aa r X i v : . [ m a t h . QA ] J un DUAL CANONICAL BASES AND QUANTUMCLUSTER ALGEBRAS
FAN QIN
Dedicated to Professor Bernard Leclerc on the occasion of his sixtieth birthday
Abstract.
Given any quantum cluster algebra arising from aquantum unipotent subgroup of symmetrizable Kac-Moody type,we verify the quantization conjecture in full generality that thequantum cluster monomials are contained in the dual canonicalbasis after rescaling.
Contents
1. Introduction 21.1. Background 21.2. Notions, main results and comments 31.3. Contents 8Acknowledgments 82. Basics of cluster algebras 82.1. Seeds 92.2. Laurent polynomial rings 122.3. Mutation 142.4. Cluster algebras 202.5. Injective-reachability 213. Prerequisites on dominance order and decomposition 223.1. Dominance order and pointedness 223.2. Unitriangularity 243.3. Dominance order decomposition 243.4. Change of seeds 264. Similar seeds and a correction technique 274.1. Similar seeds 274.2. A correction technique 295. Cluster twist automorphisms 315.1. Twist automorphisms passing through similar seeds 315.2. Twist automorphism of Donaldson-Thomas type 346. Triangular bases 356.1. Triangular functions 356.2. The triangular bases 376.3. Triangular bases and twist automorphisms 396.4. Adjacent seeds 40
Introduction
Background.
Cluster algebras were introduced by Fomin andZelevinsky around the year 2000 [FZ02]. Their work was rooted inthe desire to understand, in a concrete and combinatorial way, thetheory of total positivity [Lus94] and the dual canonical bases in quan-tum groups U q − ( g ) [Lus90, Lus91][Kas93]. Cluster algebras have dis-tinguished generators called cluster variables. As a main motivation,Fomin and Zelevinsky conjectured that the cluster monomials (certainmonomials of cluster variables) belong to the dual canonical basis.The theory of cluster algebras has enjoyed a rapid growth and ithas soon been linked to many (sometimes unexpected) other topicssuch as combinatorics, representation theory, Lie theory, discrete inte-grable systems, Poisson geometry, higher Teichm¨uller theory, algebraicgeometry. We refer the reader to the surveys [Kel08, Kel]. Yet themotivational conjecture by Fomin and Zelevinsky still remains open.Using the notion of quantum cluster algebras [BZ05], a precise state-ment for the conjecture relating cluster theory and the dual canonicalbases for quantum groups was later formulated [Kim12, Conjecture1.1]. Conjecture 1.1.1 (Quantization conjecture) . Given any symmetriz-able Kac-Moody algebra g and any Weyl group element w ∈ W , up The conjecture was called the quantization conjecture by Kimura after the works[GLS11, GLS12] on (classical) unipotent subgroups. to q -power rescaling, the corresponding quantum unipotent subgroup A q [ N − ( w )] is a quantum cluster algebra and its dual canonical basiscontains the quantum cluster monomials. Here, A q [ N − ( w )] is a quantum analogue of the coordinate ring of theunipotent subgroup N − ( w ). We postpone the precise definition andsimply recall that A q [ N − ( w )] = U q − ( g ) for semisimple Lie algebras g and the longest element w .The cluster structure part in Conjecture 1.1.1 is no longer a problem.For symmetric Kac-Moody cases, A q [ N − ( w )] are known to be quantumcluster algebras by [GLS11][Kim12][GLS13]. [GY16b] found the quan-tum cluster structure on A q [ N − ( w )] for symmetrizable semisimple Liealgebras, and their arguments remain effective in general. When theauthor is preparing this paper, an explicit treatment for general caseshas become available in [GY20].The dual canonical basis part in Conjecture 1.1.1 has been largelyopen for many years. First assume g is a symmetric Kac-Moodyalgebra. [GLS12] proved an analogous statement for the dual semi-canonical basis [Lus00]. Partial results (acyclic cases) have been ob-tained, for example by [KQ14] following [HL10][Nak11]. Symmetriccases were solved only recently. Based on completely different meth-ods, [Qin17] proved it for ( g , w ) such that g is a semisimple Lie algebra, w arbitrary, or g is a general Kac-Moody algebra but w is adaptable ;[KKKO18] proved it for all ( g , w ). But, up to now, Conjecture 1.1.1 hasbeen almost untreated when g is a symmetrizable Kac-Moody algebra.The main aim of this paper is to prove Conjecture 1.1.1 in full gen-erality for any symmetrizable Kac-Moody algebra g and any w ∈ W .1.2. Notions, main results and comments.
Quantum cluster algebras.
A seed t of a cluster algebra is a collectionof cluster variables together with some combinatorial data. The set ofall seeds of a cluster algebra is denoted by ∆ + , and one can generateone seed from another (adjacent seed) via an algorithm called muta-tion. The cluster algebra is generated by its cluster variables, and itsquantization can be constructed. See Section 2 for rigorous definitionsand treatments.The readers unfamiliar with cluster theory might prefer a more in-tuitive geometric picture. An upper cluster algebra is the coordinatering of a variety called the cluster variety, and it often agrees with thecluster algebra. The cluster variables correspond to local coordinatesand the seeds to the charts of an atlas. The combinatorial data de-scribe the gluing of adjacent charts (mutation). When an appropriate(log-canonical) 2-form is chosen, the cluster variety becomes a Poissonmanifold. Correspondingly, its coordinate ring can be quantized to thequantum cluster algebra. FAN QIN
Tropical geometry appears once we look at the cluster varieties, see[FG06][FG09][GHK15], and a breakthrough has recently been madein this direction [GHKK18]. Notice that the elements of cluster alge-bras are multivariate Laurent polynomials. In this paper, by tropicalproperties, we mean properties of the corresponding Laurent degrees,following [Qin19a].For most of this paper, we assume that the cluster algebra is injective-reachable (equivalently, there exists a green to red sequence [Kel11] or,the Donaldson-Thomas transformation is cluster). This condition issatisfied by almost all well-known cluster algebras, such as those aris-ing from Lie theory or higher Teichm¨uller theory.
Triangular bases.
The approach to Conjecture 1.1.1 in the present pa-per is based on the triangular bases defined in [Qin17].For each given seed t , we have a quantum Laurent polynomial ring LP ( t ) which contains the quantum cluster algebra. Let M ◦ ( t ) denotethe lattice of Laurent degrees. Inspired by representation theory, wecan endow M ◦ ( t ) with a partial order ≺ t called the dominance orderwith respect to t .Following [Qin17], we define the triangular basis for each single seed.It is characterized by a triangularity property with respect to the dom-inance order ≺ t , whence the name “triangular”. [Qin17] further in-troduced the common triangular basis for all seeds such that it hasthe compatibility property (certain tropical property, Definition 3.4.1)expected by the Fock-Goncharov dual basis conjecture [FG06][FG09,Section 5][GHKK18, Proposition 0.7].It is worth remarking that the common triangular basis contains allquantum cluster monomials if it exists. In general, we do not knowif the triangular basis for one seed exists or not (though we can stillconstruct a unique topological basis candidate in the formal comple-tion, see Theorem 6.1.3). Even it exists, it is unclear if it provides thecommon triangular basis for all seeds. In a word, neither the local ex-istence nor the global existence is known. Nevertheless, by Lie theory,we have the following results for a symmetric Kac-Moody algebra g .(1) The quantum unipotent subgroup A q [ N − ( w )] is a quantum clus-ter algebra such that, after rescaling and localization, the dualcanonical basis provides the common triangular basis [Qin17,Theorem 1.2.1(I), 6.1.6 ][KK19].(2) A similar statement for cluster algebras categorified by repre-sentation theory of the quantum loop algebra U q ( L g ) [HL10],where the common triangular basis arises from simple modules[Qin17, Theorem 1.2.1(II)]. Remark 1.2.1 (Positivity assumption) . Many previous arguments aboutthe triangular basis [Qin17] rely on the assumption that the basis has positive multiplication structure constants. Such a positivity assump-tion is natural, because the work [Qin17] was inspired by the monoidalcategorification of cluster algebras in the sense of [HL10] . In particular,the basis is expected to consist of the simple modules.It is known that the dual canonical basis no longer satisfies the posi-tivity assumption in symmetrizable cases . This is the main obstructionthat renders many previous arguments ineffective.Main results and comments. Theorem 1.2.2 (Main theorem, Theorem 9.5.1) . After localizationand rescaling, the dual canonical bases of quantum unipotent subgroupsbecome the common triangular bases of quantum cluster algebras. Asa consequence, Conjecture 1.1.1 holds true.
Effectively, for proving Theorem 1.2.2, we verify Conjecture 1.1.1 andFock-Goncharov dual basis conjecture (for this case) simultaneously inthis common triangular basis approach.It might come as a surprise that, unlike previous works on Conjecture1.1.1, we do not need the positivity assumption on the basis (which isfalse in general). This result is derived as a consequence of a moregeneral existence theorem for the common triangular bases (Theorem1.2.3). For proving the latter theorem, we base our arguments mostlyon an analysis of tropical properties and an additional input calledtwist automorphisms, which we shall explain below.After localization, the quantum unipotent subgroup A q [ N − ( w )] be-comes isomorphic to the quantum unipotent cell A q [ N w − ]. It possessesan automorphism η w,q called the twist automorphism, which permutesthe (localized) dual canonical basis [KO17].For general quantum cluster algebras, we define the cluster twist au-tomorphisms tw (twist automorphism for short, Section 5), such thatthey include η w,q as a special case after rescaling ((9.4), Theorem 9.4.5).We show that the common triangular basis, if it exists, must be per-muted by the cluster twist automorphism (Proposition 6.3.2). Con-versely, assuming that a basis possesses this property, we prove thefollowing existence theorem for common triangular bases. Theorem 1.2.3 (Existence theorem, Theorem 6.5.4) . Assume that thequantum cluster algebra possesses the triangular basis for an initial seedand a twist automorphism. If the triangular basis is permuted by thetwist automorphism and contains the quantum cluster monomials ap-pearing along a (green to red) mutation sequence, then it is the commontriangular basis. In particular, it contains all quantum cluster mono-mials. Non-positivity was found in Shigenori Yamane’s master thesis at Osaka Univer-sity. It can be verified by the algorithm in [Lec04]. See [Tsu10] for more details.
FAN QIN
Briefly speaking, via an analysis of tropical properties, we proveTheorem 1.2.3 based on two statements for adjacent seeds which areinspired by desired conjectures. More precisely, we consider two prop-erties of a basis: • the admissibility (Definition 6.4.1) means that the basis con-tains certain cluster monomials in the spirit of Conjecture 1.1.1; • the compatibility (Definition 3.4.1) means that it has nice trop-ical properties in the spirit of Fock-Goncharov dual basis con-jecture.Then we show that the triangular basis for one seed gives rise to acompatible triangular basis for an adjacent seed provided the admis-sibility (Proposition 6.4.3); conversely, certain compatibility conditionimplies the admissibility (Proposition 6.4.5), see Sections 6.4, 6.5. It isworth remarking that we use the dominance order decomposition from[Qin19a] (Definition 3.3.1 Proposition 3.4.2) in the proof of Proposition6.4.3. In addition, we use some results from [Qin17] such as Proposition6.2.9. Remark 1.2.4 (Main ingredients) . To summarize, let us list the mainingredients in our approach to Conjecture 1.1.1 contained in this paperand its prerequisite works.Lie theoretic side: [Kim12] showed that quantum unipotent subgroups A q [ N − ( w )] possess the dual canonical bases induced from those of U q − ,which enables us to investigate Fomin-Zelevinsky’s conjectural link be-tween the dual canonical bases and cluster theory for Kac-Moody casesand formulate Conjecture 1.1.1. Based on the theory of non-commutativeunique factorization domains, [GY16b][GY20] guaranteed the clusterstructures on A q [ N − ( w )] . [KO17] showed that the twist automorphismson quantum unipotent cells A q [ N w − ] preserve the (localized) dual canon-ical bases. Their proof was based on the fascinating representationtheoretic properties of these bases.Cluster theoretic side: We need to know fundamental and importantnotions and results in cluster algebras, such as (the sign-coherence of ) g -vectors and c -vectors following [FZ07][DWZ10] ... [Dem10][GHKK18] .In this paper, we use notions and techniques introduced in [Qin17] andfurther developed in [Qin19a] for analyzing tropical properties of ele-ments of cluster algebras. In addition, we consider an analogue of thetwist automorphism on A q [ N w − ] (the cluster twist automorphism) andverify some of its properties (Theorem 5.1.7, 9.4.5). Then we replaceprevious arguments involving geometry or positivity in [Qin17] by ananalysis of tropical properties following [Qin17][Qin19a] , improve the For our main theorem, it is enough to know fundamental properties of clusteralgebras arising from quantum unipotent subgroups of symmetrizable types. Cor-respondingly, we can rely on [Dem10], but it also follows from the general resultsof [GHKK18]. theory of triangular bases in [Qin17] (Propositions 6.3.2, 6.4.3, 6.4.5),and obtain existence criteria of common triangular bases (Theorems6.5.3, 6.5.4) without imposing the positivity assumption.
Remark 1.2.5 (Generalization of dual canonical bases) . Fomin andZelevinsky’s original motivation was ambitious, such that the mean-ing of their “dual canonical basis” includes not only the well-knowndual canonical basis for quantum groups but also (unknown) gener-alizations for other (quantized) coordinate rings in Lie theory. It istherefore important to understand all good bases for other (quantum)cluster algebras. Many works have been devoted to this topic. These in-clude but are not limited to [Dup11] [GLS12] [Pla13] [Qin19a] [MSW13][FT17] [HL10] [Qin17] [KKKO18] [Thu14] [BZ14] [GHKK18] [LLZ14][LLRZ14] .By this paper, the triangular basis in the sense of [Qin17] suggests areasonable generalization of dual canonical bases in cluster theory.
Remark 1.2.6 (Twist automorphisms) . In the literature, twist au-tomorphisms are automorphisms on the nilpotent cells N w − introducedfor solving the factorization problems (chamber ansatz) which describethe inverse of the toric chart of Schubert varieties [BFZ96][BZ97] . Inthe symmetric Kac-Moody cases, [GLS11][GLS12] studied them usingcategorification via preprojective algebras. A quantum analogue wasintroduced and studied in [KO17] .The cluster twist automorphisms in this paper are defined for seeds ofgeneral quantum cluster algebras, not necessarily with a Lie theoreticbackground. They include the twist automorphisms in Lie theory asa special case (Theorem 9.4.5). Therefore, they provide a reasonablegeneralization in cluster theory. It is desirable to understand them ona categorical level (Remark 5.2.2).In this paper, we will mainly use the cluster twist automorphisms ofDonaldson-Thomas type, but the notion makes sense more generally.Further discussions about cluster twist automorphisms will appear in aseparate paper. Remark 1.2.7 (Berenstein-Zelevinsky triangular basis) . The idea toconstruct a common triangular basis first appeared in the work of Beren-stein and Zelevinsky [BZ14] . They defined triangular bases for acyclicseeds of quantum cluster algebras, and their definition is different fromours. Based on [Qin19b] and Theorem 9.5.1, we now know that theBerenstein-Zelevinsky triangular basis agrees with our common trian-gular bases (Corollary 9.5.3 ).
We also obtain a result concerning tropical properties of the dualcanonical bases (Corollary 9.5.4). It would be desirable to further un-derstand the relation between the dual canonical bases and the tropicalgeometry of cluster varieties.
FAN QIN
Contents.
We provide detailed prerequisites with examples. Anexpert might skip Sections 2, 3, 7 and probably Sections 4, 8. Ourarguments and proofs are presented in Sections 5, 6, 9.In Section 2, we review basic notions in cluster theory.Section 3 contains prerequisites on the dominance order and thecorresponding decomposition following [Qin17] [Qin19a]. Section 4contains prerequisites on similar seeds, a correction technique [Qin14][Qin17], and a result concerning mutations of similar elements (Propo-sition 4.2.6).Sections 5 and 6 contain most crucial arguments in this paper. First,we define cluster twist automorphisms and check some necessary prop-erties in Section 5. Then, we give a general construction for triangularfunctions as candidates of basis elements (Theorem 6.1.3) and reviewnotions and properties for the desired triangular bases. In addition,we show that common triangular bases must be permuted by any twistautomorphisms (Proposition 6.3.2). Finally, by using an analysis ontropical properties (such as the compatibility) and the twist automor-phisms, we give general existence criteria for common triangular bases(Theorems 6.5.3, 6.5.4).Section 7 contains prerequisites on quantum unipotent subgroups.In Section 8, we present the quantum cluster structure on quantumunipotent subgroups, following [GY16b].In Section 9, we apply the results of the previous sections to quantumcluster algebras arising from quantum unipotent subgroups and quan-tum unipotent cells. We first discuss the integral form and localizationin Sections 9.2, 9.3. Then, we compare the twist automorphisms in Lietheory and in our sense in Section 9.4. Finally, we apply the previousdiscussion to obtain our main result (Theorem 9.5.1) and deduce someother consequences in Section 9.5.
Acknowledgments
The author thanks Yoshiyuki Kimura for many helpful discussions.He is grateful to Bernhard Keller for seeing a preliminary version ofthe paper. He also thanks Milen Yakimov for informing him about therecent work [GY20]. He thanks Peigen Cao for interesting discussions.He is grateful to Bernhard Keller and Bernard Leclerc for remarks ona preliminary version of this article.2.
Basics of cluster algebras
We recall basic notions such as seeds and mutations in cluster theory.We mostly follow the convention in [GHK15], because it is conceptuallynatural to introduce various lattices. A reader might equally choosethe convention in [FZ07]. More detailed discussions could be found in[Qin19a].
Seeds.
Fixed data.
Given I a finite set of vertices endowed with a partition I = I uf ⊔ I f (called the unfrozen and frozen vertices respectively), wefurther fix the following data:(1) a rank- | I | lattice N , endowed with a Q -valued skew-symmetricbilinear form ω ( , );(2) a rank- | I uf | saturated sublattice N uf ⊂ N , called unfrozen sub-lattice;(3) strictly positive integers ( d i ) i ∈ I with the greatest common divi-sor 1;(4) a sublattice N ◦ ⊂ N of finite index, such that ω ( N uf , N ◦ ) ⊂ Z , ω ( N, N uf ∩ N ◦ ) ⊂ Z ;(5) the dual lattices M = Hom Z ( N, Z ), M ◦ = Hom Z ( N ◦ , Z ).Notice that ω ( n, ), n ∈ N , is naturally a Q -valued linear functionon N , thus an element in M Q = M ⊗ Z Q . Correspondingly, we definethe map p ∗ : N → M Q such that p ∗ ( n ) = ω ( n, ). The assumption on N ◦ implies that p ∗ ( N uf ) ⊂ M ◦ .We always make the following assumption throughout this paper. Assumption 1 (Injectivity assumption) . We assume that the map p ∗ : N uf → M ◦ is injective. We shall soon see explicit examples for the fixed data. In fact, wewill mainly work with the lattices N uf and M ◦ throughout this paper. Remark 2.1.1.
If all symmetrizers d i are , one can understand thelattices as Grothendieck groups of some categories, in which one canalso consider bases and bilinear forms, see [Qin12, Sections 2.3, 2.4] for details.Seeds as bases. Definition 2.1.2 (Seed) . A seed for the fixed data is an I -labeled col-lection E = ( e i | i ∈ I ) such that E is a basis of N , { e k | k ∈ I uf } is a Z -basis of N uf , and { d i e i | i ∈ I } is a basis of N ◦ . Given a seed E , we shall also denote it by t as in most works in clustertheory. Let E ∗ = { e ∗ i | i ∈ I } denote the dual basis of E in M . Denote F ∗ := { f ∗ i | i ∈ I } := { d i e i | i ∈ I } . Then F := { f i | i ∈ I } := { d i e ∗ i | i ∈ I } is a basis of M ◦ . By using these bases, the abstract lattices in the fixeddata can be represented as sets of (column) vectors: M ◦ ( t ) := ⊕ i ∈ I Z f i ≃ Z I N ( t ) := ⊕ i ∈ I Z e i ≃ Z I N uf ( t ) := ⊕ k ∈ I uf Z e k ≃ Z I uf , where we identify f i , e i as i -th unit vectors respectively. Let us define ω ij = ω ( e i , e j ), b ji = ω ij d j , ∀ i, j ∈ I . Apparently,we have b ji d i = − b ij d j , ∀ i, j ∈ I . Direct computation shows that p ∗ ( e j )( f ∗ i ) = b ij , ∀ i, j ∈ I . Therefore, the linear map p ∗ : N → M Q is represented by the matrix ( b ij ) i,j ∈ I , and p ∗ ( e j ) is represented by its j -th column vector col j (( b ij ) i,j ∈ I ). Definition 2.1.3 ( e B -matrix) . The e B -matrix of the seed t is definedto be e B ( t ) = ( b ik ) i ∈ I,k ∈ I uf = ( ω ki d i ) . The corresponding B -matrix isdefined to be the I uf × I uf -submatrix B ( t ) , which is called the principalpart of e B ( t ) . We often omit the symbol t when the context is clear. Under theinjectivity assumption, e B is of full rank | I uf | , and the column vectors col k e B = p ∗ e k , k ∈ I uf , are linearly independent. Remark 2.1.4 (Different conventions: seeds) . In [FZ02, FZ07] , a seed t (with geometric coefficients) is defined to be the collection of the ma-trix e B ( t ) and the cluster variables X i ( t ) , i ∈ I . See [Qin19a, Section2.1] for changing conventions. Example 2.1.5 (Type A ) . Let us take I = I uf = { , } , d = d = 1 , N = Z , an initial seed t = E = { e , e } = { (cid:18) (cid:19) , (cid:18) (cid:19) } , and thebilinear form ω satisfies ( ω ij ) = (cid:18) − (cid:19) . Then we have N ◦ = N , M ◦ = M ≃ Z such that F = { f , f } = { (cid:18) (cid:19) , (cid:18) (cid:19) } , ( b ij ) = e B = B = (cid:18) −
11 0 (cid:19) , p ∗ e = f , p ∗ e = − f .Poisson structure. By [GSV03, GSV05], since e B is of full rank, wecan find an I × I Z -valued skew-symmetric matrix Λ = (Λ ij ) such that( e B, Λ) is a compatible pair in the sense of [BZ05], i.e. e B T Λ = (cid:0) D ′ (cid:1) for some diagonal matrix D ′ = diag( d ′ k ) k ∈ I uf , d ′ k ∈ N > . Notice that Λ isnot necessarily unique. In addition, e B T Λ e B = D ′ B is skew-symmetric,i.e., we have d ′ i b ij = − d ′ j b ji ∀ i, j ∈ I uf .The matrix Λ gives rise to a bilinear form λ on M ◦ such that λ ( f i , f j ) =Λ ij . One can use it to construct Poisson structure on the correspondingalgebraic torus, see [GSV03, GSV05] for example. Lemma 2.1.6.
The following claims are true for any i ∈ I , j, k ∈ I uf .(1) λ ( f i , p ∗ e k ) = − δ ik d ′ k .(2) λ ( , p ∗ e k ) = − d ′ k d k · e k = − d ′ k f ∗ k .(3) λ ( p ∗ e i , p ∗ e k ) = d ′ i b ik . Proof. (1) The claim follows form the compatibility Λ( − e B ) = (cid:18) D ′ (cid:19) . (2) The claim follows from (1).(3) We have λ ( p ∗ e i , p ∗ e k ) = h p ∗ e i , − d ′ k f ∗ k i = b ki · ( − d ′ k ) = d ′ i b ik . (cid:3) Definition 2.1.7 (Quantum seed) . The collection ( E, λ ) is called aquantum seed. The skew-symmetric matrix D ′ B = e B T Λ e B give rise to a skew-symmetric bilinear form on N uf , still denoted by λ , such that λ ( e i , e j ) := λ ( p ∗ e i , p ∗ e j ) = d ′ i b ij . The following property follows from definition.
Lemma 2.1.8.
For any n, n ′ ∈ I uf , we have λ ( p ∗ n ′ , p ∗ n ) = λ ( n ′ , n ) . Example 2.1.9.
In Example 2.1.5, we have d = 1 and d ∨ = d ∨ = 1 .Take any α ∈ N > and the matrix Λ = (Λ ij ) = α (cid:18) −
11 0 (cid:19) . Then (Λ , e B ) is compatible with d ′ = d ′ = α . Notice that Λ gives a skew-bilinear form λ on M ◦ . Then we have ( λ ( e i , e j )) i,j ∈ I uf = ( d ′ i b ij ) i,j = α (cid:18) −
11 0 (cid:19) = − αω. For completeness, let us discuss the relation between ω and λ , thoughit will not be needed for the purpose of this paper.Denote the least common multiplier d = lcm( d i ). Define the Lang-lands dual d ∨ i = dd i . Then they satisfy d ∨ i b ij = − d ∨ j b ji , ∀ i, j ∈ I . Denotethe scaling constant α k = d ′ k d ∨ k ∈ Q > for k ∈ I uf , then we have α k = α j whenever b kj = 0, ∀ k, j ∈ I uf . It follows that α k is constant on eachconnected component of I uf in the following sense. Definition 2.1.10 (Connected components) . Given an I uf × I uf matrix B , a connected component V of I uf with respect to B is a maximalcollection of vertices in I uf such that for any i, j ∈ V , there existsfinitely many vertices i s ∈ V , ≤ s ≤ l , such that i = i , i l = j , and b i s i s +1 = 0 for any ≤ s ≤ l − . We can choose the matrix Λ such that there exists α ∈ Q > suchthat d ′ k = αd ∨ k for all k ∈ I uf , i.e., α k = α , see [GSV05, Theorem 2.1]. Lemma 2.1.11.
The following statements hold for any n ∈ N uf .(1) λ ( , p ∗ n ) = − αd · n (2) λ = − αd · ω on N uf .Proof. (1) The claims follow from Lemma 2.1.6.(2) By Lemma 2.1.6, we have λ ( e i , e j ) = d ′ i b ij = αd ∨ i · d i ω ji = αd · ( − ω ij ). (cid:3) Laurent polynomial rings.
Let k denote a base ring and v ∈ k an invertible element. We will take ( k , v ) = ( Z ,
1) and ( k , v ) =( Z [ q ± ] , q ) for classical cluster algebra and quantum cluster algebrasrespectively, where q is a formal parameter. Later, we will also take( k , v ) = ( Q ( q ) , q ) for quantized enveloping algebras, where q is a formalparameter. Notice that k has the Z -linear (or Q -linear) bar involution( ) which sends v to v − in these cases. Ring of characters.
Let there be given any lattice L of finite rank.Denote its dual lattice L ∨ = Hom Z ( L, Z ). Define the following groupalgebra endowed with the natural addition and multiplication (+ , · ) k [ L ] = ⊕ p ∈ L k χ p such that χ p · χ p ′ = χ p + p ′ . We often omit the symbol · for simplicity. k [ L ] is called the Laurent polynomial ring associated to L , and L thelattice of Laurent degrees. We often omit the commutative multiplica-tion symbol · for simplicity.It is worth reminding that χ p can be viewed as a character on thesplit algebraic torus T L ∨ = L ∨ ⊗ Z F ∗ , where F is a chosen field and F ∗ = F \{ } its multiplicative group of invertible elements. We willassume F = C throughout this paper. Quantization.
Assume that L possesses a Z -valued skew-symmetric bi-linear form λ . We then define the quantum algebra k [ L ; λ ] such that k [ L ; λ ] is given by the commutative algebra ( k [ L ] , + , · ) as before and,in addition, endowed with the twisted product ∗ : χ p ∗ χ p ′ = v λ ( p,p ′ ) χ p + p ′ . Unless otherwise specified, by the algebra structure of k [ L ; λ ], we mean(+ , ∗ ). When the context is clear, we omit the symbol λ for simplicity.Notice that, when v = 1, the twisted product ∗ agrees with the usualproduct · . Formal Laurent series.
We have the Euclidean space L R = L ⊗ Z R .Let there be given a strictly convex rational polyhedral cone σ ∈ L R ,i.e., σ = P j R ≥ p j for finitely many generators p j ∈ L . It determinesthe monoid σ L = σ ∩ L and the abelian group Z σ L = σ L ⊗ N Z . Since σ is strictly convex, the only invertible element in σ L is 0. Denote σ + L = σ L \{ } .We have the monoid algebra k [ σ L ] = ⊕ p ∈ σ L k χ p and its maximal ideal M = k [ σ + L ] = ⊕ p ∈ σ + L k χ p . Let [k [ σ L ] denote the completion of k [ σ L ] withrespect to this maximal ideal. We then define the formal completionof k [ L ] with respect to σ L to be \k [ L ; σ L ] = k [ L ] ⊗ k [ σ L ] [k [ σ L ] . When the context is clear, we write \k [ L ; σ L ] = d k [ L ] for simplicity. Sim-ilarly, we define the formal completion of the quantum algebra k [ L ; λ ]to be \k [ L ; σ L , λ ] = k [ L ; λ ] ⊗ k [ σ L ; λ ] \k [ σ L ; λ ] . \k [ L ; σ L ] (resp. \k [ L ; σ L , λ ]) is called the algebra (resp. quantum algebra)of formal Laurent series.A possibly infinite sum will be called a formal sum. Let there begiven a collection of formal sums Z ( j ) = P m i ∈ L b ( j ) m i χ m i , b ( j ) m i ∈ k , j ∈ N ,their formal sum P j Z ( j ) is well defined if, for each m i , there are onlyfinitely many Z ( j ) with non-vanishing coefficients b ( j ) m i . Remark 2.2.1.
The formal completion d k [ L ] can be shown to be theCauchy completion with respect to an explicit topology on k [ L ] , see [DM19] . It would be interesting to understand calculation in d k [ L ] froma topological point of view, though we will not pursue this direction inthis paper.Algebras associated to seeds. Let there be given a quantum seed t =( E, λ ) as before. Recall that we have lattices M ◦ ( t ) = ⊕ i ∈ I Z f i ≃ Z I N ( t ) = ⊕ i ∈ I Z e i ≃ Z I N uf ( t ) = ⊕ k ∈ I uf Z e k ≃ Z I uf , where f i , e i are viewed as the i -th unit vectors.Consider the quantum algebras of Laurent polynomials k [ M ◦ ( t ); λ ]and k [ N uf ( t ); λ ]. Denote X i = X i ( t ) = χ f i and X m = X ( t ) m = χ m for m ∈ M ◦ ( t ). The quantum Laurent polynomial ring associated to thequantum seed t is defined as LP ( t ) := k [ X ± i ] i ∈ I = k [ X m ] m ∈ M = k [ M ◦ ( t ); λ ] . Its skew-field of fractions is denoted by F ( t ).We also define the Laurent monomials Y k = Y k ( t ) = χ p ∗ e k , Y n = Y ( t ) n = χ p ∗ n , n ∈ N uf ( t ), and the corresponding subalgebra of LP ( t ): k [ Y ± k ] k ∈ I uf = k [ Y n ] n ∈ N uf ( t ) ≃ k [ N uf ( t ); λ ] . Notice that k [ Y k ] k ∈ I uf has the maximal ideal generated by Y k , k ∈ I uf . Let \k [ Y k ] k ∈ I uf denote the corresponding completion under the adictopology. As before, the formal completion of LP ( t ) is defined by d LP ( t ) = LP ( t ) ⊗ k [ Y k ] k ∈ I uf \k [ Y k ] k ∈ I uf . Its elements will be called formal Laurent series or functions.
Definition 2.2.2 (Quantum cluster variables) . The quantum clustervariables of the seed t are defined to be X i ( t ) , i ∈ I . Their monomialsare called quantum cluster monomials. The X j ( t ) , j ∈ I f are calledthe frozen variables. The group of frozen factors is defined to be P = { X ( t ) u | u ∈ Z I f } .The Laurent monomials of the form X ( t ) m , m = P m i f i , m i ∈ Z ,such that m i ≥ whenever i ∈ I uf , are called localized quantum clustermonomials.The Y k ( t ) , k ∈ I uf , are call the (unfrozen) quantum Y -variables ofthe seed t . For simplicity, we often skip the word quantum and the symbol t .2.3. Mutation.
Seeds.
Let [ ] + denote the function max( , g i )] + = ([ g i ] + ) forany vectors ( g i ). Recall that a seed is a collection of basis elements t = E = ( e i ) i ∈ I in N . Let us define new seeds in the following procedure.We start by choosing a sign ε ∈ { + , −} . For any k ∈ I uf , define I × I transformation matrices P E,ε and P F,ε whose entries are given asfollows: ( P E,ε ) ij = δ ij k / ∈ { i, j }− i = j = k [ εb kj ] + i = k, j = k ( P F,ε ) ij = δ ij k / ∈ { i, j }− i = j = k [ − εb ik ] + i = k, j = k . Denote their I uf × I uf -submatrices by P I uf E,ε and P I uf F,ε respectively .Given any unfrozen vertex k ∈ I uf , we define the new seed t ′ = E ′ =( e ′ i ) i ∈ I such that e ′ i = P j ∈ I e j · ( P E,ε ) ji ∈ N : e ′ i = ( e i + [ εb ki ] + e k i = k − e k i = k . Recall that b ij = ω ji d i = ω ( e j , e i ) d i and it depends on the seed t = E . Definition 2.3.1 (Mutation) . We denote the above operation by t ′ = µ k,ε t and call µ k,ε the mutation in the direction k . The matrices P E,ε and P I uf F,ε are denoted by F ε and E ε in [Kel12, Section5.6][BZ05, (3.2) (3.3)] respectively. We could also compute the dual basis ( E ∗ ) ′ = { e ∗ i | i ∈ I } in M andthe corresponding basis F ′ = { d i e ∗ i | i ∈ I } in M ◦ . It is straightforwardto check that we get f ′ i = P j ∈ I f j · ( P F,ε ) ji ∈ M ◦ : f ′ i = ( f i i = k − f k + P j [ − εb jk ] + f j i = k . The new bases enable us to represent the abstract lattices as sets ofnew coordinate (column) vectors: M ◦ ( t ′ ) = ⊕ Z f ′ i ≃ Z I N ( t ′ ) = ⊕ Z e ′ i ≃ Z I N uf ( t ′ ) = ⊕ k ∈ I uf Z e ′ k ≃ Z I uf where f ′ i , e ′ i are viewed as i th unit vectors.The above basis change gives a linear transformation τ k,ε : N ( t ′ ) → N ( t ) such that τ k,ε ( e ′ i ) = P j ∈ I e j · ( P E,ε ) ji and, similarly, τ k,ε : M ◦ ( t ′ ) → M ◦ ( t ) such that τ ( f ′ i ) = P j ∈ I f j · ( P F,ε ) ji . These linear maps are rep-resented by (left multiplication of) the matrices P E,ε and P F,ε respec-tively.
Remark 2.3.2.
On the one hand, these matrices are idempotent, i.e., ( P E,ε ) = ( P F,ε ) = Id I . On the other hand, it is straightforward tocheck that µ k, − ε ( µ k,ε ( t )) = t . Notice that the matrix depends on theseeds, and µ k,ε µ k,ε ( t ) = t ! It is straightforward to compute b ′ ij = ω ′ ji d i = ω ( e ′ j , e ′ i ) d i : b ′ ij = ( − b ij k ∈ { i, j } b ij + b ik [ εb kj ] + + [ − εb ik ] + b kj k = i, j . In addition, one can check that the new matrix ( b ′ ij ) i,j ∈ I only dependson ( b ij ) i,j ∈ I and is independent of the sign ε . Consequently, we definethe mutation of matrices ( b ′ ij ) i,j ∈ I = µ k (( b ij ) i,j ∈ I ) without choosing asign. Lemma 2.3.3.
The connected components of I uf with respect to B ( t ) are the same as those with respect to B ( t ′ ) . Assume that t is a quantum seed ( E, λ ) with a chosen I × I -matrixΛ = (Λ ij ) i,j ∈ I such that Λ ij = λ ( f i , f j ), e B T Λ = (cid:0) D ′ (cid:1) , D ′ =diag( d ′ i ) i ∈ I . Let Λ ′ = (Λ ′ ij ) i,j ∈ I denote the matrix such that Λ ′ ij = λ ( f ′ i , f ′ j ). The following lemma shows that µ k t := t ′ := ( E ′ , λ ) is also aquantum seed. Lemma 2.3.4 ([BZ05]) . Take any sign ε , the following claims are true.(1) ( b ′ ij ) i,j ∈ I = P F,ε ( b ij ) i,j ∈ I P E,ε (2) Λ ′ = P F,ε Λ P F,ε (3) ( e B ′ ) T Λ ′ ij = (cid:0) D ′ (cid:1) Proof. (1)(2) The statements can be proved by calculation or usinglinear algebras.(3) The statement follows from (1)(2) and the equality P F,ε = Id I ,( P I uf E,ε ) T D ′ = D ′ ( P F,ε ). (cid:3) Example 2.3.5.
In Example 2.1.5, let us take the initial seed t = E = { e , e } = { (cid:18) (cid:19) , (cid:18) (cid:19) } , ( b ij ) = (cid:18) −
11 0 (cid:19) . Choose k = 2 .Then P E, + = (cid:18) − (cid:19) , t = µ , + t = E ′ = { e ′ , e ′ } = { e + e , − e } .It follows that ( b ′ ij ) = (cid:18) − (cid:19) . Choosing k = 2 again, we get P E ′ , − = (cid:18) − (cid:19) , µ , − ( t ) = E ′′ = { e ′ + e ′ , − e ′ } = E = t . On theother hand, P E ′ , + = (cid:18) − (cid:19) and, correspondingly, µ , + ( t ) = t !Signs of seed mutations and c -matrices. Let t denote a given initialseed. Let k = k k · · · k l denote a word whose letters are unfrozen ver-tices. We are going to apply mutations along the sequence k . In clustertheory, there is a default choice of the signs ε j for the correspondingmutations once the initial seed t is given.Let I ′ = { i ′ | i ∈ I uf } denote a copy of I uf . The principal coefficientmatrix associated to t is the ( I uf ⊔ I ′ ) × ( I uf ⊔ I ′ )-matrix given by( b prin ij ) i,j ∈ I ⊔ I ′ = (cid:18) ( b ij ) i,j ∈ I − C ( t ) C ( t ) 0 (cid:19) , where C ( t ) is the I ′ × I uf -matrixwhose non-zero entries are C ( t ) i ′ ,i = 1, ∀ i ∈ I uf . Under the naturalidentification I ′ ≃ I uf , we also view C ( t ) as the identity matrix Id I uf .For any s ∈ [0 , l ], denote the word k ≤ s = k k . . . k s . Assume thatthe sign ε j have been chosen for j ≤ s , then we have the mutationsequence ←− µ ≤ s = µ k s ,ε s · · · µ k ,ε (reading from right to left) and thecorresponding seed t s = ←− µ ≤ s t . If s = 0, k ≤ s denote an empty word, ←− µ ≤ s an empty sequence, and t s = t . Applying ←− µ ≤ s to ( b prin ij ), weobtain the matrix ( b prin ij ( t s )) i,j ∈ I ⊔ I ′ = (cid:18) ( b ij ( t s )) i,j ∈ I − C ( t s ) C ( t s ) 0 (cid:19) . Definition 2.3.6.
The matrix C t ( t s ) := C ( t s ) is called the C -matrixof t s with respect to the initial seed t . Its k -th column vectors are calledthe k -th c -vectors of t s , denoted by c k ( t s ) . Notice that the C -matrices depend on the B ( t ). Recall that a vector g = ( g i ) i is called sign-coherent , if all components g i ≥ g i ≤
0, in which case we denote g ≥ g ≤ Theorem 2.3.7. [DWZ10, Theorem 1.7][GHKK18, Corollary 5.5]
For any k ∈ I uf , the c -vector c k ( t s ) is non-zero and sign-coherent. We refer the reader to [NZ12] for more discussions.Following [Kel12, section 5.6], by using c -vectors, we choose the sign ε k s +1 = + if c k s +1 ( t s ) >
0, ; choose ε k s +1 = − if c k s +1 ( t s ) <
0. Repeatingthis process, we obtain the signs for all mutations appearing in ←− µ = µ k l ,ε l · · · µ k ,ε . Definition 2.3.8.
Let there be given an initial seed t . We use ∆ + t todenote the seeds obtain from t by applying any mutations sequences,where we choose the signs by using C -vectors. Throughout this paper, we will discuss seeds taken from the same set∆ + t for some given initial seed t except in Section 4. In particular, thesigns of seed mutations are determined by t . We will denote ∆ + = ∆ + t and µ k = µ k,ε for simplicity. Mutation birational maps.
Now, let us relate the quantum algebras ofLaurent polynomials associated to t and t ′ = µ k t .We define the isomorphism µ ∗ k : F ( t ′ ) : ≃ F ( t ) such that µ ∗ k ( X ′ i ) = ( X i i = kX − f k + P j [ − b jk ] + f j + X − f k + P i [ b ik ] + f i i = k . (2.1)Denote v k = v d ′ k . By Lemma 2.1.6, for j ∈ I , i, k ∈ I uf , we have d ′ i b ik = − d ′ k b ki and X j ∗ Y k = v − δ jk k X j · Y k ,Y i ∗ Y k = v d ′ i b ik Y i · Y k , We can rewrite µ ∗ k ( X k ) = X − f k + P j [ − b jk ] + f j · (1 + Y k )= X − f k + P j [ − b jk ] + f j ∗ (1 + v − k Y k )The following equation is called the exchange relation: X k ∗ µ ∗ k ( X k ) = v λ ( f k , P j [ − b jk ] + f j ) X P j [ − b jk ] + f j + v λ ( f k , P i [ b ik ] + f i ) X P i [ b ik ] + f i (2.2) = v λ ( f k , P j [ − b jk ] + f j ) X P j [ − b jk ] + f j · (1 + v − k Y k )= v λ ( f k , P j [ − b jk ] + f j ) X P j [ − b jk ] + f j ∗ (1 + v − k Y k ) . The mutation rule for the Y -variables Y i = X ( t ) p ∗ e i , Y ′ i = X ( t ′ ) p ∗ e ′ i , i ∈ I uf , can be computed: µ ∗ k ( Y ′ k ) = Y − k , (2.3) µ ∗ k ( Y ′ i ) = Y i · − b ki X s =0 (cid:18) − b ki s (cid:19) v k Y sk , if b ki ≤ ,µ ∗ k (( Y ′ i ) − ) = Y − i · Y − b ki k · b ki X s =0 (cid:18) b ki s (cid:19) v k Y sk , if b ki ≥ , where the quantum numbers are defined as [ a ] v = a v − a − v v − v − , [ a ] v ! =[ a ] v [ a − v · · · [1] v , (cid:18) ab (cid:19) v = [ a ] v ![ a − b ] v ! · [ b ] v ! , for 0 ≤ b ≤ a ∈ N . Example 2.3.9.
Take some c ∈ N > , I = I uf = { , } , ( ω ij ) i,j ∈ I = (cid:18) −
11 0 (cid:19) , ( d , d ) = ( c, . Define B ( t ) = B = (cid:18) c − (cid:19) suchthat b ij = ω ji d i . Then we have Y = X − , Y = X c . Take Λ( t ) = Λ = α (cid:18) − (cid:19) for α ∈ N > , then D ′ = B T Λ = α (cid:18) c (cid:19) , v = v α , v = v αc . Denote t = µ t and t − = µ t . Direct computation showsthat µ ∗ ( X ( t )) = X − f + f ∗ (1 + v − Y ) = (1 + v Y ) ∗ X − f + f µ ∗ ( X ( t − )) = ( X − ) ∗ (1 + v − Y ) = (1 + v Y ) ∗ X − For Y -variables, we have Y ∗ Y = v αc Y · Y , v − b Y ∗ Y = v − b Y ∗ Y , and X − k ∗ Y k = v k Y k ∗ X − k . The mutation rule reads as µ ∗ ( Y ( t − )) = µ ∗ ( X ( t − ))= Y ∗ (1 + v − Y )= Y · (1 + Y ) µ ∗ ( Y ( t ) − ) = µ ∗ ( X ( t ) c )= ((1 + v Y ) ∗ X − f + f ) c = (1 + v Y ) ∗ (1 + v Y v ) ∗ · · · ∗ (1 + v Y v c − ) ∗ X − cf + cf = (1 + v Y ) ∗ (1 + v Y ) ∗ · · · ∗ (1 + v c − Y ) ∗ X − cf + cf We can further compute µ ∗ ( Y ( t ) − ) = (1 + v − c Y ) · (1 + v − c Y ) · · · · · (1 + v c − Y ) · X − cf + cf = Y − · Y − c · c X s =0 (cid:18) cs (cid:19) v Y s Remark 2.3.10 (Birational map for tori) . Notice that (2.3) also givesthe isomorphism (mutation rule) between the skew-fields of fractions µ ∗ k : F ( k [ N ( t ′ )]) ≃ F ( k [ N ( t )]) in which we replace Y i , Y ′ i by χ e i , χ e ′ i respectively and take i ∈ I , see [FG09, Section 3][DM19, Section 2.4.1] for more details.We call µ ∗ k the mutation birational maps for the following reason.Consider the split algebraic tori Spec F [ M ◦ ( t )] = T N ◦ ( t ) = N ◦ ( t ) ⊗ Z F ∗ and Spec F [ N ( t )] = T M ( t ) over a characteristic field F . In the classicalcase ( v = 1 ), the maps µ ∗ k are the pullback of the birational map betweenthe tori: µ k : T N ◦ ( t ) T N ◦ ( t ′ ) µ k : T M ( t ) T M ( t ′ ) . There is no sign ε in the definition of the mutation birational map µ ∗ k . This is in contrast to the mutation of bases τ k,ε . See [GNR17] fortheir relation. Remark 2.3.11 (Different conventions: mutations) . Recall that wehave two conventions for the seeds following [GHK15] and [FZ02] re-spectively, see Remark 2.1.4. The seeds in the formal sense are mutatedas sets of vectors, while those in the latter sense contain cluster vari-ables which are mutated by birational maps. Correspondingly, we havetwo sets of seeds ∆ + and e ∆ + constructed via iterated mutations re-spectively. We have the natural but non-trivial result that the two setsare in bijection, because the cluster variables are determined by theirextended g -vectors by [FZ07, Conjecture 7.10(1)] .The verification of [FZ07, Conjecture 7.10(1)] was due to [DWZ10] for quiver cases. [Dem10] generalized [DWZ10] to some valued quivercases (such as those concerned in our main theorem), and the generalcases were solved by [GHKK18] . Alternatively, given the sign-coherenceof g -vectors, we know that the cluster monomials are always compati-bly pointed by [FZ07, Remark 7.13] . Then [FZ07, Conjecture 7.10(1)] can be deduced from [Qin17, Lem 3.4.11] for injective-reachable clusteralgebras.In this paper, we choose the convention of [GHK15] because it isconceptually more natural to introduce the lattices M ◦ ( t ) , N uf ( t ) andrelated structures. A reader might equally choose the convention in [FZ02] .Mutation sequences. For any seeds t , t ∈ ∆ + . Let ←− µ t ,t denote anychosen sequence of mutations such that t = ←− µ t ,t t . Then ( ←− µ t ,t ) − is a mutation sequence that sends t to t , which we can choose as ←− µ t ,t . The following property immediately follows from the definitionof seeds in the sense of [FZ02]. See Remark 2.3.11 for changing theconvention. Lemma 2.3.12.
For any seeds t , t , t ∈ ∆ + , we have ←− µ ∗ t ,t ←− µ ∗ t ,t = ←− µ ∗ t ,t as isomorphisms from F ( t ) to F ( t ) . Moreover, ←− µ ∗ t ,t is alwaysthe identity.Tropical transformations. Let there be given any t, t ′ ∈ ∆ + . If t ′ = µ k t for some k ∈ I uf , the tropical transformation φ t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ )is defined as the piecewise linear map such that, for any g = ( g i ) ∈ M ◦ ( t ) ≃ Z I and g ′ = ( g ′ i ) ∈ M ◦ ( t ′ ) ≃ Z I , we have g ′ i = − g k i = kg i + [ b ik ] + g k g k ≥ g i + [ − b ik ] + g k g k ≤ t ′ = ←− µ t by a mutation sequence ←− µ = µ k r · · · µ k µ k , then the trop-ical transformation φ t ′ ,t : M ◦ ( t ) ≃ M ◦ ( t ′ ) is defined as the compositionof the above defined tropical transformations for adjacent seeds. It isknown that φ t ′ ,t does not depend on the choice of ←− µ , because it canbe realized as the tropicalization of the mutation birational map forcertain cluster varieties associated to t and t ′ , see [GHK15]. Definition 2.3.13 (Tropical points [Qin19a]) . Let M ◦ denote the setof equivalent classes ⊔ t ∈ ∆ + M ◦ ( t ) under the identification φ t ′ ,t . It iscalled the set of tropical points. Remark 2.3.14.
The set M ◦ corresponds to the tropical points ofcertain cluster variety, see [GHK15, GHKK18] . By choosing a seed t ,we have chosen a specific representative M ◦ ( t ) of the set of tropicalpoints M ◦ . Cluster algebras.
Let pr I uf denote the natural projection from Z I to Z I uf .Let there be given an initial seed t and the corresponding set ofseeds ∆ + = ∆ + t . For any seeds t ∈ ∆ + , choose any mutation sequence ←− µ t,t such that t = ←− µ t,t t . Recall that we have the correspondingisomorphism ←− µ ∗ t,t : F ( t ) ≃ F ( t ) which does not depend on the choiceof ←− µ t,t . By the well known Laurent phenomenon [FZ02][BZ05], thequantum cluster variable X i ( t ), i ∈ I , is sent to its Laurent expansion ←− µ ∗ t,t X i ( t ) in LP ( t ) ⊂ F ( t ). Moreover, we have the following result. Theorem 2.4.1 ([FZ07][Tra11][DWZ10][GHKK18]) . The Laurent ex-pansion of X i ( t ) in LP ( t ) takes the following form ←− µ ∗ t,t X i ( t ) = X ( t ) g t i ( t ) · X n ∈ N I uf c n Y ( t ) n where the coefficients c n ∈ k such that c = 1 and the vector g t i ( t ) ∈ Z I uf ⊕ N I f . Moreover, pr I uf g t i ( t ) and c n are determined by B ( t ) , ( d ′ k ) k ∈ I uf and ←− µ t,t . The vector g t i ( t ) is called the i -th (extended) g -vector of the seed t with respect to the initial seed t . Notice that c = 1 is a conse-quence of the sign-coherence of c -vectors, see [FZ07, Proposition 5.6].In particular, it implies that the cluster variable ←− µ ∗ t,t X i ( t ) is pointedat g t i ( t ) in the sense of Definition 3.1.3.Recall that we have frozen variables ←− µ ∗ t,t X j ( t ) = X j ( t ) for j ∈ I uf ,which we denote by X j . Definition 2.4.2 (Quantum cluster algebras) . The (partially compact-ified) quantum cluster algebra A q ( t ) with initial seed t is defined tobe the Z [ q ± ] -subalgebra of LP ( t ) generated by the quantum clustervariables ←− µ ∗ t,t X i ( t ) , i ∈ I , t ∈ ∆ + t .The (localized) quantum cluster algebra A q ( t ) with initial seed t isthe localization of A q ( t ) at the frozen variables X j , j ∈ I uf .The quantum upper cluster algebra U q ( t ) with initial seed t is de-fined to be the intersection ∩ t ∈ ∆ + t ←− µ ∗ t,t LP ( t ) . It follows from Theorem 2.4.1 that we have A q ( t ) ⊂ A q ( t ) ⊂U q ( t ) ⊂ LP ( t ).Notice that the quantum cluster algebra A q ( t ) only depends on e B ( t ) and Λ( t ).By choosing a different initial seed t , we obtain the same set of seeds∆ + t = ∆ + t , but another quantum cluster algebra A q ( t ) contained in LP ( t ). The two algebras are isomorphic such that A q ( t ) = ←− µ ∗ t,t A q ( t ).As in literature, we sometimes identify F ( t ) and F ( t ) via ←− µ ∗ t,t andomit the symbols t and ←− µ ∗ t,t for simplicity. Then we say that the(partially compactified) quantum cluster algebra A q is generated bythe quantum cluster variables X i ( t ).2.5. Injective-reachability.Definition 2.5.1 (Injective-reachability [Qin17]) . A seed t is said tobe injective-reachable if there exists a seed t = ←− µ t ′ and a permutation σ of the unfrozen vertex I uf such that we have pr I uf g tσk ( t ′ ) = − f k (2.4) for all k ∈ I uf . In this case, we denote t ′ = t [1] , t = t ′ [ − . We say t [1] is shifted from t and vice versa. Let there be given some seed t ∈ ∆ + and assume that it is injective-reachable. Then all seeds in ∆ + are injective-reachable ([Qin17, Propo-sition 5.1.4][Mul16, Theorem 3.2.1]). In particular, µ k t is injective-reachable such that ( µ k t )[1] = µ σk ←− µ µ k ( µ k t ) for any k ∈ I uf . For k ∈ I uf , we denote the quantum cluster variable I k ( t ) : = ←− µ ∗ X σk ( t [1]) ∈ LP ( t ) . (2.5) Remark 2.5.2.
Injective-reachability is equivalent to the existence ofa green to red sequence in the sense of [Kel11] . This property onlydepends on the principal B -matrix B ( t ) . When B ( t ) is skew-symmetric,it means that the shift function [1] in the cluster category and injectivemodules of the Jacobian algebra can be constructed using mutations. Inparticular, I k ( t ) corresponds to the k -th injective modules, whence thesymbol “ I ”. See [Qin17, Section 5][Qin19a] for more details. Remark 2.5.3.
It is worth reminding that (2.4) is a special equationthat holds in M ◦ ( t ) where t plays the role of the initial seed. For ageneral seed s ∈ ∆ + , we have pr I uf g sσk ( t ′ ) = − pr I uf g sk ( t ) ∈ M ◦ ( s ) ,because the g -vector is transformed under the tropical transformations. Denote e B ( t ) T Λ( t ) = (cid:0) D ′ (cid:1) , D ′ = diag( d ′ k ) k ∈ I uf . By Lemma 2.3.4,we have e B ( t ′ ) T Λ( t ′ ) = (cid:0) D ′ (cid:1) . Recall that d ′ i b ik = − d ′ k b ki for i, k ∈ I uf . Lemma 2.5.4 ([Qin19a, Proposition 2.3.3]) . We have d ′ k = d ′ σk forany k ∈ I uf . Prerequisites on dominance order and decomposition
This section mainly follows [Qin17][Qin19a].3.1.
Dominance order and pointedness.
Fix an initial seed t and∆ + = ∆ + t . For any t = E = ( e i ) i ∈ I in ∆ + , we define the followingpartial order on M ◦ ( t ). Definition 3.1.1 (Dominance order) . For any m, m ′ ∈ M ◦ ( t ) , we say m ′ is dominated by m (or inferior than m ) with respect to t , denotedby m ′ (cid:22) t m , if there exists some n ∈ N ≥ uf ( t ) = ⊕ k ∈ I uf N e k such that m ′ = m + p ∗ n . Recall that we can represent M ◦ ( t ) and N uf ( t ) by lattices of (column)vectors: M ◦ ( t ) = ⊕ i ∈ I Z f i ≃ Z I N uf ( t ) = ⊕ k ∈ I uf Z e k ≃ Z I uf . Then the dominance order reads as m ′ = m + e B ( t ) n. Lemma 3.1.2 (Finite interval [Qin17, Lemma 3.1.2]) . For any m, m ′ in M ◦ , there exists finitely many m ′′ ∈ M ◦ such that m ′′ (cid:22) t m and m ′′ (cid:23) t m ′ . Let F ( k ) denote the fraction field of the base ring k . Let there begiven a formal sum Z = P m ∈ M ◦ ( t ) b m X m , b m ∈ F ( k ). Its Laurent degree support is defined to be supp M ◦ ( t ) Z := { m ∈ M ◦ ( t ) | b m = 0 } .Given b ∈ F ( k ), bZ is called a rescaling of Z .Likewise, for a function α ∈ Hom set ( M ◦ ( t ) , F ( k )), we define its sup-port to be supp M ◦ ( t ) α = { m ∈ M ◦ ( t ) | α ( m ) = 0 } . Definition 3.1.3 (Degrees and pointedness) . The formal sum Z issaid to have (leading) degree g Z ∈ M ◦ ( t ) , denoted by deg t Z = g Z , if g Z is the unique ≺ t -maximal element in supp M ◦ ( t ) Z . It is further saidto be pointed at g Z (or g Z -pointed) if b g Z = 1 or, equivalently, it takesthe following form Z = X g Z · (1 + X n ∈ N > uf ( t ) c n Y n ) , c n = b g Z + p ∗ n . Similarly, Z is said to have codegree η , denoted by codeg t Z = η Z , if η Z is the unique ≺ t -minimal element in supp M ◦ ( t ) Z . It is further saidto be copointed at η Z if b η Z = 1 , or, equivalently, it takes the followingform Z = X η Z · ( X n ∈ N > uf ( t ) c ′− n Y − n + 1) , c ′− n = b η Z − p ∗ n . Definition 3.1.4 (Normalization) . If Z = P b m X m ∈ d LP ( t ) has de-gree g , its normalization is defined to be the pointed element [ Z ] t = b − g Z ∈ d LP ( t ) ⊗ k F ( k ) . Definition 3.1.5 (Bidegree) . If Z has degree g and codegree η , we saythat it has bidegree ( g, η ) . If it is pointed at g and copointed at η , wesay that it is bipointed at bidegree ( g, η ) . Definition 3.1.6 ( F -function, support dimension) . Let there be given Z = X m · P n ∈ N ≥ uf ( t ) c n Y n with degree m (equivalently, c = 0 ). Its F -function is defined to be F Z = P c n Y n . We define its I -support tobe the set of vertices supp I Z = { k ∈ I uf |∃ c n = 0 , n k = 0 } .If Z has bidegree ( g, η ) , then its F -function is a polynomial F Z = P ≤ n ≤ f Z c n Y n with c = 0 , c f Z = 0 , η = g + p ∗ f Z , which we callthe F -polynomial. In this case, we define its support dimension to be suppDim Z = f Z . Following [FZ07][Tra11], we also call the leading degree g Z ∈ M ◦ ( t )the (extended) g -vector of Z . Similarly, following [FG18], we can alsocall the support dimension f Z the f -vector of Z .Let there be given a subset D ⊂ M ◦ ( t ). Definition 3.1.7 (Pointed set) . A D -labeled set S = { S g | g ∈ D } in d LP ( t ) is said to be D -pointed, if each S g is g -pointed. Unitriangularity.
Let there be given a D -pointed set S = { S g | g ∈ D } ⊂ d LP ( t ). Let m denote a chosen non-unital subring of of k . Definition 3.2.1 (Unitriangular transition) . A formal sum P m ∈ D b m S m , b m ∈ k is said to be ≺ t -unitriangular, if { m | b m = 0 } has a unique ≺ t -maximal element g and b g = 1 .The formal sum is further called ( ≺ t , m ) -unitriangular, if we furtherhave b m ∈ m for any m = g .A function Z ∈ d LP ( t ) is said to be ≺ t -unitriangular to S (respec-tively, ( ≺ t , m ) -unitriangular to S ) if it has a formal sum decompositionin S which is ≺ t -unitriangular (respectively, ( ≺ t , m ) -unitriangular).A collection of functions Z ∈ d LP ( t ) is said to be ≺ t -unitriangular to S (respectively, ( ≺ t , m ) -unitriangular to S ) if each function Z is. Notice that ≺ t restricts to a partial order on D ⊂ M ◦ ( t ). Lemma 3.2.2 (Inverse transition [Qin17, Lemma 3.1.11]) . Let therebe given two D -pointed sets S = {S g | g ∈ D } and L = { L g | g ∈ D } in d LP ( t ) . Then L is ( ≺ t , m ) -unitriangular to S if and only if S is ( ≺ t , m ) -unitriangular to L .Proof. Assume that L is ( ≺ t , m )-unitriangular to S . Then, for each g ∈ D , we have a ( ≺ t , m )-unitriangular decomposition L g = X b g,g ′ S g ′ where non-zero coefficients are given by: b g,g = 1, b g,g ′ ∈ m for g ′ ≺ t g ∈ D .Therefore, the transition matrix ( b g,g ′ ) g,g ′ ∈ D is a D × D -matrix whichis lower triangular with respect to the partial order ≺ t on D , whosediagonal entries are 1 and non-diagonal entries belong to m . We try toconstruct its inverse matrix ( c g,g ′ ) g,g ′ ∈ D . It must be a lower triangularmatrix such that c g,g = 1 for all g and, for any g ′′ ≺ t g , X g ′′ (cid:22) t g ′ (cid:22) t g c g,g ′ b g ′ ,g ′′ = 0 . Notice that this is a finite sum by Lemma 3.1.2. If c g,g ′ have been deter-mined and belong to m for g ′ ≻ t g ′′ , then c g,g ′′ is uniquely determinedand belongs to m . Recursively, we determine c g,g ′ ∈ m for all g, g ′ asin [Qin17, Lemma 3.1.11]. It follows that S is ( ≺ t , m )-unitriangular to L . (cid:3) Dominance order decomposition.
Let there be given a M ◦ ( t )-pointed set S = { S g | g ∈ M ◦ ( t ) } in d LP ( t ). For any Z ∈ d LP ( t ), [Qin19a]gives an algorithm to decompose it into a well defined sum of S g . Definition-Lemma 3.3.1 ([Qin19a, 4.1.1]) . There exists a unique de-composition Z = X g ∈ M ◦ ( t ) α t ( Z )( g ) · S g , α t ( Z ) ∈ Hom set ( M ◦ ( t ) , k ) in d LP ( t ) such that the support supp ( α t ( Z )) = { g | α t ( Z )( g ) = 0 } hasfinitely many ≺ t -maximal elements. It is called the ≺ t -decompositionof Z in terms of S . This ≺ t -decomposition will be called the dominance order decompo-sition with respect to the seed t . We recall that, by [Qin19a, 4.1.1],the maximal elements in supp ( α t ( Z )) are the same as the ≺ t -maximalLaurent degrees in the Laurent degree support supp M ◦ ( t ) Z .Notice that S is a topological k -module basis of d LP ( t ) in the senseof [DM19, Lemma 2.4].Let S ′ denote any subset of S , then it is D -pointed for some sub-set D ⊂ M ◦ ( t ). Conversely, any D -pointed set S ′ can be extendedto a M ◦ ( t )-pointed set by appending more functions such as Laurentmonomials. If the ≺ t -decomposition of Z in terms of S only has ele-ments in S ′ appearing, we call it the ≺ t -decomposition of Z in termsof S ′ . Lemma 3.3.2 ([Qin17, Lemma 3.1.10][Qin19a, 4.1.1]) . If Z ∈ d LP ( t ) has a finite decomposition in terms of S ′ , then the decomposition is the ≺ t -decomposition. In particular, if Z is pointed, then the decompositionis ≺ t -unitriangular. Let there be given a ring R containing k as a subring. Denote d LP ( t ) R = d LP ( t ) ⊗ k R . Replacing k by R in Definition-Lemma 3.3.1, wecan define the unique ≺ t -decomposition for Z ′ ∈ d LP ( t ) R in terms of agiven M ◦ ( t )-pointed set with coefficients in R . The following statementwill be convenient for latter proofs. Lemma 3.3.3. If Z ∈ d LP ( t ) has a finite decomposition in terms of S ′ in d LP ( t ) R , then it is the ≺ t -decomposition in d LP ( t ) with coefficientsin k .Proof. On the one hand, by Lemma 3.3.2, the finite decomposition of Z in terms of S ′ is the ≺ t -decomposition in d LP ( t ) R .On the other hand. Extend S ′ to a M ◦ ( t )-pointed set S ⊂ d LP ( t ).Working in d LP ( t ), Z has a ≺ t -decomposition in terms of S with coef-ficients in k . By the uniqueness of the ≺ t -decomposition, this is alsothe ≺ t -decomposition of Z in terms of S in d LP ( t ) R .The above two decomposition are the same by the uniqueness of ≺ t -decomposition. The claim follows. (cid:3) Change of seeds.
Let there be given seeds t = ←− µ t,t ′ t ′ in ∆ + . Forsimplicity, we denote g ′ = φ t ′ ,t g for any g ∈ M ◦ ( t ). Definition 3.4.1 (Compatibly pointedness) . A Laurent polynomial Z ∈ LP ( t ) ∩ ←− µ ∗ t ′ ,t LP ( t ′ ) is said to be compatibly pointed at t, t ′ if Z is g Z -pointed for some g Z ∈ M ◦ ( t ) and ←− µ ∗ t,t ′ Z is a g ′ Z -pointed Laurentpolynomial in LP ( t ′ ) .Given a subset ∆ ′ ⊂ ∆ + and t ∈ ∆ ′ , Z ∈ ∩ t ′ ∈ ∆ ′ ←− µ ∗ t ′ ,t LP ( t ′ ) is saidto be compatibly pointed at ∆ ′ if it is compatibly pointed at any pair ofseeds from ∆ ′ .Given a subset D ⊂ M ◦ ( t ) , a D -pointed set S = { S g | g ∈ D } ⊂LP ( t ) is said to be compatibly pointed at t and t ′ , if all S g do. Similarly,it is said to be compatibly pointed at ∆ ′ if all S g do.A D -pointed set S = { S g | g ∈ D } in LP ( t ) and a φ t ′ ,t D -pointed set S ′ = { S ′ g ′ | g ′ ∈ φ t ′ ,t D } in LP ( t ′ ) are said to be compatible, if ←− µ ∗ t,t ′ S g = S ′ φ t ′ ,t g for all g . It might be possible to rephrase Definition 3.4.1 for suitable formalLaurent series. But Laurent polynomial cases suffice for our purposein this paper.Now assume that t = µ k t ′ in ∆ + for some k ∈ I uf . We furtherassume that the given M ◦ ( t )-pointed set S = { S g | g ∈ M ◦ ( t ) } consistsof elements from LP ( t ) ∩ ( µ ∗ k ) − LP ( t ′ ) compatibly pointed at t, t ′ . Thenthe corresponding dominance order decomposition enjoys the followinginvariance property. Proposition 3.4.2 (Mutation-invariance [Qin19a, Proposition 4.2.1]) . For any Laurent polynomial Z ∈ LP ( t ) ∩ ( µ ∗ k ) − LP ( t ′ ) , its ≺ t -decompositionin terms of S in LP ( t ) and the ≺ t ′ -decomposition of µ ∗ k Z in terms of µ ∗ k S in LP ( t ′ ) have the same coefficients: α t ( Z )( g ) = α t ′ ( µ ∗ k Z )( φ t ′ ,t g ) , ∀ g ∈ M ◦ ( t ) . Definition 3.4.3 (Degree transformation [Qin19a]) . We define the lin-ear map ψ t ′ ,t : M ◦ ( t ) → M ◦ ( t ′ ) such that ψ t ′ ,t ( P g i f i ) = P g i φ t ′ ,t f i forany ( g i ) ∈ Z I . The map ψ t ′ ,t is bijective, see [Qin19a]. Let us explain its meaning.Denote ←− µ t,t ′ = ←− µ − t ′ ,t . By [Qin19a, Lemma 3.3.4], the mutation map ←− µ ∗ t,t ′ : F ( t ) ≃ F ( t ′ ) induces a natural embedding ι from LP ( t ) to d LP ( t ′ ) called the formal Laurent series expansion, such that for any Z ∈ LP ( t ) ∩ ←− µ ∗ t ′ ,t LP ( t ′ ), we have ι ( Z ) = ←− µ ∗ t,t ′ ( Z ). We denote ι by ←− µ ∗ t,t ′ from now on. Then we have the following result. Lemma 3.4.4 ([Qin19a, Lemma 3.3.7]) . For any Laurent monomial X ( t ) g ∈ LP ( t ) , g ∈ M ◦ ( t ) , ←− µ ∗ t,t ′ X ( t ) g is a pointed function in d LP ( t ′ ) with the degree ψ t ′ ,t g . Let there be given injective-reachable seeds t = ←− µ t [ − ψ t [ − ,t and Lemma 3.4.4, we obtain thefollowing useful results. Proposition 3.4.5 (Degree/codegre swap [Qin19a, Propositions 3.3.9,3.3.10]) . (1) For any g, η ∈ M ◦ ( t ) , we have η ≺ t g if and only if ψ t [ − ,t η ≻ t [ − ψ t [ − ,t g .(2) Let there be given Z ∈ LP ( t ) ∩ ( ←− µ ∗ ) − LP ( t [ − . Then Z iscopointed at η if and only if ←− µ ∗ Z is pointed at ψ t [ − ,t η . Definition 3.4.6 (Support dimensions [Qin19a]) . Assume t is injective-reachable. For any g ∈ M ◦ ( t ) , if there exists some n ∈ N ≥ uf ( t ) ≃ N I uf such that ψ − t [ − ,t φ t [ − ,t g = g + e B ( t ) · n, we say g has the support dimension suppDim g = n . Proposition 3.4.7 ([Qin19a, Propositions 3.4.7]) . Let there be givena g -pointed element Z ∈ LP ( t ) ∩ ( ←− µ ∗ ) − LP ( t [ − , g ∈ M ◦ ( t ) . Then Z is compatibly pointed at t, t [ − if and only if it is bipointed withsupport dimension suppDim g .Proof. Assume Z is copointed at η = g + p ∗ suppDim g . By Proposition3.4.5, we deduce that deg t [ − ←− µ ∗ Z = ψ t [ − ,t η = φ t [ − ,t g , where the lastequality follows from the definition of suppDim g .Conversely, if Z is compatibly pointed at t, t [ − Z must be copointed at some η , such that ψ t [ − ,t η = φ t [ − ,t g . It follows from definition that η = g + p ∗ suppDim g . (cid:3) In the proof of Proposition 6.4.5, we will need the fact that the clustermonomials are compatibly pointed, which is a consequence of the sign-coherence of g -vectors [DWZ10][GHKK18], see [FZ07, Remark 7.13].By Proposition 3.4.7, this fact implies that the support dimension of a g -pointed cluster monomial (Definition 3.1.6) provides suppDim g . Remark 3.4.8.
Although we do not need the following result for prov-ing the main Theorem (Theorem 1.2.2), it is worth remarking thatthe θ -basis in [GHKK18] provides the existence of suppDim g for all g ,where the initial seed t is assumed to be injective-reachable. In partic-ular, suppDim g only depends on the principal part B ( t ) .In view of the Fock-Goncharov dual basis conjecture, the dimensions suppDim g should be understood as the correct support dimension pos-sessed by good basis elements, see [Qin19a] for more details. Similar seeds and a correction technique
Similar seeds .
Following [Qin14][Qin17, Section 4], we considersimilar seeds with the same principal B -matrix. Let there be given fixed data such as I, I uf , M ◦ and N uf in Section 2. The following definition ismore restricted than those in [Qin17, Section 4] but it suffices for ourpurpose. Definition 4.1.1 (Similar seeds) . Let σ denote a permutation of I uf .Two seeds t (1) , t (2) (not necessarily related by mutations) are called sim-ilar up to σ if b σ ( i ) σ ( j ) ( t (2) ) = b ij ( t (1) ) for any i, j ∈ I uf . Two quantumseeds are called similar up to σ if we further have d ′ k ( t (1) ) = d ′ σk ( t (2) ) for all k ∈ I uf . For simplicity, we replace the symbol ( t ( r ) ) by ( r ) when describingdata associated to the seeds t ( r ) , r = 1 , σ on I uf induces a linear map σ : N (1) uf ≃ N (2) uf suchthat σe (1) k = e (2) σk where e ( r ) k are the k -th unit vector in N uf ( r ) , r = 1 , σ by identity to a permutation σ on I . Then we obtain a linearmap σ : M ◦ ( t (1) ) ≃ M ◦ ( t (2) ) such that σf (1) i = f (2) σi where f ( r ) i are the i -th unit vectors in M ◦ ( t ( r ) ), r = 1 , M ◦ ( t ( r ) ) ≃ Z I and N ( r ) uf ≃ Z I uf . Wehave the following result. Lemma 4.1.2. If η = g + e B (1) n , for n ∈ Z I uf , then ση = σg + e B (2) σn + u for some u ∈ Z I f . In particular, η ≺ t (1) g implies that ση ≺ t (2) ( σg + u ) for some u ∈ Z I f .Proof. We have that σ e B (1) e (1) k = σ P b (1) ik f (1) i = P i ∈ I b (1) ik f (2) σi = P i ∈ I b (2) σi,σk f (2) σi + u = e B (2) e (2) σk + u = e B (2) σe (1) k + u for some u ∈ Z I f . The claim followsby the linearity of σ . (cid:3) Let ←− µ = µ k r · · · µ k denote any given mutation sequence. Define σ ←− µ = µ σk r · · · µ σk . The following result is a consequence of Theorem2.4.1. Theorem 4.1.3 (Quantum F -polynomial) . For any j ∈ I , denote the(quantum) cluster variable ←− µ ∗ X j ( ←− µ t (1) ) ∈ A q ( t (1) ) by ←− µ ∗ X j ( ←− µ t (1) ) = X ( t (1) ) g (1) · X n ∈ N I uf c n Y ( t (1) ) n where g (1) ∈ Z I ≃ M ◦ ( t (1) ) , c n ∈ k . Then there exists some frozen fac-tor p ∈ P such that the quantum cluster variable ( σ ←− µ ) ∗ X σj (( σ ←− µ ) t (2) ) ∈A q ( t (2) ) takes the form ( σ ←− µ ) ∗ X σj (( σ ←− µ ) t (2) ) = p · X ( t (2) ) σg (1) · X n ∈ N I uf c n Y ( t (2) ) σn . A correction technique.
Following [Qin14][Qin17, Section 4],we explain how to translate algebraic relations from one seed to asimilar one, called a correction technique. Let there be given simi-lar quantum seeds t (1) , t (2) as before, which are not necessarily relatedby mutations.For r = 1 ,
2, given g ( r ) -pointed function Z ( r ) ∈ d LP ( t ( r ) ), where g ( r ) ∈ Z I ≃ M ◦ ( t ( r ) ). Then Z ( r ) takes the form X ( t ( r ) ) g ( r ) · P n ∈ N I uf c ( r ) n Y ( t ( r ) ) n ,where c ( r )0 = 1, c ( r ) n ∈ k . Recall that we have the natural projection pr I uf : Z I → Z I uf . Definition 4.2.1. Z (1) and Z (2) are called similar (as pointed func-tions) if pr I uf g (2) = σ pr I uf g (1) and c (2) σn = c (1) n for n ∈ N I uf . It follows from definition that similar pointed Laurent polynomialshave the same support dimension up to the permutation σ on I uf . Theorem 4.2.2 (Correction technique [Qin17, Theorem 4.2.1]) . Letthere be given pointed functions M ( r )0 , M ( r )1 , · · · , M ( r ) s and (possibly in-finitely many) Z ( r ) j in d LP ( t ( r ) ) , r = 1 , , such that M (1) i and M (2) i are similar, Z (1) j and Z (2) j are similar. Assume that all Z (1) j satisfy deg t (1) Z (1) j = deg t (1) Z (1)0 + e B ( t (1) ) u j for some u j ∈ N I uf , and we havethe following algebraic equations in d LP ( t (1) ) M (1)0 = [ M (1)1 ∗ · · · ∗ M (1) s ] t (1) c Z (1)0 = X j =0 c j Z (1) j such that c j ∈ k , c = 0 . Then we have the following algebraic equationsin d LP ( t (2) ) : M (2)0 = p M · [ M (2)1 ∗ · · · ∗ M (2) s ] t (2) c Z (2)0 = X j =0 c j p j · Z (2) j where the correction coefficients p M , p j ∈ P are determined by deg t (2) p M = deg t (2) M (2)0 − s X i =1 deg t (2) M (2) i deg t (2) p j = deg t (2) Z (2)0 + e B ( t (2) ) σu j − deg t (2) Z (2) j . Lemma 4.2.3. If Z (1) and Z (2) are similar, so are their inverses ( Z (1) ) − and ( Z (2) ) − . Proof.
For r = 1 ,
2, we have Z ( r ) = X ( t ( r ) ) g ( r ) · X c ( r ) n Y ( t ( r ) ) n = X ( t ( r ) ) g ( r ) ∗ X v P k ∈ I uf d ′ k g ( r ) k n k c ( r ) n Y ( t ( r ) ) n =: X ( t ( r ) ) g ( r ) ∗ X n ∈ N I uf b ( r ) n Y ( t ( r ) ) n =: X ( t ( r ) ) g ( r ) ∗ G ( r ) Because t (1) , t (2) are similar and Z (1) , Z (2) are similar, we obtain that G (1) and G (2) are similar. It follows that ( G (1) ) − and ( G (2) ) − are sim-ilar formal series in the variables Y k ( t (1) ) and Y σk ( t (2) ), k ∈ I uf , respec-tively. Therefore, the inverses Z ( r ) = ( G ( r ) ) − ∗ X ( t ( r ) ) − g ( r ) are similarby Theorem 4.2.2 or direct computation. (cid:3) Let there be given any mutation sequence ←− µ . Denote s (1) = ←− µ t (1) and s (2) = ( σ ←− µ ) t (2) . Lemma 4.2.4 ([Qin17, Lemma 4.2.2]) . The following statements aretrue.(1) s (1) and s (2) are similar quantum seeds up to the permutation σ .(2) The quantum cluster monomials ←− µ ∗ X ( s (1) ) m ∈ LP ( t (1) ) and ( σ ←− µ ) ∗ X ( s (2) ) σm ∈ LP ( t (2) ) , m ∈ N I , are similar. As a consequence of Lemma 4.2.4, we obtain the following result.
Lemma 4.2.5.
For any m ∈ Z I uf . The pointed functions ←− µ ∗ X ( s (1) ) m ∈ d LP ( t (1) ) and ( σ ←− µ ) ∗ X ( s (2) ) σm ∈ d LP ( t (2) ) are similar.Proof. By Lemma 4.2.4, we know that ←− µ ∗ X ( s ( r ) ) [ m ] + , r = 1 ,
2, are sim-ilar, and ←− µ ∗ X ( s ( r ) ) [ − m ] + , r = 1 ,
2, are also similar. Then Lemma 4.2.3implies that the inverse ←− µ ∗ X ( s ( r ) ) − [ − m ] + , r = 1 ,
2, are similar. Finally,Theorem 4.2.2 implies that the normalized product ←− µ ∗ X ( s (1) ) m :=[ ←− µ ∗ X ( s (1) ) [ m ] + ∗ ←− µ ∗ X ( s (1) ) − [ − m ] + ] s (1) is similar to p · [ ←− µ ∗ X ( s (2) ) [ m ] + ∗←− µ ∗ X ( s (2) ) − [ − m ] + ] s (2) for some frozen factor p ∈ P . The claim follows. (cid:3) The following natural result is not needed by the main Theorem,though it will be used in the proof of Proposition 6.5.1.
Proposition 4.2.6.
Denote t (1) = ←− µ t ′ (1) and t (2) = ( σ ←− µ ) t ′ (2) . Letthere be given similar pointed functions Z (1) ∈ LP ( t (1) ) ∩ ( ←− µ ∗ ) − LP ( t ′ (1) ) and Z (2) ∈ LP ( t (2) ) such that ←− µ ∗ Z (1) is pointed. Then ( σ ←− µ ) ∗ Z (2) iscontained in LP ( t ′ (2) ) , is pointed, and is similar to ←− µ ∗ Z (1) .Proof. Let us omit the symbol t for simplicity, and denote X ( t ′ ( i ) ), Y ( t ′ ( i ) ) by X ′ ( i ) , Y ′ ( i ) respectively. Denote Z (1) = p · P n ∈ N I uf c n ( X (1) ) m · ( Y (1) ) n for m ∈ Z I uf , p ∈ P , c n ∈ k , c = 1. Then the similar element Z (2) takes the form Z (2) = p · P c n ( X (2) ) σm · ( Y (2) ) σn , p ∈ P . Applying the mutation ←− µ ∗ , weobtain the following equality in d LP ( t ′ (1) ): ←− µ ∗ Z (1) = p · X c n ←− µ ∗ (( X (1) ) m · ( Y (1) ) n ) . Similarly, we have the following equality in d LP ( t ′ (2) ):( σ ←− µ ) ∗ Z (2) = p · X c n ( σ ←− µ ) ∗ (( X (2) ) σm · ( Y (2) ) σn ) . Denote deg t ′ ( i ) by deg ( i ) for simplicity. By [Qin19a, Proposition3.3.8], we have deg ( i ) ←− µ ∗ ( Y ( i ) ) n = deg ( i ) ( Y ′ ( i ) ) C ( i ) n , where C ( i ) := C t ′ ( i ) ( t ( i ) )is the C -matrix associated to the seed t ( i ) with respect to the initialseed t ′ ( i ) , i = 1 , C -matrices (Definition 2.3.6) and the similar-ity between t (1) , t (2) , we have C (1) ij = C (2) σi,σj . It follows that σ ( C (1) n ) = C (2) σn , n ∈ Z I uf (see the proof of Lemma 4.1.2). Denote C (1) = C and g ( i ) = deg ( i ) ( σ i − ←− µ ) ∗ ( X ( i ) ) σ i − m . Then ( σ i − ←− µ ) ∗ (( X ( i ) ) σ i − m · ( Y ( i ) ) σ i − n ) is pointed at g ( i ) + deg ( i ) ( Y ′ ( i ) ) σ i − Cn , where i = 1 , σ ←− µ ) ∗ (( X (2) ) σm · ( Y (2) ) σn ) are similar to ←− µ ∗ (( X (1) ) m · ( Y (1) ) n ) by Lemma 4.2.5. Let V denote a pointed function in LP ( t ′ (2) )similar to ←− µ ∗ Z (1) . Then the correction technique (Theorem 4.2.2) im-plies that there exists some p ∈ P such that the following holds in d LP ( t ′ (2) ): V = p · X c n ( σ ←− µ ) ∗ (( X (2) ) σm · ( Y (2) ) σn ) . It follows that we have ( σ ←− µ ) ∗ Z (2) = V · p − · p , which is pointed andsimilar to the Laurent polynomial ←− µ ∗ Z (1) . (cid:3) Cluster twist automorphisms
Twist automorphisms exist on unipotent cells and is related to Cham-ber Ansatz problem, see [KO17, Section 1.3] for more details and aquantum analogue on quantum unipotent cells. In this section, forgeneral quantum cluster algebras not necessarily having a Lie theoreticbackground, we consider an analogous notion. We call our constructioncluster twist automorphisms or twist automorphisms for short.5.1.
Twist automorphisms passing through similar seeds.
Letthere be given quantum seeds t ′ = ←− µ t ∈ ∆ + such that they are similarup to a permutation σ . Assume that we have an algebra homomor-phism var t : F ( t ) ≃ F ( t ′ ) which preserves the twisted product. In view of Theorem 4.1.3 and the correction technique in Section 4.2, we wantit to send Y -variables to Y -variables as follows. Definition 5.1.1.
The algebra isomorphism var t : F ( t ) ≃ F ( t ′ ) iscalled a variation map if, for any k ∈ I uf , there exists some p k ∈ P such that var t ( X k ) = p k · X σk var t ( Y k ( t )) = Y σk ( t ′ ) var t ( P ) = P . By definition, if var t is a variation map, so does its inverse.We will be interested in the case that var t is the map induced froma linear bijection var t : M ◦ ( t ) ≃ M ◦ ( t ′ ) such that var t ( f k ) = f ′ σk + u k var t ( X i ∈ I b ik f i ) = X i ∈ I b ′ σi,σk f ′ σi var t ( ⊕ j ∈ I f Z f j ) = ⊕ j ∈ I f Z f ′ j where f i , f ′ i are the i -th unit vectors in M ◦ ( t ) and M ◦ ( t ′ ) respectively,such that u k ∈ Z I f and p k = X u k . This linear map has the followingproperty by definition. Lemma 5.1.2. If g ′ ≺ t g in M ◦ ( t ) , then var t g ′ ≺ t ′ var t g in M ◦ ( t ′ ) . Recall that we have the mutation birational map ←− µ ∗ : F ( t ′ ) ≃ F ( t ). Definition 5.1.3 (Cluster twist automorphism) . The composition tw t = ←− µ ∗ var t : F ( t ) ≃ F ( t ) is called a cluster twist automorphism passingthrough t ′ (twist automorphism for short). Here, var t : F ( t ) ≃ F ( t ′ ) iscalled its variation part and ←− µ ∗ : F ( t ′ ) ≃ F ( t ) its mutation part. Notice that t = ←− µ − t ′ . If var t is a variation map, then var t ′ :=( var t ) − : F ( t ′ ) ≃ F ( t ) is a variation map too. Similar to the definitionof the twist automorphism tw t , it is natural to consider the compositionin the opposite order var t ′ ( ←− µ − ) ∗ : F ( t ) ≃ F ( t ). It turns out that var t ′ ( ←− µ − ) ∗ = ( tw t ) − , since ( ←− µ − ) ∗ = ( ←− µ ∗ ) − . Lemma 5.1.4.
A twist automorphism tw t is a permutation on the setof the localized quantum cluster monomials.Proof. Let there be given any localized quantum cluster monomial Z ∈A ( t ) ⊂ LP ( t ). By the definition of a variation map and Theorem 4.1.3, var t ( Z ) is a localized quantum cluster monomial in A ( t ′ ) ⊂ LP ( t ′ ).Then ←− µ ∗ var t ( Z ) becomes its Laurent expansion in A ( t ) = ←− µ ∗ A ( t ′ ) ⊂LP ( t ). The map tw t is a permutation because its inverse ( tw t ) − = ( var t ) − ( ←− µ ∗ ) − sends localized quantum monomials to localized quantum cluster mono-mials by similar arguments as above. (cid:3) Remark 5.1.5.
By definition, given a seed t , a twist automorphismmight not be unique, and its existence is not yet clear. We will discussgeneral twist automorphisms elsewhere. Next, assume that a twist automorphism tw t on F ( t ) is given, wewant to show that it gives rise to twist automorphisms for all seeds.For any k ∈ I uf and t ∈ ∆ + , consider seeds s = µ k t, s ′ = µ σk t ′ ∈ ∆ + . Then we have s ′ = µ σk ←− µ µ k ( s ) =: ←− ν s and the correspondingmutation birational map ( ←− ν ) ∗ : F ( s ′ ) ≃ F ( s ). Recall that we alsohave isomorphisms µ ∗ k : F ( s ) ≃ F ( t ), µ ∗ σk : F ( s ′ ) ≃ F ( t ′ ). Then tw t gives rise to an automorphism tw s = ( µ ∗ k ) − tw t µ ∗ k on F ( s ). Moreover,we have the decomposition tw s = ( ←− ν ) ∗ var s where we define the algebraisomorphism var s := ( µ ∗ σk ) − var t ( µ ∗ k ). We summarize our constructionin the following commutative diagrams. s ′ ←− ν ←− s ↑ µ σk ↑ µ k t ′ ←− µ ←− t F ( s ) var s −−→ F ( s ′ ) ←− ν ∗ −−→ F ( s ) ↓ µ ∗ k ↓ µ ∗ σk ↓ µ ∗ k F ( t ) var t −−→ F ( t ′ ) ←− µ ∗ −−→ F ( t ) Proposition 5.1.6.
The map var s : F ( s ) ≃ F ( s ′ ) for s = µ k t is avariation map in the sense of Definition 5.1.1.Proof. We check the defining properties of a variation map one by one,using the mutation rules for X -variables and Y -variables.Take any i ∈ I uf . By the mutation rule of Y -variables, µ ∗ k Y i ( s )is a function of Y i ( t ) and Y k ( t ). Similarly, µ ∗ σk Y σi ( s ′ ) is a function of Y σi ( t ′ ) = var t Y i ( t ) and Y σk ( t ′ ) = var t Y k ( t ). Moreover, the two functionsare the same because they only depends on the principal B -matricesassociated to the similar seeds t and t ′ , see (2.3). It follows that var t µ ∗ k Y i ( s ) = µ ∗ σk Y σi ( s ′ ). Therefore, Y σi ( s ′ ) = ( µ ∗ σk ) − var t µ ∗ k Y i ( s ) = var s Y i ( s ).If i = k , we get var t µ ∗ k X i ( s ) = var t X i ( t ) = p i · X σi ( t ′ ) = µ ∗ σk ( p i · X σi ( s ′ )). Therefore, var s X i ( s ) = p i · X σi ( s ′ ) where p i ∈ P . For i = k , we have var t µ ∗ k X k ( s ) = var t ( X ( t ) − f k + P j ∈ I [ − b jk ] + f j ∗ (1 + v − k Y k ( t ))= var t X ( t ) − f k + P j ∈ I [ − b j,k ] + f j ∗ var t (1 + v − k Y k ( t ))= X ( t ′ ) u − f ′ σk + P j ∈ I [ − b ′ σj,σk ] + f ′ σj ∗ (1 + v − k Y σk ( t ′ ))for some u ∈ Z I f determined by var t . Notice that we have d ′ k = d ′ σk .Then the above result turns out to be var t µ ∗ k X k ( s ) = X ( t ′ ) u − f ′ σk + P j ∈ I [ − b ′ σj,σk ] + f ′ σj ∗ (1 + v − σk Y σk ( t ′ ))= p · µ ∗ σk X σk ( s ′ )where p = X u . It follows that var s X k ( s ) = p · X σk ( s ′ ).Finally, we have var t µ ∗ k ( P ) = var t P = P = µ ∗ σk P , which implies var s P = P . (cid:3) As a consequence, tw s is a twist automorphism on F ( s ). Therefore,once we are given a twist automorphism tw t on F ( t ) for some seed t ,we obtain twist automorphisms tw s on F ( s ) for all seeds s ∈ ∆ + byrepeatedly using Proposition 5.1.6. We deduce the following result. Theorem 5.1.7.
The twist automorphism tw t on F ( t ) gives rise toa twist automorphism tw s = ( ←− µ ∗ t,s ) − tw t ←− µ ∗ t,s on F ( s ) for any seed s = ←− µ s,t t ∈ ∆ + . Notice that the twist automorphisms tw s in Theorem 5.1.7 becomethe same automorphism once we identify the isomorphic fraction fields F ( s ) by mutations, s ∈ ∆ + . Correspondingly, we can use tw to denotethe twist automorphism.5.2. Twist automorphism of Donaldson-Thomas type.
In thispaper, we will mainly be interested in the case when the pair of similarseeds is given by t ′ = ←− µ t such that t ′ = t [1]. Notice that t [1] and t aresimilar as quantum seeds by Lemma 2.5.4.Assume we are given a twist automorphism tw t = ←− µ ∗ var t on F ( t )passing through t ′ = t [1], where var t : F ( t ) ≃ F ( t ′ ) is the variationpart. Then for any k ∈ I uf and s = µ k t , s ′ = µ σk t ′ , we have s ′ = s [1], see [Qin17, Proposition 5.1.4][Mul16, Theorem 3.2.1]. Therefore,the twist automorphism tw s passes through s [1] as well. Repeat thisargument, we obtain that for all s ∈ ∆ + , the twist automorphisms tw s pass through s [1].We say such a twist automorphism tw to be of Donaldson-Thomastype (DT-type for short). Lemma 5.2.1.
Let there be given seeds t [1] = ←− µ t and an associatedtwist automorphism tw that passes through t [1] . Then, for any k ∈ I uf , there exists a frozen factor p k ∈ P such that tw X k ( t ) = p k · I k ( t )(5.1) where the cluster variable I k ( t ) is given by (2.5) .Proof. The variation map var t identify X k ( t ) with p k · X σk ( t [1]) for some p k ∈ P . The claim follows from definitions of tw and I k ( t ). (cid:3) Remark 5.2.2.
We call a twist automorphism tw = ←− µ ∗ var t on F ( t ) passing through t [1] to be of Donaldson-Thomas type for the followingreason. When B ( t ) is skew-symmetric, there is an associated Jacobianalgebra of a quiver with potential [DWZ10] in the categorification ap-proach to cluster algebras. Then the mutation part ←− µ ∗ of tw is oftencalled the Donaldson-Thomas transformation, and it is categorified bythe shift functor in the associated -Calabi-Yau category, see the in-spiring work [Nag13] for more details.It would be desirable to further investigate the twist automorphismin categories. In general, it is unclear how to categorify ←− µ ∗ for skew-symmetrizable B ( t ) . Moreover, we do not know the categorical meaningof the variation map. Triangular bases
In this section, we prove some general results concerning triangularbases and their relation with twist automorphisms. We will apply theresults to cluster algebras arising from quantum unipotent subgroupsin Section 9.We consider the quantum case ( k , v ) = ( Z [ q ± ] , q ), where q is aformal parameter. Let m = v − Z [ v − ] denote the chosen non-unitalsubring of k . Recall that we have the automorphism of k as a Z -module such that v = v − , which extends to the bar involution ( Z -linear anti-automorphism) on the quantum algebra d LP ( t ) such that vX m = v − X m . By bar-invariance we mean invariant under this barinvolution.Let t denote a given quantum seed in ∆ + .6.1. Triangular functions.
Let there be given a M ◦ ( t )-pointed set S = { S g | g ∈ M ◦ ( t ) } in the formal Laurent polynomial ring d LP ( t ).Then any Z ∈ d LP ( t ) has a unique ≺ t -decomposition in terms of S (Definition 3.3.1). Notice that S is a topological basis in the sense of[DM19].We choose S to be the set of distinguished functions in d LP ( t ). Definition 6.1.1 (Triangular funcitons) . For any g ∈ M ◦ ( t ) , the tri-angular function L g with respect to the set of distinguished functions S is the g -pointed bar-invariant element in d LP ( t ) , such that its ≺ t -decomposition in terms of S is ( ≺ t , m ) -unitriangular. Before constructing the triangular functions, we first prove the fol-lowing general statement for constructing a bar-invariant difference.
Proposition 6.1.2 (Bar-invariant difference) . For any Z = P b m X m ∈ d LP ( t ) , there exists a unique formal sum P α g S g , α g ∈ m , with finitelymany ≺ t -maximal degrees in { g | α g = 0 } , such that Z − P α g S g isbar-invariant.Proof. Similar to the construction of ≺ t -decomposition [Qin19a, Sec-tion 4], we give an algorithm for the construction by tracking the Lau-rent degrees from larger ones to smaller ones.For any Z = P b m X m ∈ d LP ( t ), denote its (finitely many) ≺ t -maximal degrees by g ( i ) . For each corresponding coefficient b g ( i ) , thereexists a unique α g ( i ) ∈ m such that b g ( i ) − α g ( i ) is bar-invariant. Moreprecisely, α g ( i ) is the unique solution in m = v − Z [ v − ] such that b g ( i ) − b g ( i ) = α g ( i ) − α g ( i ) . Notice that, α g ( i ) = 0 if and only if b g ( i ) is not bar-invariant.The ≺ t -maximal degree terms in the difference Z − P α g ( i ) S g havebar-invariant coefficients b g ( i ) − α g ( i ) . Then we obtain a bar-invariantterm P ( b g ( i ) − α g ( i ) ) X g ( i ) . Let us proceed with the difference Z ′ = Z − P α g ( i ) S g ( i ) − P ( b g ( i ) − α g ( i ) ) X g ( i ) ∈ d LP ( t ). Repeating this (possiblyinfinite) process, we obtain a formal sum Z − P α g S g which equals theformal sum of the bar-invariant terms, where each term is of the form P ( b g ( i ) − α g ( i ) ) X g ( i ) . (cid:3) Theorem 6.1.3 (Existence of triangular functions) . For any g ∈ M ◦ ( t ) , the triangular function L g with respect to S exists.Proof. By applying Proposition 6.1.2 to S g and the M ◦ ( t )-pointed set S , we can find unique α g ′ ∈ m for all g ′ (cid:22) t g such that the difference S g − P g ′ α g ′ S g ′ is bar-invariant. Moreover, since the leading term of S g has the bar-invariant coefficient 1, we must have α g = 0 ∈ m . Nowwe obtain the triangular function L g = S g − P g ′ ≺ t g α g ′ S g ′ , α g ′ ∈ m . (cid:3) Lemma 6.1.4 (Uniqueness) . For any g ∈ M ◦ ( t ) , the triangular func-tion L g with respect to S is unique.Proof. Let there be given two such triangular functions L g and L ′ g .Then L g − L ′ g = P g ′ ≺ t g β g ′ S g ′ for some β g ′ ∈ m . If the difference doesnot vanish, the coefficients of its ≺ t -maximal degrees take the form0 = β g ′ ∈ m . But L g − L ′ g is also bar-invariant. It follows that β g ′ = 0,a contradiction. (cid:3) Remark 6.1.5.
Recursive constructions similar to those for Theorem6.1.3 have been used for studying well-known bases in representationtheory [Nak04, Lemma 8.4][Lus90, 7.10] .Notice that the construction of the triangular functions depends onthe chosen set S of the distinguished functions. For the purpose inthis paper, our chosen set of distinguished functions will be I t definedin (6.1) . One might also consider other distinguished functions. Forexample, by choosing the dual PBW bases or the standard modules ofquantum affine algebras, one can show that the corresponding triangularfunctions are the dual canonical bases or the characters of the simplemodules, see [Qin17, Section 9.1] .It is worth reminding that, in our situation, the chosen set S is oftenNOT a basis of the algebra that we consider, unlike the dual PBW basesor the bases consisting of standard modules. The triangular bases.
Given a subalgebra A ( t ) of the quantumupper cluster algebra U q ( t ), we have A ( t ) ⊂ U q ( t ) ⊂ LP ( t ). Underthe isomorphism ←− µ ∗ t,t ′ : F ( t ) ≃ F ( t ′ ) for any given t ′ = ←− µ t ′ ,t t ∈ ∆ + ,we obtain that A ( t ′ ) := ←− µ ∗ t,t ′ A ( t ) satisfies A ( t ′ ) ⊂ U q ( t ′ ) ⊂ LP ( t ′ ).For simplicity, we might omit ←− µ ∗ t,t ′ and the choice of initial seed, andsimply write A ⊂ U q .We further assume that t is injective-reachable (Section 2.5). Definition 6.2.1 (Triangular bases [Qin17]) . We say a k -basis L ofthe subalgebra A ( t ) ⊂ U q ( t ) is a triangular basis with respect to the seed t , if the following conditions hold:(1) (Bar invariance) The basis elements in L are invariant underthe bar involution.(2) The quantum cluster monomials in t and t [1] are contained in L .(3) (Parametrization) L is M ◦ ( t ) -pointed, i.e., it takes the form L = { L g | g ∈ M ◦ ( t ) } , such that L g is g -pointed.(4) (Triangularity) For any basis elements L g and i ∈ I , the de-composition of the normalized product [ X i ( t ) ∗ L g ] t in terms of L is ( ≺ t , m ) -unitriangular: [ X i ( t ) ∗ L g ] t = L g + f i + X g ′ ≺ t g + f i b g ′ L g ′ , b g ′ ∈ m . A basis L of A ( t ) naturally gives rise to a basis ←− µ ∗ t,t ′ L of A ( t ′ ) forany seeds t = ←− µ t,t ′ t ′ . Definition 6.2.2 (Common triangular bases [Qin17]) . The triangularbasis L of A ( t ) with respect to the seed t is said to be the commontriangular basis, if ←− µ ∗ t,t ′ L are the triangular bases of A ( t ′ ) for all seeds t ′ ∈ ∆ + respectively and they are pairwise compatible. The triangular basis satisfies following factorization property.
Lemma 6.2.3 (Factorization [Qin17, Lemma 6.2.1]) . Let L = { L g | g ∈ M ◦ ( t ) } be the triangular basis with respect to t , where deg t L g = g .Take any g ∈ M ◦ ( t ) .(1) For any i ∈ I such that X i ( t ) ∗ L g = v α L g ∗ X i ( t ) , α ∈ Z , wehave [ X i ( t ) ∗ L g ] t = [ L g ∗ X i ( t )] t = L g + f i ∈ L .(2) For any j ∈ I f , we have [ X j ( t ) ∗ L g ] t = [ L g ∗ X j ( t )] t = L g + f j ∈ L . Let pr I uf and pr I f denote the natural projection from M ◦ ( t ) ≃ Z I to Z I uf and Z I f respectively. For any g = ( g i ) i ∈ I ∈ M ◦ ( t ) = ⊕ Z f i , let [ g ] + denote ([ g i ] + ) i . For any m ∈ N I uf , let I m denote the normalization ofthe twisted product [ Q I m kk ( t )] t in LP ( t ), where the cluster variables I k ( t ) are given by (2.5). Then the twisted product X [ pr I uf g ] + ∗ I [ − pr I uf g ] + has degree g + u for some u ∈ Z I f . We define the following g -pointedfunction in LP ( t ): I tg := [ p g ∗ X ( t ) [ pr I uf g ] + ∗ I ( t ) [ − pr I uf g ] + ] t (6.1)where p g = X ( t ) − u ∈ P .For the rest of the paper, we choose the set of distinguished functionsin d LP ( t ) to be I t = { I tg | g ∈ M ◦ ( t ) } . Definition 6.2.4 (Weakly triangular basis [Qin17, Definition 6.3.1]) . Let L t denote the set of triangular functions { L g | g ∈ M ◦ ( t ) } withrespect to the set of distinguished functions I t . If L t is a k -basis of A ( t ) ⊂ U q ( t ) , we call it the weakly triangular basis of A ( t ) with respectto the seed t . By Lemma 6.1.4, the weakly triangular basis is unique if it exists.Notice that a weakly triangular basis contains the quantum clustermonomials in t, t [1]. If it further satisfies condition (4) in Definition6.2.1, it becomes the triangular basis.
Lemma 6.2.5 ([Qin17, Lemma 6.3.2]) . If A has a triangular basis L for seed t , then L is the weakly triangular basis L t . In particular, thetriangular basis is unique. Lemma 6.2.3 and the correction technique (Theorem 4.2.2) impliesthe following result.
Lemma 6.2.6 ([Qin17, Lemma 6.3.4]) . Let there be given two similarquantum seeds t and t ′ (not necessarily related by mutations). Assumethat a subalgebra A ( t ) ⊂ U q ( t ) possesses the weakly triangular basis(resp. triangular basis) L t with respect to the seed t . Let L t ′ denotethe set of all pointed functions in LP ( t ′ ) similar to the elements of L t . Then L t ′ spans a k -subalgebra A ( t ′ ) ⊂ U q ( t ′ ) such that it is the weaklytriangular basis (resp. triangular basis) of A ( t ′ ) with respect to t ′ . We recall some useful properties of the set of distinguished functions I t and the (weakly) triangular basis L t . Lemma 6.2.7 (Substitution [Qin17, Lemma 6.2.4] ) . Assume that [ p ∗ X ( t ) d X ∗ I ( t ) d I ] t is ( ≺ t , m ) -unitriangular to I t for any p ∈ P , d X , d I ∈ N I uf . If a pointed function Z ∈ d LP ( t ) is ( ≺ t, m ) -unitriangular to I t ,so is the product [ p ∗ X ( t ) d X ∗ Z ∗ I ( t ) d I ] t . Lemma 6.2.8 ([Qin17, Lemma 5.4.2]) . For any k ∈ I uf , the set ofdistinguished functions I t is compatibly pointed at t and µ k t . Finally, the following result gives a sufficient condition for strength-ening a weakly triangular basis into a triangular basis.
Proposition 6.2.9 ([Qin17]) . Let there be given k ∈ I uf and seeds t = µ k t ′ . If the triangular basis L t for seed t and the weakly triangular basis L t ′ for t ′ both exist, such that they are compatible, i.e., µ ∗ k L tg = L t ′ φ t ′ ,t g ∀ g ∈ M ◦ ( t ) , then L t ′ is the triangular basis for seed t ′ .Proof. The claim is proved as Claim (ii) in the proof of [Qin17, Propo-sition 6.6.3]. Its proof is based on basic properties of triangular bases,as well as the observation that the exchange relation of quantum clustervariables gives a ( ≺ t ′ , m )-triangular decomposition of X k ( t ′ ) ∗ X k ( t ). (cid:3) Triangular bases and twist automorphisms.
Let there begiven similar quantum seeds t, t ′ = ←− µ t ∈ ∆ + . Assume that tw = tw t = ←− µ ∗ var t is a twist automorphism on F ( t ) passing through t ′ where var t : F ( t ) ≃ F ( t ′ ) is the variation map. Proposition 6.3.1.
Assume that L t and L t ′ are the weakly triangularbasis with respect to the seeds t and t ′ respectively. If L t ′ = ( ←− µ ∗ ) − L t ,then L t is permuted by the twist automorphism tw .Proof. Notice that var t L t consists of all elements similar to those of L t .Therefore, var t L t is the weakly triangular basis with respect to t ′ byLemma 6.2.6. We deduce from the uniqueness of triangular functionsthat var t L t = L t ′ = ( ←− µ ∗ ) − L t and, equivalently, tw L t = L t . (cid:3) As a consequence of Proposition 6.3.1, we obtain the following niceproperty of common triangular bases.
Proposition 6.3.2. If A possesses the common triangular basis L ,then L is permuted by any twist automorphisms. Adjacent seeds.
We prove some important statements concern-ing adjacent seeds, which will be useful for studying the existence ofcommon triangular bases.For any t ′ = µ k t , j = k ∈ I uf , recall that we have µ ∗ k X j ( t ′ ) = X j ( t )and µ ∗ k I j ( t ′ ) = I j ( t ). Definition 6.4.1 (Admissibility) . Let there be given seeds t ′ = µ k t .A triangular basis L t with respect to seed t is said to be admissiblein direction k , if the quantum cluster monomials µ ∗ k X k ( t ′ ) d , µ ∗ k I k ( t ′ ) d , d ∈ N , are contained in L t . Let t be a given injective-reachable seed. By the following Lemma, atwist automorphism could reduce the burden to check the admissibilitycondition by half. Lemma 6.4.2.
Assume that we have a DT-type twist automorphism tw . If tw L t = L t , then L t is admissible in direction k if and only if itcontains X k ( t ′ ) d , d ∈ N .Proof. If L t contains X k ( t ′ ) d for some d ∈ N , then L t = tw L t contains tw X k ( t ′ ) d . By Lemma 5.2.1, there exists some frozen factor p k suchthat tw X k ( t ′ ) d = p dk · I k ( t ′ ) d . By the factorization property (Lemma6.2.3), L t contains I k ( t ′ ) d as well. (cid:3) The following crucial result tells us that the admissibility conditionimplies the existence of the compatible triangular basis for an adjacentseed. Unlike previous works, no positivity assumption on the basis isimposed.
Proposition 6.4.3 (Adjacent compatibility) . Let there be given adja-cent seeds t ′ = µ k t related by one mutation, k ∈ I uf , and the triangularbasis L t with respect to the seed t . If L t is admissible in direction k ,then L t ′ := ( µ ∗ k ) − L t is the triangular basis with respect to t ′ . Moreover, L t and L t ′ are compatible.Proof. For any g ∈ M ◦ ( t ), denote g ′ = φ t ′ ,t g ∈ M ◦ ( t ′ ) as before.For any g ′ , the g ′ -pointed function I t ′ g ′ in LP ( t ′ ) takes the form I t ′ g ′ = ( [ p g ′ ∗ X ( t ′ ) d X ∗ X k ( t ′ ) g ′ k ∗ I ( t ′ ) d I ] t ′ g ′ k ≥ p g ′ ∗ X ( t ′ ) d X ∗ I k ( t ′ ) − g ′ k ∗ I ( t ′ ) d I ] t ′ g ′ k ≤ p g ′ ∈ P and d X , d I ∈ N I uf \{ k } . Moreover, µ ∗ k I t ′ g ′ is g -pointed in LP ( t ) by Lemma 6.2.8. Then it takes the following form in LP ( t ): µ ∗ k I t ′ g ′ = ( [ p g ′ ∗ X ( t ) d X ∗ µ ∗ k X k ( t ′ ) g ′ k ∗ I ( t ) d I ] t g ′ k ≥ p g ′ ∗ X ( t ) d X ∗ µ ∗ k I k ( t ′ ) − g ′ k ∗ I ( t ) d I ] t g ′ k ≤ . Since L t is admissible in direction k , it contains the quantum clustermonomials µ ∗ k X ( t ′ ) d , µ ∗ k I k ( t ′ ) d viewed as elements in LP ( t ). Then thesequantum cluster monomials are ( ≺ t , m )-unitriangular to I t . By theSubstitution Lemma 6.2.7, µ ∗ k I t ′ g ′ is ( ≺ t , m )-unitriangular to I t , and thus( ≺ t , m )-unitriangular to L t . Then, by the inverse transition (Lemma3.2.2), L t is ( ≺ t , m )-unitriangular to the M ◦ ( t )-pointed set µ ∗ k I t ′ = { µ ∗ k I t ′ g ′ | g ′ ∈ M ◦ ( t ′ ) } . More precisely, for any g ∈ M ◦ ( t ), L tg has thefollowing ≺ t -decomposition in d LP ( t ): Z := L tg = µ ∗ k I t ′ g ′ + X η ≺ t g b g,η µ ∗ k I t ′ η ′ where the coefficients b g,η ∈ m , η ′ = φ t ′ ,t η .Because the set of distinguished functions I t ′ is compatibly pointedat t and t ′ (Lemma 6.2.8), by the mutation-invariance property of domi-nance order decomposition (Proposition 3.4.2), the above ≺ t -decompositionof L tg in terms of I t ′ also gives the ≺ t ′ -triangular decomposition in d LP ( t ′ ): Z ′ := ( µ ∗ k ) − L tg = I t ′ g ′ + X η ≺ t g b g,η I t ′ η ′ . (6.2)We claim that (6.2) is ( ≺ t ′ , m )-unitriangular with the leading term I t ′ g ′ , i.e., ( µ ∗ k ) − L tg is g ′ -pointed. For the proof, first notice that ( µ ∗ k ) − commutes with the bar involution. It follows that Z ′ is bar-invariant.Consider the support supp I t ′ ( Z ′ ) := { g ′ } ∪ { η ′ | b g,η = 0 } . Denote b g,g = 1. Any ≺ t ′ -maximal elements m ′ ∈ supp I t ′ ( Z ′ ) contributes aLaurent monomial b g,m X ( t ′ ) m ′ with a ≺ t ′ − maximal degree in the Lau-rent expansion of Z ′ ∈ LP ( t ′ ). Notice that b g,g = 1 and b g,η ∈ m for η ≺ t g . Then, the bar-invariance of Z ′ implies that g ′ is the unique ≺ t ′ -maximal element in the supp I t ′ Z . Our claim has been verified.By the above claim, L t ′ := ( µ ∗ k ) − L t is compatible with L t , and itis ( ≺ t ′ , m )-unitriangular to I t ′ . It follows from definition that it is theweakly triangular basis for t . By Proposition 6.2.9, L t ′ is further thetriangular basis for the seed t ′ . (cid:3) We also need to verify the admissibility condition, provided somecompatibility condition (tropical properties). Let there be given any k ∈ I uf and d ∈ N . Denote t ′ = µ k t and t ′ [1] = µ σk ( t [1]). Lemma 6.4.4.
Let there be given g = d deg t µ ∗ k X k ( t ′ ) and g -pointedelement Z ∈ LP ( t ) ∩ µ ∗ k LP ( t ′ ) . If Z is bipointed and its support di-mension is the same as that of µ ∗ k X k ( t ′ ) d , then Z = µ ∗ k X k ( t ′ ) d .Proof. We can view Z and µ ∗ k X k ( t ′ ) d as elements of the (type A ) quan-tum cluster algebra LP ( t ) ∩ µ ∗ k LP ( t ′ ), where k is viewed as the only unfrozen vertex. Then Z and µ ∗ k X ( t ′ ) d are compatibly pointed at theset of all seeds { t, t ′ } , and we have Z = µ ∗ k X k ( t ′ ) d by Proposition 3.4.7and [Qin19a, Lemma 3.4.11].Alternatively, let us give a fundamental proof. Notice that Z takesthe form X g ∗ (1 + P c r ( Y ′ k ) r + 1)where c r ∈ k , R ∈ N , g ′ = deg t ′ (( µ ∗ k ) − X g ∗ ( Y ′ k ) − d ). By the samecomputation, ( µ ∗ k ) − ( µ ∗ k X k ( t ′ ) d ) = X k ( t ′ ) d also has this degree. Conse-quently, X ′ g ′ = ( X ′ k ) d .Now consider µ ∗ k Z ′ ∈ LP ( t ). Similar to the above computation, itsdegree is given by deg t ( µ ∗ k ( X ′ k ) d ∗ Y kR )). Since deg t µ ∗ k Z ′ = deg t Z = deg t µ ∗ k ( X ′ k ) d , we must have R = 0. Consequently, Z ′ = ( X ′ k ) d . (cid:3) Proposition 6.4.5 (Admissiblity by compatibility) . (1) Take g = d deg t µ ∗ k X k ( t ′ ) . Let there be given g -pointed element Z ∈ LP ( t ) ∩←− µ ∗ t [ − ,t LP ( t [ − . If Z is compatibly pointed at t, t [ − , then Z = µ ∗ k X k ( t ′ ) d .(2) Take g = d deg t ←− µ ∗ k I k ( t ′ ) . Let there be given g -pointed element Z ∈ LP ( t ) ∩ ←− µ ∗ t [1] ,t LP ( t [1]) . If Z is compatibly pointed at t [1] , t , then Z = ←− µ ∗ k I k ( t ′ ) d .Proof. Denote t ′ = µ k t = ←− µ t ′ ,t t . Recall that ←− µ ∗ t ,t ←− µ ∗ t ,t = ←− µ ∗ t ,t forany seeds t , t , t ∈ ∆ + (Lemma 2.3.12).(1) Since Z is compatibly pointed at t, t [ − Z has support dimension suppDim g . The same holds for the clustermonomial µ ∗ k X k ( t ′ ) d . We deduce that L tg and X k ( t ′ ) d have the samesupport dimension. Lemma 6.4.4 implies that L tg = µ ∗ k X k ( t ′ ) d .(2) Notice that we have ←− µ ∗ t,t [1] ←− µ ∗ k I k ( t ′ ) = ←− µ ∗ t,t [1] ←− µ ∗ t ′ ,t I k ( t ′ )= ←− µ ∗ t,t [1] ←− µ ∗ t ′ ,t ←− µ ∗ t ′ [1] ,t ′ X σk ( t ′ [1])= ←− µ ∗ t ′ [1] ,t [1] X σk ( t ′ [1])= µ ∗ σk X σk ( t ′ [1]) . Since Z is compatibly pointed at t [1] , t , we obtain that Z ′ := ←− µ ∗ t,t [1] Z is compatibly pointed at t [1] , t , such that deg t [1] Z ′ = φ t [1] ,t deg t Z = φ t [1] ,t deg t ←− µ ∗ k I k ( t ′ ) d = deg t [1] ←− µ ∗ t,t [1] ←− µ ∗ k I k ( t ′ ) d = deg t [1] µ ∗ σk X σk ( t ′ [1]) d . Itfollows from (1) that Z ′ = µ ∗ σk X σk ( t ′ [1]) d . Therefore, Z = ( ←− µ ∗ t,t [1] ) − Z ′ =( ←− µ ∗ t,t [1] ) − ←− µ ∗ t,t [1] ←− µ ∗ k I k ( t ′ ) d = ←− µ ∗ k I k ( t ′ ) d . (cid:3) Criteria for the existence of the common triangular bases.
In this section, we give several criteria that guarantee the existence ofthe common triangular bases. Let there be given seeds t [1] = ←− µ t = ←− µ t [1] ,t t as before.Define seeds t [ r ] = ( σ r − ←− µ ) t [ r −
1] recursively for r ∈ Z . Thenwe have t [ r ] = t [ r − { t [ r ] , r ∈ Z } is called aninjective-reachable chain, see [Qin17] for more details. Notice that wehave ( µ k t )[ r ] = µ σ r k ( t [ r ]), ←− µ t [ r +1] ,t [ r ] = σ r ←− µ t [1] ,t . Denote ←− µ t [ r ] ,t [ r +1] = ←− µ − t [ r +1] ,t [ r ] . Proposition 6.5.1.
Assume that a given subalgebra A ( t ) ⊂ U q ( t ) pos-sesses the triangular bases L t . The following statements are true.(1) The triangular bases L t [ r ] for the seeds t [ r ] exist, r ∈ Z , and theyconsist of similar pointed functions.(2) If L t [1] is compatibly pointed at t, t [1] , then L t [ r ] is compatiblypointed at t [ r ] , t [ r − , for any r ∈ Z .(3) If ←− µ ∗ t [1] ,t L t [1] is the triangular basis L tg , then ←− µ ∗ t [ r ] ,t [ r − L t [ r ] is thetriangular basis L t [ r − , r ∈ Z . Proposition 6.5.2.
Assume that a given subalgebra A ⊂ U q possessescompatible triangular bases at { t [ r ] , r ∈ Z } . Then, for any k ∈ I uf , itpossesses compatible triangular bases at { ( µ k t )[ r ] , r ∈ Z }∪{ t [ r ] , r ∈ Z } . We postpone the proofs of Propositions 6.5.1, 6.5.2 to the end ofsection.
Theorem 6.5.3 (Existence by compatibility) . Assume that a givensubalgebra A ( t ) ⊂ U q ( t ) possesses the triangular bases L t . If L t is com-patibly pointed at t, t [1] and, moreover, ( ←− µ ∗ t [1] ,t ) − L t is the triangularbasis L t [1] g for the seed t [1] , then L t is the common triangular basis.Proof. Proposition 6.5.1 implies that the triangular bases L t [ r ] , r ∈ Z , exist and are compatible. Using Proposition 6.5.2 recursively foradjacent seeds starting from { t [ r ] , r ∈ Z } , we obtain that L t is thecommon triangular basis. (cid:3) Recall that if the common triangular basis exists, then it is permutedby all twist automorphisms (Proposition 6.3.2). As an inverse result, wehave the following existence theorem which implies the main theorem (Theorem 1.2.2). Denote ←− µ t [1] ,t = ←− µ = µ k l · · · µ k µ k . Denote t s = ←− µ ≤ s t = µ k s · · · µ k µ k t for 1 ≤ s ≤ l . Theorem 6.5.4 (Existence by twist automorphisms) . Assume that agiven subalgebra A ( t ) ⊂ U q ( t ) has the triangular basis L t with respectto the seed t . Further assume that L t satisfies the following properties:(1) There exists a twist automorphism tw of Donaldson-Thomastype on F ( t ) such that tw permutes L t .(2) The quantum cluster monomials ←− µ ∗≤ s X k s ( t s ) d , d ∈ N , obtainedalong the mutation sequence ←− µ t [1] ,t starting from t are containedin L t .Then L t is the common triangular basis. In particular, it containsall quantum cluster monomials.Proof. Because L t is permuted by tw and contains all cluster monomialsappearing along the mutation sequence ←− µ , we deduce from Lemma6.4.2 and Proposition 6.4.3 that it gives rise to the triangular basis L t s for the seeds t s , which are compatible with L t . In particular, L t and L t [1] are compatible. We deduce the claim from Theorem 6.5.3 .Let us give an alternative proof which does not depend on Theorem6.5.3 nor Proposition 6.5.1. For any r ∈ Z , by Lemma 6.2.6 for similarseeds t, t [ r ], we can construct the triangular bases L t [ r ] for seed t [ r ]as the set of elements similar to the elements in L t . Then it containsthe quantum cluster monomials appearing along the sequence σ r ←− µ from t [ r ] to t [ r + 1] by Lemma 4.2.4. It follows that L t [ r ] and L t [ r +1] are compatible by Lemma 6.4.2 and Proposition 6.4.3. Therefore, weobtain compatible triangular bases at { t [ r ] , r ∈ Z } . The claim followsby using Proposition 6.5.2 repeatedly for adjacent seeds. (cid:3) Proof of Proposition 6.5.2.
Denote t ′ = µ k t . Recall that µ σ r k t [ r ] = t ′ [ r ], r ∈ Z . Since L t [ r ] is compatibly pointed at t [ r ] , t [ r − µ ∗ σ r k X σ r k ( t ′ [ r ])) by Proposition 6.4.5(1). Similarly, L t [ r +1] con-tains Z := µ ∗ σ r +1 k X σ r +1 k ( t ′ [ r +1]). In addition, since L t [ r +1] and L t [ r ] arecompatible, L t [ r ] = ←− µ ∗ t [ r +1] ,t [ r ] L t [ r +1] also contains the Z ′ := µ ∗ t [ r +1] ,t [ r ] Z .Lemma 2.3.12 implies that we have Z ′ = µ ∗ t [ r +1] ,t [ r ] µ ∗ t ′ [ r +1] ,t [ r +1] X σ r +1 k ( t ′ [ r + 1]))= µ ∗ t ′ [ r +1] ,t [ r ] X σ r +1 k ( t ′ [ r + 1]))= µ ∗ t ′ [ r ] ,t [ r ] µ ∗ t ′ [ r +1] ,t ′ [ r ] X σ r +1 k ( t ′ [ r + 1]))= µ ∗ t ′ [ r ] ,t [ r ] I σ r k ( t ′ [ r ])= µ ∗ σ r k I σ r k ( t ′ [ r ]) . Therefore, L t [ r ] is admissible in direction σ r k . Then Proposition 6.4.3implies that L t [ r ] gives rise to a compatible triangular basis L t ′ [ r ] in theadjacent seed t ′ [ r ]. The claim follows. (cid:3) Proof of Proposition 6.5.1. (1) We define L t [ r ] to be the set of pointedfunctions in LP ( t [ r ]) similar to the elements of L t . The claim followsfrom Lemma 6.2.6.(2) By Proposition 3.4.7, L t [1] g has the support dimension suppDim g ∈ N I uf ≃ N ≥ uf ( t [1]) for any g ∈ Z I ≃ M ◦ ( t [1]). By (1), L t [ r ] and L t [1] havesimilar elements. Let us relabel the set of vertices I uf by ( σ r − ) − when working with the seed t [ r ], such that b ij ( t [1]) = b ij ( t [ r ]) (or,equivalently, we create a new seed σ − r t [ r ] such that e k ( σ − r t [ r ]) := e σ r − k ( t [ r ])). Then we have suppDim L t [ r ] g = suppDim L t [1] g = suppDim g as elements in Z I uf .Since the support dimension suppDim g associated to g ∈ Z I onlydepends on the principal B -matrix (see Remark 3.4.8), suppDim g for t [1] and t [ r ] are the same. Using Proposition 3.4.7 again, we deducethat L t [ r ] g is compatibly pointed at t [ r ] , t [ r −
1] for any r ∈ Z .(3) For any g ∈ Z I , L t [ r ] σ r − g and L t [1] g are similar by (1). If follows that ←− µ ∗ t [ r ] ,t [ r − L t [ r ] σ r − g and ←− µ ∗ t [1] ,t L t [1] g are similar by Proposition 4.2.6. Since ←− µ ∗ t [1] ,t L t [1] g ∈ L t , its similar function ←− µ ∗ t [ r ] ,t [ r − L t [ r ] σ r − g must belong to L t [ r − by (1). By the same argument, we have ( ←− µ ∗ t [ r ] ,t [ r − ) − L t [ r − g ′ ∈ L t [ r ] for any g ′ ∈ Z I . The claim follows. (cid:3) Prerequisite for quantum unipotent subgroup
We collect useful notions and results for quantum groups, mostlyfollowing [KO17][Kim12]. Notice that different conventions have beenused in literature. We use an easy model to explain ours, see Examples7.2.7, 7.3.2. We choose the field k = Q ( q ) with a formal parameter q .Notice that k has a bar involution such that q = q − .7.1. Quantized enveloping algebras.
Root datum.
A root datum consists of the following.(1) A finite set [1 , r ] := { , , . . . , r } .(2) A finite dimensional Q -vector space h , its dual space h ∗ , andthe canonical pairing h , i .(3) A lattice P ⊂ h ∗ called the weight lattice.(4) The dual lattice P ∨ = Hom Z ( P, Z ) ⊂ h called the coweightlattice.(5) A linearly independent subset { α i | i ∈ [1 , r ] } ⊂ P , called the setof simple roots.(6) A linearly independent subset { α ∨ i | i ∈ [1 , r ] } ⊂ P ∨ , called theset of simple coroots.(7) A Q -valued symmetric Z -bilinear form ( , ) on P , such that • d i := ( α i , α i ) ∈ Z > , • h α ∨ i , µ i = α i ,µ )( α i ,α i ) for any µ ∈ P , • Denote C ij = h α ∨ i , α j i . Then C = ( C ij ) ≤ i,j ≤ r is a gen-eralized symmetrizable Cartan matrix, i.e., C ii = 2 and,for any i = j , we have C ij ∈ Z ≤ , C ij = 0 if and only if C ji = 0.Notice that we have d i C ij = d j C ji . We define the root lattice Q = ⊕ i Z α i ⊂ P , the coroot lattice Q ∨ = ⊕ i Z α ∨ i . Define Q ± = ± ⊕ i N α i ,the semigroup of dominant weights P + = { λ ∈ P |h α ∨ i , λ i ≥ , ∀ i } .Assume that we have fundamental weights ̟ i ∈ P + such that h α ∨ i , ̟ j i = δ ij , ∀ i, j . Denote ρ = P i ∈ [1 ,r ] ̟ i .In addition, for any i ∈ [1 , r ], we define the simple reflection s i on h ∗ such that, for any µ ∈ h ∗ , s i ( µ ) = µ − h α ∨ i , µ i α i . The group W generated by the simple reflections is called the Weylgroup. Let there be given any w ∈ W . Given a word i = i · · · i l with symbols from i j ∈ [1 , r ], l ∈ N , we denote s i = s i · · · s i l , and thelength | i | = l . Recall that, if l = min {| i ′ || s i ′ = w } , we call the word i a reduced word for w , the product s i a reduced expression of w , and l ( w ) = l the length of w . Choosing a reduced word i for w , we definethe support of w in [1 , r ] to be supp w := supp s i := supp i := { i | i ∈ i } . One can check that supp w does not depend on the choice of the reducedword.Similarly, as in [Lus93, 2.2.6], we define reflections s i on h such that,for any h ∈ h , we have s i ( h ) = h − h α i , h i α ∨ i . Then h s i ( h ) , µ i = h h, s i ( µ ) i .For any k ∈ [1 , l ], define its successor and predecessor by s ( k ) := k [1] := min( { j > k | i j = i k } ∪ { + ∞} ) p ( k ) := k [ −
1] := max( { j < k | i j = i k } ∪ {−∞} )We also define k [0] = k , and, recursively define k [ d ±
1] = k [ d ][ ±
1] for d ∈ Z if k [ d ] ∈ [1 , l ]. In addition, denote k max := max { j ∈ [1 , l ] | i j = i k } k min := min { j ∈ [1 , l ] | i j = i k } For any a ∈ supp w , we denote min a = min { k ∈ [1 , l ] | i k = a } max a = max { k ∈ [1 , l ] | i k = a } Similarly, for b ∈ supp ( i i · · · i j ), j ∈ [1 , l ], define min , ≥ j b = min { k ∈ [ j, l ] | i k = b } . max , ≤ j b = max { k ∈ [1 , j ] | i k = b } . Finally, for a ∈ [1 , r ], j, k ∈ [1 , l ], denote the multiplicity: m ( a, [ j, k ]) = |{ s ∈ [ j, k ] | i s = a }| m ( a ) = m ( a, [1 , l ]) m + k = m ( i k , [ k + 1 , l ]) m − k = m ( i k , [1 , k − . Definition of quantized enveloping algebras U q . Denote q i = q d i , [ a ] i =[ a ] q i = q ai − q − ai q i − q − i for i ∈ [1 , r ], a ∈ N . The quantized enveloping algebra U q is the non-commutative k -algebra generated by the generators E i , F i , K h , i ∈ [1 , r ], h ∈ P ∨ subject to the following relations. Denote K i = K d i α ∨ i and the divided power F ( k ) i = F ki [ k ] i ! and E ( k ) i = E ki [ k ] i . For any i, j ∈ [1 , r ], h ∈ P ∨ , we have K h K h ′ = K h + h ′ , K = 1 ,K h E i K − h = q h h,α i i E i ,K h F i K − h = q h h, − α i i F i ,E i F j − F j E i = δ ij K i − K − i q i − q − i , − C ij X k =0 ( − k E ( k ) i E j E (1 − C ij − k ) i = 0 ,i = j, − C ij X k =0 ( − k F ( k ) i F j F (1 − C ij − k ) i = 0 ,i = j. In particular, K i E j = q C ij i E j K i , K i F j = q − C ij i F j K i .Recall that we have a k -algebra triangular decomposition U q = U q + ⊗ U q ⊗ U q − = h E i i i ∈ [1 ,r ] ⊗h K h i h ∈ P ∨ ⊗h F i i i ∈ [1 ,r ] . We have a natural Q -grading wt ( ) on U q such that wt E i = α i , wt F i = − α i , wt K h = 0 re-spectively. For any weight γ ∈ Q , the homogeneous weight γ elementsform a k -vector space( U q ) γ = { u ∈ U q | K h uK − h = q h h,γ i u, ∀ h ∈ P ∨ } . Automorphisms and bilinear forms.
Definition 7.1.1.
Define the k -algebra involution ( ) ∨ on U q such that E ∨ i = F i , F ∨ i = E i , ( K h ) ∨ = K − h . Extend the bar involution ( ) on k to an involution on U q such that q = q − , E i = E i , F i = F i , K h = K − h . Define the k -algebra anti-involution ∗ on U q such that ∗ ( E i ) = E i , ∗ ( F i ) = F i , ∗ ( K h ) = K − h . Definition 7.1.2.
For i ∈ [1 , r ] , define k -linear endomorphisms e ′ i and i e ′ on U q − such that, for any homogeneous x, y , we have e ′ i ( xy ) = e ′ i ( x ) y + q h α ∨ i , wt ( x ) i i xe ′ i ( y ) e ′ i ( F j ) = δ iji e ′ ( xy ) = q h α ∨ i , wt ( y ) i i i e ′ ( x ) y + x i e ′ ( y ) i e ′ ( F j ) = δ ij Following the convention in [KO17], we consider Lusztig’s pairing( , ) L on U q − . It is the k -bilinear form such that(1 , L = 1( F i x, y ) L = 11 − q i ( x, e ′ i ( y )) L ( xF i , y ) L = 11 − q i ( x, i e ′ ( y )) L It is known that ( , ) L is symmetric and non-degenerate. Notice thatone could also work with Kashiwara’s bilinear form ( , ) K , and theresults only differ by scalars depending on the Q -grading, see [Kim12,Lemma 2.12]. Definition 7.1.3.
Define the dual bar involution σ on U q − such that ( σ ( x ) , y ) L = ( x, y ) L . Definition 7.1.4.
Define the k -linear isomorphism c tw : U q − → U q − such that, for homogeneous x , we have c tw ( x ) = q ( wt x, wt x ) − ( wt x,ρ ) x. For µ = P µ i α i ∈ Q , define Tr ( µ ) = P µ i ∈ Z . Proposition 7.1.5 ([KO17, Proposition 3.10]) . For a homogeneouselement x ∈ U q − , we have σ ( x ) = ( − Tr ( wt x ) c tw (( ) ◦ ∗ )( x ) . In particular, for homogeneous x, y , we have σ ( xy ) = q ( wt x, wt y ) σ ( y ) σ ( x ) . For example, we compute that ( F i , F i ) L = − q i , σ ( F i ) = − q i F i , for i ∈ [1 , r ].Define the twisted dual bar involution σ ′ = c − tw ◦ σ on U q − . Then wehave σ ′ ( x ) = ( − Tr ( wt x ) (( ) ◦ ∗ )( x ) for homogeneous x . In particular, σ ( x ) = x if and only if σ ′ ( x ) = c − tw ( x ) = q − ( wt x, wt x )+( wt x,ρ ) x . It followsthat σ ′ ( xy ) = σ ′ ( y ) σ ′ ( x ).In order to compare quantum groups with the quantum cluster al-gebras, a rescaling will be needed. We extend the base field from k = Q ( q ) to k ( q ) = Q ( q ), denote U q − Q ( q ) = U q − ⊗ Q ( q ) Q ( q ). Definethe Q ( q )-module endomorphism cor of U q − Q ( q ) such that cor x = q − ( wt x, wt x )+ ( wt x,ρ ) x for homogeneous x . Then σ ( x ) = x if and only if σ ′ ( cor x ) = cor x . Weunderstand cor as a scalar correction.7.2. Quantum unipotent subgroups.
Take any w ∈ W . Choosea reduced expression −→ w = s i = s i · · · s i l of w . We have the naturalfunction i ( ) : [1 , l ] → [1 , r ] such that i ( j ) = i j . For any j < k ∈ [1 , l ],denote i [ j, k ] = i j i j +1 · · · j k −→ w [ j,k ] = s i [ j,k ] −→ w ≤ j = s i [1 ,j ] −→ w ≥ k = s i [ k,l ] Define the roots β k = −→ w ≤ k − α i k , k ∈ [1 , l ].Following the convention in [KO17, Definition 3.23](see T ′′ i, , T ′ i, − in[Lus93, Section 37.1.3][Kim12, Section 4.2.1][Sai94, Proposition 1.3.1]),for any i ∈ [1 , r ], we define the k -algebra automorphism T i on U q (braidgroup action) such that T i ( K h ) = K s i ( h ) T i ( E j ) = ( − F i K i i = j P r + s = − C ij ( − r q − ri E ( s ) i E j E ( r ) i i = jT i ( F j ) = ( − K − i E i i = j P r + s = − C ij ( − r q ri F ( r ) i F j F ( s ) i i = j . Its inverse map is given by T − i ( K h ) = K s i ( h ) T − i ( E j ) = ( − K − i F i i = j P r + s = − C ij ( − r q − ri E ( r ) i E j E ( s ) i i = jT − i ( F j ) = ( − E i K i i = j P r + s = − C ij ( − r q ri F ( s ) i F j F ( r ) i i = j . Notice that T i (( U q ) γ ) = ( U q ) s i ( γ ) for γ ∈ Q . Define T −→ w = T i T i · · · T i l .Then it is known that T −→ w only depends on w ∈ W , which we denoteby T w . Proposition 7.2.1 ([Lus93, Section 37.2.4]) . We have ∗ ◦ T i ◦ ∗ = T − i . Choose any sign e ∈ { +1 , − } . We might omit e when e = +1.For any m ∈ N , let us define F e ( mβ k ) = T ei · · · T ei k − ( F ( m ) i k ) . Notice that wt ( F e ( mβ k )) = − mβ k .For any multiplicity vector c = ( c , . . . , c l ) ∈ N [1 ,l ] , define the corre-sponding ordered product ([Kim12, Section 4.3]): F e ( c, −→ w ) = ( F e ( c β ) · · · F e ( c l β l ) e = 1 F e ( c l β l ) · · · F e ( c β ) e = − ∗ F e ( c, −→ w ) = F − e ( c, −→ w ). Proposition 7.2.2 ([Kim12, Theorem 4.9] [Lus92, Proposition 40.2.1,41.1.3]) . { F e ( c, −→ w ) | c ∈ N [1 ,l ] } forms a basis of a k -subspace of U q − which does not depend on the choice of −→ w . Denote the subspace by U q − ( w, e ) . Definition 7.2.3. { F e ( c, −→ w ) | c ∈ N [1 ,l ] } is called the PBW basis (Poincar´e-Birkhoff-Witt basis) of U q − ( w, e ) . When the context is clear, we could simply write F e ( c ) instead of F e ( c, −→ w ).For any j ≤ k such that i j = i k , set β [ j,k ] = X s ∈ [ j,k ] ,i s = i j β s ∈ Q. It is straightforward to check that β [ j,k ] = ( −→ w For any j < k , we have F +1 ( β k ) F +1 ( β j ) − q − ( β j ,β k ) F +1 ( β j ) F +1 ( β k ) = X c ∈ N [ j +1 ,k − f c F +1 ( c, −→ w ) F − ( β j ) F − ( β k ) − q − ( β j ,β k ) F − ( β k ) F − ( β j ) = X c ∈ N [ j +1 ,k − f c F − ( c, −→ w ) such that f c ∈ k , and the equations are Q − -grading homogeneous. It follows that U q − ( w, e ) is the k -subalgebra of U q − generated by { F e ( β j ) | j ∈ [1 , l ] } . Proposition 7.2.5 ([Kim12, Proposition 4.22][Lus93, 38.2.3]) . Forany c, c ′ ∈ N [1 ,l ] , we have ( F e ( c, −→ w ) , F e ( c ′ , −→ w )) L = l Y k =1 δ c k ,c ′ k c k Y s =1 − q si k . Definition 7.2.6 ([KO17, Definition 3.26]) . We denote U q + ( w ) = ( U q − ( w )) ∨ A q [ N − ( w ))] = ∗ ( U q − ( w )) . The algebra U q − ( w ) := U q − ( w, +1) is called the quantum nilpotentsubalgebra associated to w ∈ W . The algebra A q [ N − ( w )] is called thequantum unipotent subgroup associated to w . Notice that A q [ N − ( w )] = U q − ( w, − A q [ N − ( w )] has a Q − -grading induced from that of U q − . It isa quantum analogue of the coordinate ring of the unipotent subgroup N − ( w ), see [KO17] for more details. Example 7.2.7 ( sl type) . Consider the Cartan matrix C = (cid:18) − − (cid:19) and the reduced word −→ w = s s s . We get β = α , β = s α = α + α , β = α . Under our convention, we compute that F ( β ) = F , F ( β ) = T F = F F − qF F , F ( β ) = T T F = F . The LS-lawnow reads as F ( β ) F ( β ) − qF ( β ) F ( β ) = F ( β ) . Notice that F ( β ) F ( β ) = F ((1 , , , −→ w ) = F [1 , . By applying the ∗ anti-involution, we get F − ( β ) = F F − qF F = T − ( F ) , F − [1 , 3] = F − ( β ) F − ( β ) = F F , and the LS-law reads as F − ( β ) F − ( β ) − qF − ( β ) F − ( β ) = F − ( β ) . Dual canonical bases. Take a sign e ∈ { +1 , − } . We define thedual PBW basis of U q − ( w, e ) to be F up e := { F up e ( c, −→ w ) := F e ( c, −→ w )( F e ( c, −→ w ) , F e ( c, −→ w )) L | c ∈ N I } . The corresponding dual LS-law takes the following form. Proposition 7.3.1 ([Kim12, Proposition 4.28]) . We have q ( β j ,β k ) F up − ( β j ) F up − ( β k ) − F up − ( β k ) F up − ( β j ) = X c ∈ N [ j +1 ,k − f ∗ c F up − ( c, −→ w ) such that f ∗ c ∈ Q [ q ± ] , and the equation is Q − -grading homogeneous. As a consequence, we get σF up e ( c, −→ w ) − F up e ( c, −→ w ) ∈ X c ′ Continue Example 7.2.7. Recall that σ ( F i ) = − q F i for i = 1 , , σ ( F F ) = q − σ ( F ) σ ( F ) = q F F . In addition, wecan check ( F ( β i ) , F ( β i )) L = − q . The dual PBW basis elements readas F up ( β ) = (1 − q ) F , F up ( β ) = (1 − q ) F , F up ( β ) = (1 − q )( F F − qF F ) . It is straightforward to check that σ ( F up ( β i )) = F up ( β i ) , ∀ i ∈ [1 , . Notice that F up [1 , 3] = (1 − q ) F [1 , 3] = (1 − q ) F F , σF up [1 , 3] = (1 − q − ) q F F . The LS-law now reads as σ ( F up [1 , − F up [1 , 3] = ( q − − q ) F up ( β ) . We can calculate the corresponding dual canonical basis element B up [1 , 3] = F up [1 , − qF up ( β ) . We then obtain the following exchange relation in U q − ( w ) : F up ( β ) F up ( β ) = B up [1 , 3] + qF up ( β ) . Apply the anti-involution ∗ , we obtain the following exchange relationin A q [ N − ( w )] : F up − ( β ) F up − ( β ) = B up − [1 , 3] + qF up − ( β ) . Apply the dual bar involution σ , we get q − F up − ( β ) F up − ( β ) = B up − [1 , 3] + q − F up − ( β ) . Notice that all factors appearing are dual canonical basis elements ofthe form B up − ( c, −→ w ) for some c . Explicitly, we have F up − ( β ) = (1 − q ) F F up − ( β ) = (1 − q ) F F up − ( β ) = (1 − q )( F F − qF F ) B up − [1 , 3] = (1 − q )( F F − qF F ) . Subalgebras and unipotent quantum minors. For any j ≤ k ∈ [1 , l ], we define the subalgebras U q − ( w, e ) [ j,k ] = k [ F e [ β i ]] i ∈ [ j,k ] of U q − ( w, e ), e = ± 1, respectively. Notice that they depend on the re-duced expression −→ w of w . By the LS-law, U q − ( w, e ) [ j,k ] has the k -basis F up e : = { F up e ( c, −→ w ) | c ∈ N [ j,k ] } , called the dual PBW basis.As before, denote U q − ( w ) [ j,k ] = U q − ( w, +1) [ j,k ] and A q [ N − ( w )] [ j,k ] = U q − ( w, − [ j,k ] . Then we have we have ∗ U q − ( w ) [ j,k ] = A q [ N − ( w )] [ j,k ] .The braid group action T ≤ j − = T −→ w [1 ,j − gives a k -algebra isomor-phism T ≤ j − : U q − ( −→ w [ j,k ] ) ≃ U q − ( w ) [ j,k ] , because T ≤ j − sends the quantum root vector T i j · · · T i s − F i s in U q − ( −→ w [ j,k ] )to ( T i · · · T i j − ) T i j · · · T i s − F i s = F ( β s ) ∈ U q − ( w ) [ j,k ] , ∀ s ∈ [ j, k ]. It fol-lows that we have a k -algebra isomorphism ∗ T ≤ j − ∗ : A q [ N − ( −→ w [ j,k ] )] ≃ A q [ N − ( w )] [ j,k ] which identifies the dual PBW bases.If we have i j = i k = a ∈ [1 , r ], then the dual canonical basis element B up − [ j, k ] = B up − ( c [ j,k ] , −→ w ) is called a unipotent quantum minor, whichwe denote by D [ j, k ]. See D [ j, k ] = D −→ w ≤ k ̟ a , −→ w Let us recall certain localization of quantum alge-bras following [KO17, Section 4].We refer the reader to [Kim12, 5.1.3][KO17, Definition 3.37] for thedefinition of the quotient algebra A q [ N − ∩ X w ] of U q − , where X w denotethe Schubert variety and ˚ X w the Schubert cell. By [Kim12, Theorem5.13], the natural composition ι w : A q [ N − ( w )] ֒ → U q − ։ A q [ N − ∩ X w ]is an embedding. For each basis element b ∈ B up − , denote the image ι w ( b ) = [ b ]. For any λ = P a ∈ supp w λ a ̟ a ∈ ⊕ a N ̟ a ≃ N supp w , denote c λ = P λ a c [ min a, max a ] . Following [Kim12, Corollary 6.4][KO17, Proposition3.47], denote D wλ,λ = B up − ( c λ , −→ w ) and D w = { q s D wλ,λ | s ∈ Z , ∀ λ } ,[ D w ] = ι w D w . Following [KO17, Section 2.6, Section 4], define thefollowing localization A q [ N − ( w ) ∩ wG min0 ] := A q [ N − ( w )][ D − w ] A q [ N w − ] := A q [ N − ∩ ˚ X w ] := A q [ N − ∩ X w ][[ D w ] − ] . They have natural Q -grading. A q [ N w − ] is called the quantum unipotentcell associated to w .By [KO17, Theorem 4.13], ι w induces a Q -graded algebra isomor-phism ι w : A q [ N − ( w ) ∩ wG min0 ] ≃ A q [ N w − ] called the De Concini-Procesiisomorphism. By definition, A q [ N − ( w ) ∩ wG min0 ] has the following basis˚ B up − ( w ) = { q ( λ, wt S + λ − wλ ) D − wλ,λ S | S = B up − ( c, −→ w ) , ∀ c ∈ N l , ∀ λ ∈ ⊕ a N ̟ a } . It follows from the isomorphism ι w that A q [ N w − ] has the following basis˚ B up ,w − = { q ( λ, wt S + λ − wλ ) [ D wλ,λ ] − [ S ] | [ S ] = [ B up − ( c, −→ w )] , ∀ c ∈ N l , ∀ λ ∈ ⊕ a N ̟ a } . We call the bases ˚ B up − ( w ) and ˚ B up ,w − the (localized) dual canonical bases.The twisted dual bar involution σ ′ naturally extends to a Q -anti-involution on the localization A q [ N − ( w ) ∩ wG min0 ] ≃ A q [ N w − ]. As be-fore, define the Q ( q )-module endomorphism c tw such that c tw ( x ) = q ( wt x, wt x ) − ( wt x,ρ ) x for homogeneous x . Define the dual bar involution σ = c tw σ ′ on the localization. Then we still have σ ( xy ) = q ( wt x, wt y ) σ ( y ) σ ( x )for homogeneous x, y . Moreover, the dual canonical bases ˚ B up − ( w ) and˚ B up ,w − are σ -invariant, see [KO17, Proposition 4.9].As before, define the Q ( q )-module endomorphism cor on A q [ N − ( w ) ∩ wG min0 ] Q ( q ) ≃ A q [ N w − ] Q ( q ) such that cor ( x ) = q − ( wt x, wt x )+ ( wt x,ρ ) x forhomogeneous x . Then x is σ -invariant if and only if cor x is σ ′ -invariant.8. Cluster structures on quantum unipotent subgroup For our purpose, we need to know that a quantum unipotent sub-group of symmetrizable Kac-Moody type is a quantum cluster algebra,such that the unipotent quantum minors are quantum cluster variables.In this section, we explain how [GY16b] implies the cluster structureon quantum unipotent subgroups in general. An explicit and detailedtreatment could be found in the recent work [GY20] with a differentconvention.8.1. CGL extension. Definition 8.1.1 (Twist derivation) . Let there be given a k -algebra S and σ ∈ End S . A k -module endomorphism δ is called a σ -derivationof S if it satisfies δ ( xy ) = δ ( x ) y + σ ( x ) δ ( y )Let there be given a k -algebra S and its σ -derivation δ . We definethe Ore extension S [ x ; σ, δ ] = ⊕ d ∈ N x d S as a free S -module, where x isan indeterminate. We endow S [ x ; σ, δ ] with the algebra structure byrequiring xs = σ ( s ) x + δ ( s ).Take the base field k = Q ( q ) as before. We can identify Q with thecharacter lattice of the algebraic split torus H = Hom Z ( Q, Z ) ⊗ Z k ∗ ≃ ( k ∗ ) r such that γ = P γ i α i ∈ Q is identified with χ γ , whose action on h = ( h , . . . , h r ) ∈ H is given by χ γ ( h ) = Q h γ i i . Let H act on ( U q ) γ such that h · u = χ γ ( h ) u .Recall that, by the dual LS-law (Proposition 7.3.1), we have q ( β j ,β k ) F up − ( β j ) F up − ( β k ) − F up − ( β k ) F up − ( β j ) ∈ X c ∈ N [ j +1 ,k − Q [ q ± ] F up − ( c, −→ w )for j < k . Let us describe it as a twist derivation. Lemma 8.1.2. Let there be given k ∈ [1 , r ] . We can always choosesome h ( k ) , h ( k ) ∗ ∈ H such that χ β j ( h ( k ) ) = q ( β j ,β k ) , χ β i ( h ( k ) ∗ ) = q − ( β i ,β k ) ,for any j ≤ k , i ≥ k .Proof. (1) We show the existence of h ( k ) . Denote β j = P ri =1 ( β j ) i α i for j ≤ k .Assume h ( k ) = ( h , . . . , h r ) = q a = ( q a , . . . , q a r ), a = P ri =1 a i α ∨ i .Then χ β j ( h ( k ) ) = q P ( β j ) i a i . We need to find a such that P i ∈ [1 ,r ] ( β j ) i a i =( β k , β j ), ∀ j ≤ k .Denote β k = P ri =1 ( β k ) i α i . Since h α ∨ i , β j i = ( α i ,β j ) d i , we have( β k , β j ) = X s ∈ [1 ,r ] (( β k ) s α s , β j )= X s h ( β k ) s d s α ∨ s , X i ( β j ) i α i i = X s X i ( β k ) s d s C si ( β j ) i = X s X i ( β k ) s d i C is ( β j ) i = X i a i ( β j ) i where a i = P s ∈ [1 ,r ] C is ( β k ) s d i . We can set a = P a i α ∨ i .(2) Similar to (1), we can show that ( h ( k ) ) − = q − a is a solution for h ( k ) ∗ . (cid:3) From now on, choose h ( k ) , h ( k ) ∗ ∈ H as in the proof of Lemma 8.1.2.Denote the action of h ( k ) on A q [ N − ( w )] ≤ k by σ k and the action of h ( k ) ∗ on A q [ N − ( w )] ≥ k by σ ∗ k . Define the following k -module endomorphisms: δ k : A q [ N − ( w )] [1 ,k − → A q [ N − ( w )] [1 ,k − u F up − ( β k ) u − σ k ( u ) F up − ( β k ) δ ∗ k : A q [ N − ( w )] [ k +1 ,l ] → A q [ N − ( w )] [ k +1 ,l ] u F up − ( β k ) u − σ ∗ k ( u ) F up − ( β k )One can check that δ k is a σ k -derivation and δ ∗ k a σ ∗ k -derivation in thesense of Definition 8.1.1.By the above discussion, we can view A q [ N − ( w )] as an iterated Oreextension: A q [ N − ( w )] = k [ F up − ( β )][ F up − ( β ); σ , δ ] · · · [ F up − ( β l ); σ l , δ l ] . It is further equipped with the action by the torus H , such that(1) F up − ( β i ) are H -eigenvectors. In fact, we have hF up − ( β i ) = χ β i ( h ) F up − ( β i ) ∀ h ∈ H .(2) δ k are locally nilpotent σ k -derivation (by the LS-law).(3) For any k ∈ [1 , l ], the endomorphism σ k on U q − ( w ) ≤ k − is therestriction of the action by an element h ( k ) ∈ H . Moreover, h ( k ) F up − ( β k ) = λ k F up − ( β k ) for some λ k non-root of unit. (In fact,we have λ k = χ β k ( h ( k ) ) = q ( β k ,β k ) = q d k .)Such an iterated Ore extension with torus action is called a Cauchon-Goodearl-Letzter (CGL) extension, see [GY16a, Defintion 2.2].For j < k , denote h ( k ) F up − ( β j ) = q Λ kj F up − ( β j ). It follows that q Λ kj = χ β j ( h ( k ) ) = q ( β k ,β j ) . Define Λ jk = − Λ kj , Λ kk = 0.We observe that A q [ N − ( w )] also satisfies the following property(1) ∀ j < k , δ k ( x j ) ∈ A q [ N − ( w )] [ j +1 ,k − (2) ∀ j < k , h ( j ) ∗ ( F up − ( β k )) = q − Λ kj F up − ( β k ), and h ( k ) ∗ ( F up − ( β k )) = λ ∗ k for some λ ∗ k non-root of unit. (In fact, we have λ ∗ k = χ β k ( h ( k ) ∗ ) = q − ( β k ,β k ) = q − d k .)We further observe that A q [ N − ( w )] can also be written as the fol-lowing iterated Ore extension A q [ N − ( w )] = k [ F up − ( β l )][ F up − ( β l − ); σ ∗ l − , δ ∗ l − ] · · · [ F up − ( β ); σ ∗ , δ ∗ ] . Correspondingly, A q [ N − ( w )] is called a symmetric CGL extension, see[GY16a, Defintion 2.6]. Quantum cluster structure. Define the set of vertices I = [1 , l ], I uf = { k ∈ I | s ( k ) = + ∞} , I f = I \ I uf . By a general result of Goodearland Yakimov [GY16b], the symmetric CGL extension A q [ N − ( w )] is aquantum cluster algebra. Let us explain its cluster structures in moredetails. Definition 8.2.1. Let there be given j ∈ [1 , l ] . An element Z inthe quantum unipotent subgroup R = A q [ N − ( −→ w ≤ j )] is said to be H -homogeneous prime if it has the following properties:(1) It is a homogeneous element under the H -action.(2) It is normal: ZR = RZ .(3) It is prime: R/ ( ZR ) is a domain. Recall that, for any j ≤ k ∈ [1 , l ] such that i j = i k = a , we have theunipotent quantum minor D [ j, k ] = B up − ( c [ j,k ] , −→ w ). Proposition 8.2.2 ([GY16b]) . Let there be given any j ≤ k ∈ [1 , l ] .(1) The unipotent quantum minors D [ min , ≥ j a, max , ≤ k a ] where a ∈ supp −→ w [ j,k ] , are H -homogeneous prime elements in A q ( N − ( w )) [ j,k ] .(2) All H -homogeneous prime elements in A q ( N − ( w )) [ j,k ] take theform ξD [ min , ≥ j a, max , ≤ k a ] for some ξ ∈ k .Proof. (1) It follows from Theorem 7.4.1 that the unipotent quan-tum minors D [ min , ≥ j a, max , ≤ k a ] are homogeneous normal elements in A q ( N − ( w )) [ j,k ] ≃ A q ( N − ( −→ w [ j,k ] )). Notice that their decomposition intothe dual PBW basis takes the form (7.1). We further deduce that theyare H -homogeneous prime elements by [GY16b, Proposition 3.10].(2) The claim follows from (1) and [GY16b, Theorem 5.1]. (cid:3) Following [Kim12, Section 4.8], define bilinear forms c −→ w , N −→ w on N [1 ,l ] such that c −→ w ( c, c ′ ) = X j Continue Example 7.2.7 7.3.2. We can compute that ̟ i + −→ w ≤ ̟ i = ̟ + ̟ − β [1 , = 2 ̟ − α − α .Then Proposition 8.2.4 implies that D [1 , D [1 , 1] = q (2 ̟ − α − α , − α ) D [1 , D [1 , q − D [1 , D [1 , .D [1 , D [2 , 2] = q (2 ̟ − α − α , − α − α ) D [2 , D [1 , D [2 , D [1 , . As an example for (8.1) , we also compute D [1 , D [3 , 3] = q ((1+ s s s ) ̟ , − β ) D [3 , D [1 , qD [3 , D [1 , . We can also calculate the q -power using the bilinear form N −→ w . Given any j ∈ [1 , l ], recall that we have j = j min [ m − j ]. Applying therescaling map cor to (7.1), we get cor D [ j min , j ] = ξ j cor F up − ( β j ) · · · cor F up − ( β j min [1] ) cor F up − ( β j min ) + Z for some Z ∈ A q [ N − ( w )] [ j min +1 ,j − , where the rescaling constant is givenby ξ j = q − P m − j ≥ a>b ≥ ( β j min[ a ] ,β j min[ b ] ) = q − P m − j ≥ a>b ≥ N −→ w ( c [ j min[ a ] ,j min[ a ]] ,c [ j min[ b ] ,j min[ b ]] ) . Notice that ξ j is the same as the rescaling constant in [GY16b, (4.18)].We have the following results. Theorem 8.2.6 ([GY16b, Theorem 8.2] [GY20, Theorem B]) . (1)There exists a unique matrix e B ( −→ w ) = ( b ik ) i ∈ I,k ∈ I uf such that e B ( −→ w ) is compatible with L and P i ∈ I b ik wt X i = 0 for k ∈ I uf .(2) Let t = t ( −→ w ) denote an initial seed associated to −→ w , suchthat e B ( t ) = e B ( −→ w ) , Λ( t ) = L , for j ∈ [1 , l ] . Then we have an al-gebra isomorphism κ from the partially compactified quantum clusteralgebra A ( t ) Q ( q ) := A ( t ) ⊗ Q ( q ) to the quantum unipotent sub-group A q [ N − ( w )] Q ( q ) , such that the initial quantum cluster variables X j ( t ) , j ∈ I , are identified with the rescaled unipotent quantum minors X j = cor D [ j min , j ] . Notice that cor D [ j min , j ] is σ ′ -invariant and X j ( t ) = κ cor D [ j min , j ]is invariant under the bar involution ( ) on A q ( t ). We obtain thefollowing result. Lemma 8.2.7. The isomorphism κ identifies the twisted dual bar in-volution σ ′ on A q [ N − ( w )] Q ( q ) and ( ) on A q ( t ) Q ( q ) , i.e. for any Z ∈ A ( t ) Q ( q ) , we have σ ′ ( κZ ) = κZ . Remark 8.2.8. By [BFZ05] , the I × I uf -matrix e B ( −→ w ) = ( b ik ) i ∈ I,k ∈ I uf isgiven by the following: b jk = j = p ( k ) − j = s ( k ) C i j i k j < k < s ( j ) < s ( k ) − C i j i k k < j < s ( k ) < s ( j )0 else . For verifying that such a matrix satisfies the condition in Theorem8.2.6, see [BZ05, Theorem 8.3] (or arguments in [GY16b, Section 10.1] ). Example 8.2.9. Continue Example 7.2.7 7.3.2. Notice that we have cor F i = q − F i and cor F F = q − F F , cor F F = q − F F . We get cor ( F up − ( β )) cor ( F up − ( β )) = q cor ( B up − [1 , q − cor ( F up − ( β )) . Denote X = cor F up − ( β ) , X ′ = cor F up − ( β ) , X = cor F up − ( β ) , X = cor B up − [1 , . By using quantum Serre relation, it is straightforwardto check that (knowing that F F F = F F F ) we have X X = q − X X , X X = qX X , X X = X X , see also Example 8.2.5.Denote I = [1 , and I uf = { } . We compute explicitly the matrix L = − − . Take ( b ij ) = − − − , then its I × I uf -submatrix e B is the same as in Remark 8.2.8. We have e B T L = (cid:0) (cid:1) .By the above computation, A q [ N − ( w )] Q ( q ) is a partially compactifiedquantum cluster algebra with initial seed t such that e B ( t ) = e B ( −→ w ) and Λ( t ) = L . Its mutation rule reads as X ∗ X ′ = q Λ( f , − f + f ) X + q Λ( f , − f + f ) X = q X + q − X . Quantum cluster variables. As before, denote A q [ N − ( w )] [ j,k ] = k [ cor F up − ( β s )] s ∈ [ j,k ] for j < k ∈ [1 , l ]. Lemma 8.3.1. Let there be given any j ≤ k ∈ [1 , l ] such that i j = i k .The rescaled unipotent quantum minor cor D [ j, k ] is a quantum clustervariable in A ( t ) .Proof. By Theorem 7.4.1, D [ j, k ] is a homogeneous prime elements in A q [ N − ( w )] [ j,k ] that do not lie in A q [ N − ( w )] [ j +1 ,k ] or A q [ N − ( w )] [ j,k − . Inaddition, its decomposition (7.1) by the dual PBW basis has the leadingterm F up − ( c [ j,k ] , −→ w ). Such a homogeneous prime element is denoted by y [ j,k ] in [GY16b, Theorem 5.1]. Moreover, by [GY16b, Theorem 5.3Theorem 8.2], there exists a scalar ξ ∈ q Z such that κξD [ j, k ] is aquantum cluster variable in some seed t τ ∈ ∆ + , where t τ is determinedby a permutation τ of [1 , l ]. Finally, since a quantum cluster variablemust be bar-invariant, ξD [ j, k ] must agree with the σ ′ -invariant element cor D [ j, k ], where we identify the bar-involution and σ ′ by Lemma 8.2.7. (cid:3) Theorem 8.3.2 ([GY16b, Theorem 5.3, Theorem 8.2(c)]) . There isa seed t [1] ∈ ∆ + whose quantum cluster variables are cor D [ j, j max ] , j ∈ [1 , l ] . Moreover, there exists a mutation sequence ←− µ from t to t [1] such that the quantum cluster variables obtained along the mutationsequence take the form cor D [ j, k ] for j ≤ k ∈ [1 , l ] , i j = i k . By Lemma 9.1.3, the seed t [1] in Theorem 8.3.2 is shifted from t in the sense of Definition 2.5.1. Remark 8.3.3. The e B -matrix e B ( t [1]) is given by [GY16b, Section10.1] as the following ( [GY16b] treated finite dimensional Lie algebras,but their statements concerning combinatorics such [GY16b, Proposi-tion 10.4] are valid for a general root datum).Define the permutation w on [1 , l ] such that w ( j min [ d ]) = j max [ − d ] , j ∈ [1 , l ] , ≤ d ≤ m ( i j ) . Denote j ′ := w ( j ) . Then I f = { j | p ( j ′ ) = −∞} . The ( j, k ) -entry of e B ( t [1]) is given by b jk = j ′ = p ( k ′ ) − j ′ = s ( k ′ ) C i j i k p ( j ′ ) < p ( k ′ ) < j ′ < k ′ − C i j i k p ( k ′ ) < p ( j ′ ) < k ′ < j ′ . The Λ -matrix of t [1] is determined by the q -commutativity relationamong the quantum cluster variables cor D [ j, j max ] , j ∈ I . By [GY16b,Theorem 9.5] , for any j < k , we have N −→ w ( c [ j,j max ] , c [ k,k max ] ) = (( −→ w By the combinatorial algorithm in [GLS11, Section13.1] , we can construct a mutation sequence for Theorem 8.3.2 as fol-lows. For k ∈ [1 , l ] , denote the following mutation sequence (read fromright to left) ←− µ k : = µ k min [ m ( i k , [ k,l ]) − · · · µ k min [1] µ k min . Notice that the sequence is trivial when m ( i k [ k, l ]) − < . Define themutation sequence ←− µ −→ w := ←− µ l · · · ←− µ ←− µ . Then t [1] = ←− µ −→ w t .Moreover, t and t [1] are similar up to the permutation σ on on [1 , l ] such that σ ( k ) = k, k = k max σ ( k min [ d ]) = k max [ − d − , < d < m ( i k ) . See Remarks 8.2.8, 8.3.3. Example 8.3.5 ([GLS11, Example 13.2]) . Let us consider a more com-plicated example. Take the Cartan matrix C = − − − − − − , thereduced word i = (1 , , , , , , , , , . Correspondingly, I = [1 , , I f = { , , , } . We choose an initial seed t = t ( −→ w ) such that its e B -matrix is given by Remark 8.2.8. We associate to its (skew-symmetric)matrix ( b ij ) i,j ∈ I a quiver e Q ( t ) , which is a directed graph with the set ofvertices I and [ b ij ] + -many arrows from i to j for all i, j ∈ I . We canchoose t such that the e Q ( t ) is drawn as in Figure 8.1 (frozen verticesare diamond nodes and unfrozen ones are circle nodes; the number onthe arrow means the multiplicity).The mutation sequence ←− µ −→ w applies successively on the sequence ofvertices (1 , , , , , , , , , , , , (read from left to right). Wehave the seed t [1] = ←− µ −→ w t . Its quiver e Q ( t [1]) is drawn as Figure8.2. Relabeling the vertices by the permutation w of I given in Re-mark 8.3.3, we obtain Figure 8.3. Figure 8.1. The quiver e Q ( t ) associated to the reducedword i .9. Dual canonical bases are common triangular bases In this section, we apply previous discussion of triangular basesand cluster twist automorphisms to quantum unipotent subgroups andquantum unipotent cells.Let there be given any partially compactified quantum cluster al-gebra A q ( t ) Q ( q ) ≃ A q [ N − ( w )] Q ( q ) as in Theorem 8.2.6. Recall that I = [1 , l ], I f = { j ∈ I | j = j max } .9.1. Parametrization. Recall that the initial quantum cluster vari-ables X j ( t ) are identified with the rescaled unipotent quantum mi-nors cor D [ j min , j ], j ∈ I . Moreover, the dual canonical basis elements D [ j min , j ] = B up − ( c [ j min ,j ] , −→ w ) are parametrized by c [ j min ,j ] ∈ N I and Figure 8.2. The quiver e Q ( t [1]) associated with the re-duced word i Figure 8.3. The quiver e Q ( t [1]) associated to the re-duced word i with vertices relabeled. X j ( t ) by their leading degrees f j ∈ M ◦ ( t ) ( j -th unit vector). Follow-ing [Qin17], we translate the multiplicity vector c ∈ N [1 ,l ] to a vectorin M ◦ ( t ) by defining a linear map θ − from N [1 ,l ] to M ◦ ( t ) = Z I suchthat θ − ( c [ j,j ] ) = ( f j − f p ( j ) p ( j ) = −∞ f j p ( j ) = −∞ It follows that θ − is injective. We further define the bijective linearmap θ : M ◦ ( t ) ≃ Z [1 ,l ] , called the parametrization map, such that θ ( f j ) = c [ j min ,j ] . Let us make the identification A q ( t ) ≃ A q [ N − ( w )]. Recall that,by Theorem 8.2.6, the rescaled unipotent quantum minors cor D [ j, k ], i j = i k , are quantum cluster monomials. In particular, the rescaled dualPBW generators cor F up − ( β s ), s ∈ [1 , l ], are quantum cluster variables. Proposition 9.1.1. The following statements are true.(1) The rescaled dual PBW basis cor F up − is a θ − ( N [1 ,l ] ) -pointed basisof A q ( t ) such that cor F up − ( c ) is θ − ( c ) -pointed.(2) The rescaled dual canonical basis cor B up − is a θ − ( N [1 ,l ] ) -pointedbasis of A q ( t ) such that cor B up − ( c ) is θ − ( c ) -pointed.(3) The rescaled dual canonical basis cor B up − is ( ≺ t , q Z [ q ]) -unitriangularto the rescaled dual PBW basis cor F up − .Proof. It is possible to prove the statements based on computation ofthe bilinear N −→ w . Let us give a more abstract proof based on Lemma8.2.7, which identifies the twisted dual bar involution σ ′ and the barinvolution ( ).Recall that the rescaled dual PBW generators cor F up − ( β s ), s ∈ [1 , l ],are quantum cluster variables and, in particular, pointed. It followsthat, for any cor F up − ( c ), there exists a scalar ξ c ∈ q Z such that e F up − ( c ) := ξ c cor F up − ( c ) is pointed. Denote e F up − = { e F up − ( c ) | c } , then it is a pointedset. We have a ( <, q Z [ q ])-decomposition cor B up − ( c ) = cor F up − ( c ) + X c ′ For any k ∈ I uf , we have I k ( t ) = cor D [ k [1] , k max ] and deg t I k ( t ) = − f k + f k max . (9.1) Proof. Notice that cor D [ k [1] , k max ] is a quantum cluster variable byTheorem 8.3.2. In addition, we have deg t cor D [ k [1] , k max ] = θ − ( c [ k [1] ,k max ] ) = − f k + f k max . It follows that cor D [ k [1] , k max ] = ←− µ ∗ t [1] ,t X σk ( t [1]) = I k ( t ) by definition. (cid:3) For completeness, we compare the partial orders on both sides of themap θ , though it will not be used in this paper. Define the reverselexicographical order < ′ on Z [1 ,l ] such that c ′ < ′ c if there exists some s ∈ [1 , l ] such that c ′ s < c s and c ′ j = c j for all j > s . Lemma 9.1.4. For any g ′ ≺ t g in M ◦ ( t ) , we have θ ( g ′ ) < ′ θ ( g ) in Z [1 ,l ] .Proof. It suffices to check that θ ( deg Y k ) < ′ k ∈ I uf . Wehave deg Y k = − f s ( k ) + P j 0. Theclaim follows. (cid:3) Integral form. Denote A = Q [ q ± ] . Recall that A q [ N − ( w )] is the Q ( q )-algebra generated by the dual PBW generators F up − ( β i ), i ∈ I over Q ( q ), subject to the Q -grading homogeneous relations given bythe LS-law: q ( β j ,β k ) F up − ( β j ) F up − ( β k ) − F up − ( β k ) F up − ( β j ) = X c ∈ N [ j +1 ,k − b j,k ( c ) F up − ( c, −→ w )for j ≤ k , where the coefficients b j,k ( c ) ∈ Q [ q ± ].The A -algebra A q [ N − ( w )] A generated by the dual PBW generators F up − ( β i ), i ∈ I , is called the integral form of A q [ N − ( w )]. See [Kim12]for more details.Work with the (partially compactified) quantum cluster algebra A q ( t ) Q ( q ) ≃ A q [ N − ( w )] Q ( q ) . By applying the linear map cor to each PBW genera-tor, we get the following rescaled LS-law: q ( β j ,β k ) cor F up − ( β j ) cor F up − ( β k ) − cor F up − ( β k ) cor F up − ( β j )(9.2) = X c ∈ N [ j +1 ,k − b ′ j,k ( c ) cor F up − ( c, −→ w )where b ′ j,k ( c ) ∈ b j,k ( c ) · q Z .Notice that cor F up − ( β i ) are quantum cluster variables. So it is not sur-prising that the coefficients appearing in (9.2) should belong to Z [ q ± ],which we give a rigorous proof below. Lemma 9.2.1. We have b ′ j,k ( c ) ∈ Z [ q ± ] .Proof. Denote Z = q ( β j ,β k ) cor F up − ( β j ) cor F up − ( β k ) − cor F up − ( β k ) cor F up − ( β j ).Then Z ∈ LP ( t ). Notice that cor F up − ( c ) is a pointed set in LP ( t ) byProposition 9.1.1. Then the finite decomposition in (9.2) is the ≺ t -decomposition in LP ( t ) with coefficients in Z [ q ± ] by Lemma 3.3.3. (cid:3) Corollary 9.2.2. We have b j,k ( c ) ∈ Z [ q ± ] .Proof. We have b j,k ( c ) ∈ Z [ q ± ] ∩ Q [ q ± ]. The claim follows. (cid:3) Correspondingly, the dual PBW generators F up − ( β i ), i ∈ I , generatea Z [ q ± ]-algebra, which we denote by A q [ N − ( w )] Z [ q ± ] . Then the dualPBW basis { F up − ( c, −→ w ) | c ∈ N [1 ,l ] } is a Z [ q ± ]-basis of A q [ N − ( w )] Z [ q ± ] .Consider the extension A q [ N − ( w )] Z [ q ± ] = A q [ N − ( w )] Z [ q ± ] ⊗ Z [ q ± ] Z [ q ± ].Theorem 8.2.6 could be strengthened as the following, which was alsoproved in the recent work [GY20, Theorem B]. Theorem 9.2.3. Take the initial seed t = t ( −→ w ) as before. We havea Z [ q ± ] -algebra isomorphism κ : A q ( t ) ≃ A q [ N − ( w )] Z [ q ± ] such thatthe initial quantum cluster variables X j ( t ) , j ∈ I , are identified withthe rescaled unipotent quantum minors cor D [ j min , j ] .Proof. Make the identification A q ( t ) Q ( q ) ≃ A q [ N − ( w )] Q ( q ) by Theo-rem 8.2.6.Recall that the rescaled dual PBW basis cor F up − is a Z [ q ± ]-basis of A q [ N − ( w )] Z [ q ± ] . In addition, the rescaled dual PBW generators arequantum cluster variables. Therefore, A q [ N − ( w )] Z [ q ± ] is a subalgebraof the quantum cluster algebra A q ( t ).Moreover, the rescaled dual PBW basis cor F up − is a Q ( q )-basis of A q ( t ) Q ( q ) ≃ A q [ N − ( w )] Q ( q ) . It follows that any Z ∈ A q ( t ) ⊂ LP ( t )has a finite decomposition in terms of cor F up − . By Lemma 3.3.3, thisdecomposition is the ≺ t -decomposition in LP ( t ) with coefficients in Z [ q ± ]. Therefore, the rescaled dual PBW basis cor F up − is a Z [ q ± ]-basis of A q ( t ). (cid:3) Applying Theorem 9.2.3 to symmetric Kac-Moody cases, we get thefollowing result. Corollary 9.2.4 ([Qin17, Theorem 9.1.3]) . Conjecture [GLS13, Con-jecture 12.7] holds true. Since the dual PBW basis is a Z [ q ± ]-basis of the algebra A q [ N − ( w )] Z [ q ± ] ,so is the dual canonical basis B up − . In particular, the multiplicationstructure constants of B up − take value in Z [ q ± ]. Correspondingly, let A q [ N − ( w ) ∩ wG min0 ] Z [ q ± ] and A q [ N w − ] Z [ q ± ] denote the free Z [ q ± ]-modulesspanned by the localized dual canonical bases ˚ B up − ( w ) and ˚ B up ,w − re-spectively. Then they are isomorphic Z [ q ± ]-algebras: ι w : A q [ N − ( w ) ∩ wG min0 ] Z [ q ± ] ≃ A q [ N w − ] Z [ q ± ] . In addition, notice that A q [ N − ( w ) ∩ wG min0 ] Z [ q ± ] can also be viewed asthe localization of the A q [ N − ( w )] Z [ q ± ] with respect to D w . By Theorem9.2.3, it is isomorphic to the (localized) quantum cluster algebra A q ( t )after an extension to Z [ q ± ]. We obtain the following. Theorem 9.2.5. Take the initial seed t = t ( −→ w ) as before. We havea Z [ q ± ] -algebra isomorphism ι w κ : A q ( t ) ≃ A q [ N w − ] Z [ q ± ] such thatthe initial quantum cluster variables X j ( t ) , j ∈ I , are identified withthe rescaled unipotent quantum minors cor [ D [ j min , j ]] . Remark 9.2.6 (Rescaled quantum cluster algebra) . We remark thatthe extension to include q is a technical trick needed for changing fromthe dual bar involution σ to the twisted dual bar involution σ ′ , where σ ′ is identified with the usual bar involution ( ) for quantum clusteralgebras (Lemma 8.2.7). For readers do not want to introduce q , onecan instead rescale the quantum cluster algebras as below (see [GLS13,Section 10.4] ).Notice that every quantum cluster monomial x is a homogeneous el-ement. Correspondingly, we can define the rescaled quantum clustermonomials cor − x , which correspond to dual canonical basis elementsby Theorem 9.5.2. The rescaled partially compactified quantum clusteralgebra can be defined to be the Z [ q ± ] -algebra generated by the rescaledquantum cluster variables. Then it is isomorphic to A q [ N − ( w )] Z [ q ± ] .Correspondingly, the rescaled (localized) quantum cluster algebra de-fined as a localization is isomorphic to A q [ N w − ] Z [ q ± ] . Localization and the initial triangular basis. Taking the lo-calization at frozen variables X j = κ − D [ j min , j ], j ∈ I f , we obtainthe (localized) quantum cluster algebra A q ( t ) = A q ( t )[ X − j ] j ∈ I f . Us-ing Theorem 7.4.2, we deduce that by taking the localization of thedual canonical basis at frozen variables, A q ( t ) has a M ◦ ( t )-pointed Z [ q ± ]-basis L := { [ X d ∗ S ] t | S ∈ κ − B up − , d ∈ Z I f } . (9.3) Proposition 9.3.1 ([Qin17, Corollary 9.1.9]) . The basis L is the tri-angular basis with respect to the initial seed t .Proof. We refer the reader to [Qin17, Section 9.1] for a detailed proof.The key step is [Qin17, Proposition 9.1.8] which verifies the ≺ t -unitriangularityproperty of the dual canonical basis by an induction on the length of w with the help of the dual PBW basis.. (cid:3) Theorem 9.3.2. The isomorphism ι w κ : A q ( t ) ≃ A q [ N w − ] Z [ q ± ] identi-fies the initial triangular basis L and the rescaled localized dual canon-ical basis cor ˚ B up ,w − .Proof. Recall that we denote ι w b = [ b ] for dual canonical basis elements b ∈ B up − .By construction, elements of L take the form b ( λ, S ) = [ X − λ ∗ S ] t for S ∈ κ − B up − , λ ∈ N I f . The correspondingly elements b ′ ( λ, S )in ˚ B up ,w − take the form q ( λ, wt S + λ − wλ ) [ D wλ,λ ] − [ S ], where we identify N I f ≃ ⊕ a ∈ supp w N ̟ a such that D wλ,λ = κX λ . Then we have κb ( λ, S ) = ξ cor b ′ ( λ, S ) for some ξ ∈ q Z . On the one hand, since b ( λ, S ) is invari-ant under the bar involution ( ), κb ( λ, S ) must be invariant under thetwisted dual bar involution σ ′ . On the other hand, since b ′ ( λ, S ) is σ -invariant, cor b ′ ( λ, S ) is σ ′ -invariant. Because σ ′ and the bar involution( ) are identified, we have ξ = 1. It follows that ι w κ L = cor ˚ B up ,w − . (cid:3) Twist automorphism. [KO17] introduced a quantum analogueof the twist automorphism for quantum unipotent cells A q [ N w − ], whichwas denoted by η w,q . Theorem 9.4.1 ([KO17, Theorem 6.1]) . There exists a Q ( q ) -algebraautomorphism η w,q on A q [ N w − ] such that, for k ∈ [1 , l ] , we have η w,q [ D [ k min , k ]] = q − ( ̟ ik , − β [ k min ,k ] ) ( D [ k min , k max ]) − D [ k [1] , k max ] , k ∈ I uf ,η w,q [ D [ k min , k max ] − ] = q − ( ̟ ik , − β [ k min ,k max] ) D [ k min , k max ] . In particular, wt η w,q x = − wt x for homogeneous x . Moreover, η w,q restricts to a permutation on the (localized) dual canonical basis ˚ B up ,w − and commutes with the dual bar involution σ . The rescaling map cor decomposes as cor = cor cor = cor cor forthe Z [ q ± ]-module endomorphisms cor , cor on A q [ N w − ] Z [ q ± ] such that,for any homogeneous x , we have cor ( x ) = c ( x ) x := q ( wt x,ρ ) x cor ( x ) = c ( x ) x := q − ( wt x, wt x ) x. Then cor is an algebra automorphism of the Q -graded algebra A q [ N w − ] Z [ q ± ] .Notice that η w,q is well-defined on A q [ N w − ] Z [ q ± ] . We rescale η w,q to thefollowing automorphism on A q [ N w − ] Z [ q ± ] : tw w := η w,q cor − . (9.4) Lemma 9.4.2. The rescaled twist automorphism tw w commutes with σ ′ .Proof. For any homogeneous x , we have ση w,q x = η w,q σx by Theo-rem 9.4.1. Since wt η w,q x = − wt x , we have c ( η w,q x ) = c ( x ) − and c ( η w,q x ) = c ( x ). Recall that σ = cor − σ ′ . We compute ση w,q x = cor − cor − σ ′ η w,q ( x )= c ( η w,q x ) − c ( η w,q x ) − σ ′ η w,q x = c ( x ) − c ( x ) σ ′ η w,q x = c ( x ) − σ ′ ( c ( x ) − η w,q x )= c ( x ) − σ ′ ( η w,q c ( x ) − x )= c ( x ) − σ ′ tw w x Similarly, we have η w,q σx = η w,q c ( x ) − c ( x ) − σ ′ x = c ( x ) − η w,q c ( σ ′ x ) − σ ′ x = c ( x ) − tw w ( σ ′ x ) . It follows that σ ′ tw w x = tw w σ ′ x . (cid:3) The automorphism tw w on A q [ N w − ] Z [ q ± ] induces an automorphism tw w on A q ( t ) via the identification A q ( t ) ≃ A q [ N w − ] Z [ q ± ] . We translateTheorem 9.4.1 for cluster algebras as the following. Theorem 9.4.3. Let there be given t = t ( −→ w ) as before. The algebraautomorphism tw w on A q ( t ) satisfies tw w ( X k ( t )) = [ X − k max ∗ I k ( t )] t , k ∈ I uf (9.5) tw w ( X j ( t )) = X j ( t ) − , j ∈ I f . (9.6) Moreover, tw w commutes with the bar involution ( ) on A q ( t ) , and itrestricts to a permutation on the initial triangular basis L .Proof. Since σ ′ is identified with ( ) by Lemma 8.2.7, Lemma 9.4.2implies that tw w commutes with ( ).Next, for any S ∈ ˚ B up ,w − , we have η w,q S ∈ ˚ B up ,w − by Theorem 9.4.1.We compute tw w cor S = η w,q cor − cor cor S = η w,q c ( S ) − c ( S ) S = c ( S ) − c ( S ) η w,q S = c ( η w,q S ) c ( η w,q S ) η w,q S = cor ( η w,q S )It follows that tw w preserves cor ˚ B up ,w − ≃ L .Finally, applying the rescaling cor to the unipotent quantum minorsappearing in Theorem 9.4.1 we get tw w ( X k ( t )) = ξ k X − k max ∗ I k ( t ) , k ∈ I uf , tw w ( X j ( t )) = ξ j X j ( t ) − , j ∈ I f , for some scalars ξ k , ξ j ∈ q Z . Since tw w commutes with the bar invo-lution ( ) and quantum cluster variables are bar-invariant, the scalarsare chosen such that the right hand side are bar-invariant. It followsthat the right hand side are normalization of the ordered product. (cid:3) Next, we want to show that the automorphism tw w is a DT-typecluster twist automorphism defined in Section 5.The automorphism tw w naturally extends to an automorphism on F ( t ). Recall that the mutation sequence induces a k -algebra isomor-phism ←− µ ∗ : F ( t [1]) ≃ F ( t ). Let us define var w : F ( t ) ≃ F ( t [1]) by var w = ( ←− µ ∗ ) − tw w . From Lemma 9.1.3, we deduce that var w ( X k ( t )) = X − k max · X σk ( t [1]) , k ∈ I uf , var w ( X j ) = X − j , j ∈ I f . In particular, we see that var w can be induced from the linear homo-morphism var w : M ◦ ( t ) ≃ M ◦ ( t [1]) such that var w ( f k ) = f ′ σ ( k ) − f ′ k max , , k ∈ I uf var w ( f j ) = − f ′ j , j ∈ I f , where f i , f ′ i denote the i -th unit vector in M ◦ ( t ) and M ◦ ( t [1]) respec-tively. Proposition 9.4.4. The map var w is a variation map.Proof. It remains to check that var w Y k ( t ) = Y σk ( t [1]) for any k ∈ I uf ,or, equivalently, var w deg t Y k ( t ) = deg t [1] Y σk ( t [1]), i.e., var w ( P i ∈ I b ik f i ) = P i ∈ I b ′ ik f ′ i , where b ik = b ik ( t ), b ′ ik = b ik ( t [1]), f i , f ′ i denote the i -thunit vectors in M ◦ ( t ) and M ◦ ( t [1]) respectively.It might be possible to verify the claim by working with the matricesdirectly (Remarks 8.2.8, 8.3.3). Let us give a more conceptual proof.On the one hand, we deduce from the definition of tw w and Lemma9.1.3 that deg t tw w X j ( t ) = − f j , j ∈ I . It follows that deg t tw w Y k ( t ) = − deg t Y k ( t ).On the other hand, since t and t [1] are similar up to σ , we have b ik = b ′ σi,σk for i, k ∈ I uf . It follows from the definition of var w that var w ( P b ik f i ) − P b ′ ik f ′ i = u k for some u k ∈ Z I f . Therefore, var w Y k ( t ) = Y σk ( t [1]) · p k where p k = X u k ∈ P . We deduce that tw w Y k ( t ) = ←− µ ∗ var w Y k ( t ) = ←− µ ∗ Y σk ( t [1]) · p k . In addition, we know that deg t ←− µ ∗ Y σk ( t [1]) = − deg t Y k ( t ) by properties of c -vectors, see [Qin19a, Proposition 3.3.8].Therefore, deg t tw w Y k ( t ) = − deg t Y k ( t ) + u k .Combining the above discussions, we obtain u k = 0. It follows that var w deg t Y k ( t ) = deg t [1] Y σk ( t [1]). (cid:3) Consequently, we obtain the following result. Theorem 9.4.5. The automorphism tw w decomposes as tw w = ←− µ ∗ var w such that var w : F ( t ) ≃ F ( t [1]) is a variation map. In particular, tw w is a twist automorphism of Donaldson-Thomas type in the sense of Sec-tion 5.2. Example 9.4.6. Continue Example 7.2.7 7.3.2. Notice that ←− µ ∗ ( X ) = I ( t ) = X ′ = X − f + f + X − f + f . The automorphism tw w on A q ( t ) takes the following form tw w ( X ) = [ X − ∗ I ( t )] t = [ X − ∗ X ′ ] t = X − f + X − f + f − f tw w ( X ) = X − tw w ( X ) = X − Define var w : F ( t ) ≃ F ( t [1]) such that var w ( X ) = q − Λ ′ X ′ ∗ X − = q Λ X ′ ∗ X − = q X ′ ∗ X − var w ( X ) = X − var w ( X ) = X − Then it is straightforward to check that var w is a k -algebra automor-phism, tw w = ←− µ ∗ var w and var w ( Y ) = X f − f = Y ′ . Consequently, tw w is an twist automorphism in our previous sense. Notice that we have tw w ( X ′ ) = tw w ( X − ∗ ( q X + q − X ))= ((1 + qY ) ∗ X − ) − ∗ q − ( qX − + X − )= X ∗ (1 + qY ) − ∗ q − X − ∗ ( qY + 1)= X f − f . Consequences. Quantization conjecture. Theorem 9.5.1. The rescaled localized dual canonical basis L in (9.3) is the common triangular basis of A q ≃ A q [ N w − ] Z [ q ± ] . In particular, itcontains all quantum cluster monomials.Proof. The rescaled localized dual canonical basis L is the initial trian-gular basis for t by Proposition 9.3.1. Thanks to its invariance underthe twist automorphism (Theorems 9.4.3, 9.4.5) and the admissibil-ity along the mutation sequence in Theorem 8.3.2, we can apply theexistence Theorem 6.5.4 to the initial triangular basis and the claimfollows. (cid:3) The quantization conjecture follows as a consequence. Theorem 9.5.2 ([Kim12, Conjecture 1.1]) . All quantum cluster mono-mials of A ( t ) are contained in the dual canonical basis B up − of U q − ( w ) up to q -powers.Comparing triangular bases. Berenstein-Zelevinsky defined triangularbasis for acyclic seeds in [BZ14], and their definition is different fromours. Take w = c where c is a Coxeter word. The correspondingquantum unipotent cell is an acyclic quantum cluster algebra. Wehave the following natural result. Corollary 9.5.3 ([Qin19b, Conjecture 1.2.1]) . The Berenstein-Zelevinskytriangular basis for acyclic seeds is the same as the common triangularbasis in [Qin17] . Consequently, it contains all quantum cluster mono-mials.Proof. [Qin19b] verifies the statements for symmetric Kac-Moody cases.Now we have seen the existence of common triangular bases for generalKac-Moody algebras. With this existence, the previous proof [Qin19b]is also valid for symmetrizable Kac-Moody cases. (cid:3) Tropical properties of dual canonical bases. We refer the readers to[Qin19a] for necessary definitions and results relating bases and trop-ical points (Definition 2.3.13), see also [GHKK18] for a sophisticatedgeometric view. 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