear map p * : N → M • determines a monomial map LP X → LP A sending X n to A p * n , which is still denoted by p * for simplicity.Then p * is an algebra homomorphism preserving the v-twisted products.
Let T denote the subalgebra k[X k ] k∈I uf of the quantum torus algebra LP X .We introduce the k-algebra LP X = LP X ⊗ T T , where T is the completion of T with respect to its maximal ideal generated by X k , k ∈ I uf .Similarly, define LP

Mutation maps
Denote [ ] + = max( , 0).For any vector (g i ), denote [(g Let t denote a given seed and k any chosen unfrozen vertex.Choose any sign ε ∈ {1, −1}.We define a new seed t Let F • denote the fraction fields of the Laurent polynomial rings LP • for classical cases and the skew fields of fractions of the quantum torus algebras LP • for quantum cases.We always call F • fraction fields for simplicity.
We further relate the cluster variables for t ′ with those for t, by introducing isomorphisms for the fraction fields µ Following [45, equation (2.6)] and [47, equations (2.2) and (2.4)], for quantum cases, we define where • denotes the commutative product.Here, we denote The maps (µ • k ) * are called the mutation maps, where • stands for A or X.For classical cases, they induce birational maps µ It is straightforward to check the well-known fact that B ′ does not depend on the choice of the sign ε.Moreover, mutation is an involution on the seeds, and the compositions are the identity.One can also check that mutations commute with the monomial map p * : For simplicity, we might omit the symbol • when there is no confusion.

Hamiltonian formalism
For classical cases, let us recall the Hamiltonian formalism for the mutation maps following [18] (see also [11,25]).Recall that the Euler dilogarithm function is given by One can check that In particular, we have where the commutative products are used.Then the mutation map (µ [5], if t is endowed with a compatible Poisson structure, then we have a unique compatible Poisson structure for t ′ such that the homomorphism ψ A k,ε is a Poisson homomorphism.In addition, it is independent of ε.We still use λ to denote the corresponding skew-symmetric bilinear form on (M • ) ′ , and denote the corresponding matrix by

Mutation sequences
Let k = (k 1 , . . ., k l ) denote a finite sequence of unfrozen vertices.A sequence of mutations We deduce the following equality in F A (t) from (2.2):

Formal Laurent series expansions
We recall the maps ι • taking formal Laurent series expansions following [45, Section 3.3].They will only be used in Lemma 6.5.Take t ′ = µt for any mutation sequence µ.The mutation map (µ (1) (2) We prove it for the X-side, and the proof for the A-side is the same.Our proof is similar to that of [45,Lemma 3.3.7 (2)].

Cluster algebras
Let there be any given initial seed t 0 .We use ∆ + = ∆ + t 0 to denote the set of seeds t = ((A i (t)) i∈I , (X i (t)) i∈I , B(t)) obtained from t 0 by sequences of mutations.If we work with the quantum cases, we also choose a compatible Poisson structure and consider the associated quantization as in Section 2.3.Recall that the cluster variables A j (t 0 ), j ∈ I f , are unchanged by mutations, which are denoted by A j and are called the frozen variables.We use Z to denote the multiplicative group generated by A ± j , j ∈ I f .Using the v-twisted product * , the partially compactified (quantum) cluster algebra A(t 0 ) with the initial seed t 0 is defined to be the k-subalgebra of F A (t 0 ) generated by all the cluster variables µ * A i (t), i ∈ I, t = µt 0 ∈ ∆ + t 0 .The (localized) cluster algebra A(t 0 ) is defined to be its localization A(t 0 ) A −1 j j∈I f .The upper cluster algebra (or upper cluster A-algebra) U A (t 0 ) with the initial seed t 0 is defined as the intersection ∩ t=µt 0 ∈∆ + µ * LP A (t) inside F A (t 0 ).By the Laurent phenomenon [5,13], it contains the cluster algebra A(t 0 ).For classical cases, if a compatible Poisson structure on LP A (t 0 ) is given, then U A (t 0 ) inherits the Poisson structure.
The cluster Poisson algebra (or upper cluster X-algebra) U X (t 0 ) with the initial seed t 0 is defined as the intersection ∩ t=µt 0 ∈∆ + µ * LP X (t) inside F X (t 0 ).For classical cases, it inherits the canonical Poisson structure from that of LP X (t 0 ).
We often identify fraction fields F • (t) and F • (t 0 ) via the mutation map µ * for simplicity.Correspondingly, we omit the symbol t 0 and µ * in the above notations, and we can write U A = ∩ t LP A (t) and U X = ∩ t LP X (t).

Cluster varieties
Let us work at the classical cases.Given two seeds t ′ = µt.Using the mutation birational maps µ : T A (t) T A (t ′ ), we can glue the tori T A (t), t ∈ ∆ + , into a scheme A (see [24, Proposition 2.4]), which is called the cluster A-variety or cluster K2 variety.Note that the upper cluster algebra U A is the ring of global sections of its structure sheaf.A choice of compatible Poisson structure on LP A (t 0 ) gives rise to a Poisson structure on A. It often happens that A is a smooth manifold, for example, for many well-known the cluster algebra arising from Lie theory (unipotent cells [15], double Bruhat cells [3,22]).
Similarly, using the mutation birational maps µ : T X (t) T X (t ′ ), we can glue the tori T X (t), t ∈ ∆ + , into a scheme X called the cluster X-variety or the cluster Poisson variety.The cluster Poisson algebra U X is the ring of global sections of its structure sheaf.Note that X has the canonical Poisson structure.

Transition matrices
We use ( ) T to denote matrix transpose.
Given seeds t ′ = µ k t and a mutation sign ε ∈ {+, −}.Let us describe the monomial part of mutation using transition matrices. 3efine the following I × I-matrix P N k,ε (t): Then it represents the linear map We see that ψ N k,ε induces the monomial part ψ X k,ε of the mutation from F X (t ′ ) to F X (t).Similarly, define the following I × I-matrix P M k,ε (t): Then it represents the linear map We see that ψ M k,ε induces the monomial part ψ A k,ε of the mutation from F A (t ′ ) to F A (t).The following results were known by [5,41], see [31,Section 5.6] for a summary.Proposition 2.9.We have the following equalities: ( , where Λ is the matrix of the Poisson structure.Remark 2.10.It is straightforward to check that we have and similarly
Recall that a vector is said to be sign-coherent if its coordinates are all non-negative or all non-positive.
such that the following statements hold.
(1) Any cluster A-variable A i (t), i ∈ I, has the following Laurent expansion in LP A (t 0 ): such that c 0 = 1.
(2) Any cluster X-variable X i (t), i ∈ I, has the following expression in F X (t 0 ): where P , Q are polynomials in k[X k ] k∈I uf with constant term 1.
(3) The row vectors of F (t) are sign-coherent.
(4) The column vectors of E(t) are sign-coherent.
The I uf × I uf -submatrices C(t) := E(t) uf and G(t) := F (t) uf are usually called the C-matrix and the G-matrix of t (with respect to the initial seed t 0 ), respectively.Definition 2.12 ( [45]).For any pair of seeds t, t ′ ∈ ∆ + , let t be the initial seed and E(t ′ ) (resp.F (t ′ )) denote the corresponding E-matrix (resp.F -matrix) of t ′ .We define the linear map: Example 2.13.Take the index set I = {1, 2} such that 1 is the only unfrozen vertex.Choose The seed t ′ = µ 1 t is the only non-initial seed.Using the commutative product •, we can write or, equivalently, Note that, for any k ∈ I uf , ψ • t,µ k t is represented by the matrix P • k,+ (t), where • stands for M or N .In particular, ψ • t,µ k t ψ • µ k t,t might not be the identity by Remark 2.10.Identify N (t) with N (t 0 ) by using the linear map ψ N t 0 ,t .Then the basis vector e i (t) has the coordinate vector col i E(t) in N (t 0 ) ≃ Z I .Similarly, identify M (t) with M (t 0 ) by using the linear map ψ M t 0 ,t .Then the basis vector f i (t) has the coordinate vector col i F (t) in M • (t 0 ) ≃ Z I .We will often work with vectors, linear maps and bilinear forms in the fixed lattices N (t 0 ) and M (t 0 ).We refer the reader to [24] for more details on the fixed data.

Canonical mutation signs
For any k ∈ I uf , recall that the k-th c-vector col k C(t) is sign coherent.Then we define the canonical mutation sign ε for the mutation of t at the vertex k to be the sign of col k C(t).From now on, we always choose the canonical mutation sign unless otherwise specified.
For k = (k 0 , k 1 , . . ., k r ), denote k ≤s = (k 0 , . . ., k s ) for any 0 ≤ s ≤ r − 1 and t s = µ k ≤s−1 t 0 .Let ε s denote the canonical sign for the mutation of t s at the direction k s .

Principal coefficients
Let us recall the seeds with principal coefficients and their relation with the C-matrix and the G-matrix.This part will only be used in Sections 5.2 and 6.3.Denote a copy of I uf by I ′ uf = {k ′ |k ′ ∈ I uf }.We extend the corresponding principal B-matrix B uf to the ( , which is called the principal coefficient B-matrix, where Id I uf represents the natural isomorphism I uf ≃ I ′ uf .The corresponding diagonal matrices are We denote by t prin 0 the seed obtained from t 0 by changing the fixed data as above.Let k denote a sequence of vertices and t = µ k t 0 .Then it is known that the C-matrix and the G-matrix can be computed using principle coefficients: where Then, using Lemma 2.15, we obtain

Degrees and pointedness
We endow N uf (t) = ⊕Ze k ≃ Z I uf with the natural partial order such that n It is further said to be pointed at n or n-pointed if c 0 = 1.Now assume that t is a seed such that ker p * ∩ N ≥0 uf (t) = 0.This condition is satisfied when the seed t satisfies the full rank assumption (see Assumption 2.4).We recall the degrees and pointedness introduced in [44,45].Definition 2.17 (dominance order [44]).For any m, m ′ ∈ M • (t), we say m ′ is dominated by m, denoted by m ′ ⪯ t m, if m ′ = m + p * n for some n ∈ N ≥0 uf (t).

Definition 2.18 (pointedness [44]). A formal Laurent series Z ∈ LP
A (t) is said to have degree m, denoted by for some c 0 ̸ = 0, c n ′′ ∈ k, i.e., its ≺ t -maximal Laurent degree is m.Its F -function is defined as It is further said to be pointed at m or m-pointed if c 0 = 1.Theorem 2.11 implies that deg t 0 X i (t) = col i E(t) and deg t 0 A i (t) = col i F (t).By definition, ψ N t 0 ,t is the linear map sending e i (t) = deg t X i (t) to deg t 0 µ * X i (t).Similarly, ψ M t 0 ,t is the linear map sending 2.12 Injective-reachable Definition 2.19 (injective-reachable [44, Section 2.3]).A seed t 0 is said to be injective-reachable, if there exists another seed t 0 [1] = µt 0 ∈ ∆ + t 0 and a permutation σ of I uf , such that for any k ∈ I uf , we have ψ N t 0 ,t 0 [1] e σk (t 0 [1]) = −e k (t 0 ). (2.5) The sequence µ is also called a green to red sequence from t 0 to t 0 [1] in the sense of Keller [30].Assume that t 0 is injective-reachable.Then all t ∈ ∆ + t 0 are injective-reachable, i.e., we can always find a seed t [1], see [44,Proposition 5.1.4]or [38].
By [44, Proposition 2.3.3], for any k ∈ I uf , we have We will see that t 0 [1] ∈ ∆ + is similar to t 0 up to σ in the sense of Definition 3.1.

Twist endomorphisms for upper cluster A-algebras
In this section, we introduce the notion of a twist endomorphisms for a pair of similar seeds t, t ′ , which is defined as the composition of a mutation map with a monomial map called a variation map.

Similar seeds and variation maps
We first recall the definition of similar seeds.
Definition 3.1 (similar seeds [43,44]).Two seeds t, t ′ are called similar, if there is a permutation σ of I uf such that for any i, j ∈ I uf , we have b ij (t) = b σi,σj (t ′ ) and skew-symmetrizers If t and t ′ are similar, in our choices of compatible Poisson structures, we automatically have ).Note that we can always trivially extends σ to a permutation on I. Proposition 3.2.Assume that B uf is connected Definition 2.6.Let t = µt 0 be a given seed.If there is a permutation σ of I uf such that b ij (t) = b σi,σj (t 0 ) for any i, j ∈ I uf , then d k = d σk for any k ∈ I uf .
Proof .Let P σ be the permutation matrix associated to σ of rank I uf , i.e., the i-th column of P σ is the σi-th unit vector.Notice that P On the other hand, we have B(t) = F (t) −1 B(t 0 )E(t) by Lemma 2.15.Using DF (t) −1 = E(t) T D by (2.4), we obtain Since DB(t 0 ) is skew-symmetric, so is DB(t).It follows that D uf B(t) uf is also skew-symmetric.
Then Lemma 2.7 implies that P −1 σ D uf P σ = αD uf for some α ̸ = 0. Since D uf is of full rank, by taking the determinant, we see that α = 1.The claim follows.■ Assume t, t ′ are similar up to a permutation σ.In view of the correction technique [43], it is often useful to compare pointed elements in LP A (t) with those in LP A (t ′ ).As in [44], we define a variation map sending pointed elements in LP A (t) to those in LP A (t ′ ), which differ only by the frozen variables.
In this paper, we will allow roots of frozen variables.For any given integer number r > 0, we define Z 1 r to be the multiplicative group generated by the r-th roots A ± 1 r j , j ∈ I f .Define Z Q to be the multiplicative group generated by any A for some p σk , p j ∈ Z Q .For classical cases, if t and t ′ are similar seeds with compatible Poisson structures, then var A t is called a Poisson variation map if it further preserves the Poisson structures: Remark 3.5.The variation map var A t is a k-algebra homomorphism such that it sends a pointed element Z ∈ LP A (t) to a similar element in LP A (t ′ ) Z Q .
The variation map in Definition 3.3 is the monomial map associated to the following linear variation map.
Denote M • uf (t) = ⊕ k∈I uf Zf k , M • f (t) = ⊕ j∈I f Zf j .Definition 3.6.Let t, t ′ be two seeds similar up to a permutation σ.A linear map var M t : M for some Note that (3.1) can be written as Remark 3.7.Assume the existence of compatible Poisson structures, i.e, Assumption 2.4 holds.
Then the following statements are equivalent: For the quantum case k = Z v ± 1 d ′ , the monomial map Φ v associated to Ψ is a variation map.In particular, it preserves the v-twisted products.
For the classical case k = Z, the monomial map Φ 1 associated to Ψ is a Poisson variation map.
Therefore, a Poisson variation map is the classical limit of a quantum variation map at v = 1, see Section 2.3.Conversely, a Poisson variation map gives rise to a quantum variation map by the above equivalent statements.
Let P σ denote the permutation matrix associated to σ of rank I uf .Lemma 3.8.
(1) The variation map var M t has the following matrix representation in the bases f , f ′ : where (2) Moreover, equation (3.1) is equivalent to the following Proof .
(1) Recall that col k (HP σ ) = col σk H for any matrix H.The first statement follows.
(2) For any k ∈ I uf , we have We have Thus var M t −1 is a variation map with permutation σ −1 .
Since var M t is a variation map, we have var Proof .Take any mutation sequence µ = µ kr • • • µ k 1 , which is the identify if the sequence (k r , . . ., k 1 ) is empty.Denote σµ = µ σkr • • • µ σk 1 .Then s = µt and s ′ = (σµ)t ′ are similar up to σ.To prove the claim, it suffices to show that, for any z ∈ LP A (t) ∩ µ * LP A (s), we have var A t (z) ∈ (σµ) * LP A (s ′ ).If so, we obtain that, for any By the Laurent expansion of z in the seed s, we have z * µ * A(s [44,Lemma 4.2.2 (ii)], the variation map var A t sends cluster monomials of s to the cluster monomials of s ′ up to frozen factors: we have

Existence of variation maps
Assume that there is a linear variation map var M t : ) for some r > 0. Similarly, the corresponding variation map var A t : The restriction of a linear variation map to sublattices (resp.the restriction of a monomial variation maps to subalgebras) will be still called a variation map.Proposition 3.11.Let there be given two seeds t, t ′ similar up to a permutation σ.Assume that the full rank assumption holds.
(1) There exists a variation map var M t : (3) If an order |I uf | minor of B is ±1, then there exists a variation map var M t : Proof .Relabeling the vertices σk by k for t ′ if necessary, we assume σ = Id in this proof.
(1) We need to construct a linear map var M t : 3) holds.The unknown matrix U := U f,uf U f needs to satisfy the following inhomogeneous equation: is of full rank.Then we can always choose a (not necessarily unique) size-|I uf | subset J ⊂ I such that the submatrix B J,I uf is full rank.Its inverse matrix (B J,I uf ) −1 has entries in Taking u i = 0 for i ∈ I\J and letting u i , i ∈ J equal the columns of the , we obtain a special solution U 0 for (3.4), whose entries lie in (2) Let U 0 denote the special solution in (1).Let Z Q denote the set of solutions of the homogeneous linear system U B = 0, or equivalently (3) It is a direct consequence of the above argument since det B J,I uf is ±1.■

Change of seeds
We treat lattices M • (t) and the corresponding quantum torus algebras LP A (t) in this subsection.
Our arguments and results remain valid after the extension to M • (t) Q and LP A (t) Z Q .Let t ′ , t ∈ ∆ + denote seeds similar up to a permutation σ.Let there be given a linear variation map var M t : ), which are represented by the matrices P M k,+ (t) and P M σk,+ (t ′ ), respectively.Define so that the following diagram commutes: The following properties for var M s are analogous to those in [47, Proposition 5.1.6].
(1) The linear map var M s is a variation map.
Proof .As before, we can assume σ = Id by relabeling the vertices for t ′ and s ′ .Then P M k,+ (t) uf coincides with P M k,+ (t ′ ) uf by the similarity between t and t ′ .It is of the block matrix form (1) The statement can be translated from [47, Proposition 5.1.6]by using Lemma 3.13.Let us give a direct proof.The linear map var M s has the following matrix representation: We see that the matrix is block diagonal, such that its I uf × I uf submatrix is Id uf (by the similarity between t, t ′ ) and its I uf × I f submatrix is 0.
We need to check (3.2).Let e uf denote the matrix (e i ) i∈I uf where e i are column vectors.Then P N k,+ (t) uf coincides with P N k,+ (t ′ ) uf by the similarity between t and t ′ .Recall that, for j ∈ I uf , we have We deduce that col j e(t) P N k,+ (t) = col j e(t) uf P N k,+ (t) uf .Using Lemma 2.15, we have We also have Recall that var M t (p * e j (t)) = p * e j (t ′ ) for any j ∈ I uf .Then We define the monomial map var A s : LP A (s) → LP A (s ′ ) to be the monomial map associated to var M s = ψ M t ′ ,s ′ −1 var M t ψ M t,s .Since var A t is a variation map, it is also a variation map, see Remark 3.7.
Proposition 3.14.The map var A t µ A k * agrees with µ A σk * var A s , i.e., the following diagram commutes: k * A i (s) and var A s A i (s) are cluster variables in s ′ up to a frozen factor by [44,Lemma 4.2.2].By Lemma 3.13, they have the same degree.The claim follows.■ We can given a different proof for Proposition 3.14 by straightforward computation (see Proposition 4.12).
Take any mutation sequence µ and denote s = µt, s ′ = (σµ)t ′ .By using Proposition 3.14 recursively, var A t and µ uniquely determine a variation map var A s such that the following diagram is commutative: By tracking the degrees of cluster variables, we see that the variation map var M s associated to var A s is determined by var M t and µ.It still makes (3.6) commutative.

Twist endomorphisms
We are now ready to define twist endomorphisms.Notice that the mutation maps always preserves the compatible Poisson structures and the v-twisted products.Let t and t ′ = µt be two seeds similar up to σ as before.Let there be given a variation map Correspondingly, we can simply denote the twist endomorphisms tw A t by tw A when we do not want to choose a specific seed t.
Proposition 3.17.The twist endomorphism tw A restricts to an endomorphism on A Z Q .

Proof . Recall that var
Proof .We have ■ Proposition 3.11 implies the following result.
Theorem 3.19.Consider the classical case k = Z.Assume that t, t ′ ∈ ∆ + are similar and the full rank assumption holds, then the following statements are true.
(1) There exists a twist endomorphism tw A t on U A Z Q passing through the seed t ′ .The set of such twist endomorphisms is in bijection with a Q-vector space of dimension (2) If an order |I uf | minor of B is ±1, then there exists a twist endomorphism tw A t on U A passing through the seed t ′ .The set of such twist endomorphisms is in bijection with a lattice of rank 4 Twist endomorphisms for upper cluster X-algebras

Variation maps
Let t, t ′ be similar seeds as before.In the following calculation, we choose t as the initial seed.We want to construct twist endomorphism on U X (t ′ ) preserving the canonical Poisson structure (2.1).
Let us first investigate linear maps between lattices N (t) which arise as the pullback of those between M • (t).Let var M t denote a Z-linear map M • (t) → M • (t ′ ) Q , its pullback between the dual lattices gives rise to a Z-linear map var M t * from N (t ′ ) to N (t) Q .Recall that the diagonal entries of D are 1 d i , i ∈ I.
Lemma 4.1.The pullback var M t * : N (t ′ ) → N (t) Q is represented by the matrix with respect to the bases e and e ′ .
Proof .By assumption, we have Pσ 0 0 Id f .Then its pullback with respect to the dual bases is represented by the transpose: In view of Lemma 4.1, we propose the following definition.
Definition 4.2.A linear map var N t ′ : N (t ′ ) Q → N (t) Q is said to be a variation map, if it has the following Q-valued matrix representation in the basis e ′ = {e ′ i | i ∈ I}, e = {e i | i ∈ I}: and it satisfies for some n (j) ∈ Z I satisfying, ∀n ′ ∈ Z I , where X n := var X t ′ (X ′ ) n ′ .For classical cases, var X t ′ is called a Poisson variation map if it further preserves the Poisson structures: Definition 4.4.In LP X (t ′ ), take any pointed formal Laurent series Z ′ = (X ′ ) n ′ • F , where We say Z ′ and Z ∈ LP X (t) are similar if Z takes the form Remark 4.5.The variation map var X t ′ is a k-algebra homomorphism such that it sends a pointed element Z ′ ∈ LP X (t ′ ) to a similar element in LP X (t).
Remark 4.6.Let var X t ′ be a monomial variation map and var N t ′ the associated linear map.Then var N t ′ must satisfies (4.1).We can study condition (4.3) by taking n ′ to be e ′ i and consider the k-th rows of both sides for all i ∈ I, k ∈ I uf .Recall that ω(n, e σ −1 k ) = 1 We see that this equation is equivalent to (4.2) by using e σ −1 k = var N t ′ e ′ k .Remark 4.7.By using Remark 4.6, we obtain the following equivalent statements: A linear map Ψ : N (t ′ ) → N (t) is a Poisson variation map.
For the quantum case k = Z v ± 1 d ′ , the monomial map Φ v associated to Ψ is a variation map.
For the classical case k = Z, the monomial map Φ 1 associated to Ψ is a Poisson variation map.
Therefore, a Poisson variation map is the classical limit of a quantum variation map at v = 1, see Section 2.3.Conversely, a Poisson variation map gives rise to a quantum variation map by the above equivalent statements.Proposition 4.8.A linear map var N t ′ : Proof .Relabeling the vertices σk by k for t ′ if necessary, we assume that σ is the identity.
Note that, by taking the matrix presentations, the commutative diagram (4.4) is represented by a quadratic equation on the entries of var N t ′ .
Proof .Take any i, j ∈ I. On the one hand, we have p * e ′ i (e ′ j ) = ω(e ′ i , e ′ j ).On the other hand, we have Therefore  ( The statement is obvious. ( s ′ is also a variation map.Let T denote the subalgebra k[X k ] k∈I uf of the quantum torus algebra LP X (t).Let F X uf (t) denote the subalgebra of the fraction field F X (t) such that its elements take the form P * Q −1 with P ∈ LP X (t), Q ∈ T .Then it is easy to check µ X k * F X uf (t ′ ) = F X uf (t) by using the invertibility of µ X k .We naturally extend the variation map var X t ′ : LP X (t ′ ) → LP X (t) to an algebra homomorphism var Proposition 4.12.We have µ X k * var X s ′ = var X t ′ µ X σk * , i.e., the following diagram commutes: Proof .We will denote µ * k = µ X k * and ψ = ψ N below for simplicity.Relabeling the vertices for s ′ , t ′ if necessary, we can assume σ = Id.It suffices to show var We set ε = 1 if −ω(e k (s ′ ), e i (s ′ )) ≥ 0 and ε = −1 otherwise.Then we have Similar to (4.6), we have where we set ε ′ = 1 if −ω e k (s), var N s ′ e i (s ′ ) ≥ 0 and ε ′ = −1 otherwise.It follows that ε = ε ′ and var ) in polynomial expansions of the right hand sides of (4.7) and (4.8) by the quantum numbers ( a b ) v k .Therefore, we still have The following statement follows from Proposition 4.12 by tracking the degrees of X-variables.It is an analog of Lemma 3.13.
Take any mutation sequence µ and denote s = µt, s ′ = (σµ)t ′ .By using Proposition 4.12 recursively, var X t ′ and µ uniquely determine a variation map var X s ′ such that the following diagram is commutative: Let x 3 denote the minor . By [4], there exists a unique element y in N − ∩ B + P w 0 x T when x ∈ N w 0 − .The twist automorphism η w 0 on N − is defined such that η w 0 (x) = y.
Let us compute y explicitly.Take any element Note that We have

Dehn twists for surface cases
Let Σ denote a triangulable surface S with finitely many marked point.For any of its (tagged) triangulation ∆, one can construct a seed t ∆ , whose frozen vertices are contributed from curves on the boundary ∂S.We refer the reader to [12,39] for details.The punctures are the marked point in the interior S • of S. By [39], the seed t ∆ satisfies the full rank condition when Σ has no punctures.
Let L denote any closed loop in the interior of S which does not pass a marked point or is contractible to a marked point.A Dehn twist tw L around L produces a new (tagged) triangulation tw L ∆ from ∆.By [12], we have t tw L ∆ = µt ∆ for some mutation sequence µ.By construction, B(t ∆ ) = B(t tw L ∆ ).In particular, t ∆ and t tw L ∆ are similar.
Then one can construct the variation maps var M t ∆ : M • (t ∆ ) → M • (t tw L ∆ ) and var N t ∆ : N (t ∆ ) → N (t tw L ∆ ) represented by the identity matrices, respectively.The corresponding twist automorphisms are determined by They give rise to automorphisms on U A (t ∆ ) and U X (t ∆ ) associated to the Dehn twist tw L .

Once-punctured digon
Example 7.2.The following cluster Poisson algebra arises from PGL 3 -local systems on a oncepunctured digon [49].Take I = {1, 2, 3, 4}, with I uf = {2, 4} and I f = {1, 3}.Define the initial seed t such that its B-matrix is Note that B uf = 0.So we cannot endow a compatible Poisson structure λ for LP A (t).We have W = −B.Let t ′ denote the seed µt where µ = µ 2 µ 4 .Its B-matrix is B ′ = −B.We have W ′ = −W .We can check the following mutation rule: Then t ′ = t with row indices {1, 3} and column indices {2, 4}.We want to construct a linear isomorphism var N t := : N (t) → N (t ′ ) represented by an Z-valued invertible matrix The desired equation is equivalent to By computing its block submatrices, we reduce the equation to the equations For the first equation (for var N t to be a variation map), the solution takes the form V f = λ−1 µ λ µ−1 for any λ, µ.Since we are looking for a bijection var N t between lattices, we must have λ + µ = 0, 2.

A:=
LP A ⊗ p * T p * T .The elements in LP • will be called formal Laurent series.Note that p * extends to a homomorphism from LP X to LP A .

Theorem 2 .
11 ([7, 25, 52]).There exist I × I invertible Z-matrices 1 det B J,I uf Z.Let u i , i ∈ I, denote the (unknown) columns of U and b i , i ∈ I, denote the rows of B, then U B = i∈I u i b i .

. 5 )
Equation (3.5) has a unique solution u i , i ∈ I, for any given u i , i ∈ I\J.Hence the set of Q-solutions is a vector space of dimension |I f | • |I uf |.It follows that the solutions for (3.4) take the form U + U 0 , U ∈ Z Q .
and the diagonal entries of D are 1 d i ., we have f * = eD −1 and, similarly, (f ′ ) * = (e ′ )D −1 .So we obtainvar M t * e ′ D −1 = eD −1 • P σ −1 0 0 Id f Id uf U T f,uf 0 U T f ,and, equivalently,var M t * (e ′ ) = eD −1 P σ −1 0 0 Id f Id uf U T f,uf 0 U T f D. ■ Since tand t ′ are similar, we have d k = d σk by definition.Then it follows that

Lemma 4 . 13 .
The variation map var N s

(4. 9 )
Let us denote an element x ∈ N − by
and t ′ are similar seeds with compatible Poisson structures λ, then it is called a Poisson variation map if we have Lemma 3.10.A variation map var A t : LP A (t) → LP A (t ′ ) restricts to an algebra homomorphism from U A (t) to U A (t ′ ).
M t p * e k = p * e ′ σk for any k ∈ I uf , it follows that var M t −1 p * e ′ k = p * e σ −1 k .So (3.2) holds.Finally, if var M t preserves the compatible Poisson structures, so does its inverse.■ The statement is obvious.(3)By Lemma 2.15, ψ M t,s and ψ M t ′ ,s ′ preserve the bilinear form λ. The claim follows.■ ′ ) denote the monomial map associated to var M t .We further assume that it is a variation map (equivalently, var M t needs to be Poisson if we work at the quantum level, see Remark 3.7).
t : LP A (t) → LP A (t a twist endomorphism passing through the seed t ′ = µt, which is denoted by tw A t .For classical cases, if t and t ′ are equipped with compatible Poisson structures which are preserved by tw A t , then tw A t is called a Poisson twist endomorphism.Corollary 3.16.A twist endomorphism tw A t on U A (t) Z Q (resp.on U A (t)) gives rise to twist endomorphisms tw A s on U A (s) Z Q (resp.on U A (s)) for all seeds s ∈ ∆ + via the mutation maps.For classical cases, if tw A t is Poisson, so are tw A s .
If var At is an invertible variation map, then we have a twist automorphismtw A t := µ * var A t on U A (t) Z Q .Lemma 3.9 implies that var A t −1 is still a variation map.So η := (µ * ) −1 var A is a twist automorphism on U A (t ′ ) Z Q .Lemma 3.18.We have µ * η(µ * ) −1 = tw A t −1 .Namely, by identifying the fraction fields by mutations, the twist automorphisms associated to invertible variation maps var A A t sends cluster variables to cluster variables up to a frozen factor, see [44, Lemma 4.2.2].The claim follows.■ t : LP .2) It is said to be Poisson if it further preserves the Poisson structures: We will see in Proposition 4.8 that (4.2) is natural.Definition 4.2 will be naturally deduced from the definition of the following monomial variation map, see Remark 4.6.Note that we always have var N t ′ (e ′ σk ) = e k for k ∈ I uf .In particular, (4.2) is an inhomogeneous linear system of equations on var N t ′ .Definition 4.3.A k-algebra homomorphism var X t ′ : LP X (t ′ ) → LP X (t) is called a (monomial) variation map if, for any k ∈ I uf , j ∈ I f , we have Denote the pullback of var N t ′ by var M t .By Lemma 4.1, var N t ′ takes the form of (4.1) if and only if var M t takes the form of (3.2).It remains to show that var M t p The following result is analogous to Lemma 3.9.An invertible linear map var N t ′ : N (t) → N (t ′ ) is a (Poisson) variation map if and only if its inverse is.
* (e k ) = p * e ′ k is equivalent to ω var N t ′ e ′ i , var N t ′ e ′ k = ω(e ′ i , e ′ k ), ∀i ∈ I, k ∈ I uf .Recall that p * e i ( ) = ω(e i , ), we have i ,e k = −p * (e k ) var N t ′ e ′ i = − var M t p * (e k )(e ′ i ).* (e k ) = p * (e ′ k ).■ ′ : N (t ′ ) → N (t)is represented by an invertible matrix Lemma 4.10.Assume that var N t ′ : N (t ′ ) Q → N (t) Q is a linear map.Let var M t denote its pullback.The following diagram commutes if and only if var N t ′ preserves ω M also of the form in (4.1).The other conditions in Definition 4.2 can be checked easily.■ , p * = var M t p * var N t ′ if and only if var N t ′ preserves ω. ■ 4.2 Change of seeds Let t ′ , t ∈ ∆ + denote seeds similar up to a permutation σ.For any k ∈ I uf , consider the seeds s = µ k t and s ′ = µ σk t ′ .We define the linear map var N s ′ : N (s ′ ) → N (s) to be ψ N (3)3)As in the proof of Proposition 3.12 (1), straightforward computation shows that var N s ′ is a block triangular matrix of the form in (4.1).By Lemma 2.15, ψ N s,t andψ N s ′ ,t ′ preserve the Poisson structure ω.So if ω var N t ′ e i (t ′ ), var N t ′ e j (t ′ ) = ω(e i (t ′), e j (t ′ )) for some i, j, then ω var N s ′ e i (s ′ ), var N s ′ e j (s ′ ) = ω(e i (s ′ ), e j (s ′ )).The desired claims follow.■ Note that var N t ′ ψ t ′ ,s ′ e i (s ′ ) = ψ t,s var N s ′ e i (s ′ ) by the definition of var N s ′ .Moreover, since var N s ′ is a variation map by Proposition 4.11, we have ω e k (s), var N s ′ e i (s ′ ) = ω var N s ′ e k (s ′ ), var N s ′ e i (s ′ ) = ω(e k (s ′ ), e i (s ′ )).