Skein and cluster algebras of marked surfaces without punctures for \mathfrak{sl}_3
aa r X i v : . [ m a t h . G T ] M a r SKEIN AND CLUSTER ALGEBRAS OF MARKED SURFACESWITHOUT PUNCTURES FOR sl TSUKASA ISHIBASHI AND WATARU YUASA
Abstract.
For a marked surface Σ without punctures, we introduce a skein algebra S q sl , Σ consisting of sl -webs on Σ with the boundary skein relations at marked points.We realize a subalgebra A q sl , Σ of the quantum cluster algebra quantizing the functionring O cl ( A SL , Σ ) inside the skein algebra S q sl , Σ . We also show that the skein algebra S q sl , Σ is contained in the corresponding quantum upper cluster algebra by giving a wayto obtain cluster expansions of sl -webs. Moreover, we show that the bracelets and thebangles along an oriented simple loop in Σ give rise to quantum GS-universally positiveLaurent polynomials. Contents
1. Introduction 12. Quantum cluster algebras 73. Skein algebras of marked surfaces without punctures for sl Introduction
Quantizations of the SL ( C ) -character variety Hom( π (Σ) , SL ( C )) (cid:12) SL ( C ) for a sur-face Σ have been studied in several different ways. One is via the Kauffman bracketskein algebra , first defined in [BFK99] for a closed surface; [PS19, BW11] for a surfacewith boundary; [RY14] for a punctured surface; [Mul16] for a marked surface withoutpunctures. In each of these works the Kauffman bracket skein relation provides a non-commutative deformation of the trace identity among the SL ( C ) -matrices, and there-fore the skein algebra can be regarded as a deformation quantization of the SL ( C ) -character variety (or its suitable variants). More generally, connections between the SL ( C ) -character variety and the Kauffman bracket skein algebra for a -manifold hasbeen studied in [Bul97, PS00].Another one is via the quantum cluster algebra introduced by Berenstein–Zelevinsky[BZ05]. More precisely, for a marked surface Σ ( i.e. , an oriented compact surface with Mathematics Subject Classification.
Key words and phrases.
Cluster algebra; Skein algebra; Positivity. boundary together with a finite set of marked points), the moduli space A SL , Σ of twisteddecorated SL -local systems has a canonical cluster K -structure [FG06a] encoded in aseed pattern s ( sl , Σ) , via which we identify a subring of the ring of regular functions on A SL , Σ with the corresponding cluster algebra A s ( sl , Σ) [FZ02]. Forgetting the decorationwe get the moduli space of twisted SL ( C ) -representations, which is identified with the SL ( C ) -character stack by fixing a spin structure on Σ . The general framework of thequantum cluster algebra provides non-commutative deformations of the cluster algebra A s associated with a seed pattern s depending on the choice of a quantum seed pattern s q (or the choice of compatibility matrices) “quantizing” s , whenever the seed pattern hasfull-rank exchange matrices. For the seed pattern s ( sl , Σ) , the latter condition forces Σ to have no punctures.A suitable choice of a quantum seed pattern s q ( sl , Σ) quantizing s ( sl , Σ) has beenmade by Muller [Mul16], who showed that the two quantization schemes via skein andcluster algebras give the same result for this choice. Indeed, he defined a skein algebra S A sl , Σ on a marked surface without punctures by imposing certain boundary skein relations and obtained the following comparison result: Theorem 1 (Muller [Mul16]) . For a (triangulable) marked surface with at least twomarked points on each of its connected components, the quantum cluster algebra A s q ( sl , Σ) and the quantum upper cluster algebra U s q ( sl , Σ) are both isomorphic to the skein algebra S A sl , Σ [ ∂ − ] localized at boundary intervals with q = A . Moreover, • the isomorphisms are M C (Σ) -equivariant; • the bar-involution on U s q ( sl , Σ) coincides the mirror-reflection on S A sl , Σ [ ∂ − ] ; • the ensemble grading ( a.k.a. the universal grading) on U s q ( sl , Σ) coincides with theendpoint grading on S A sl , Σ [ ∂ − ] . Indeed, he first realized each quantum cluster for s q ( sl , Σ) inside the skein algebra S A sl , Σ , then showed ([Mul16, Theorem 7.15]) the inclusions A s q ( sl , Σ) ⊂ S A sl , Σ [ ∂ − ] ⊂ U s q ( sl , Σ) , and finally proved the coincidence A s q ( sl , Σ) = U s q ( sl , Σ) ([Mul16, Theorem 9.8]). In thiscase, all the mutations are realized as flips of triangulations, the corresponding quantumexchange relations being identified with the skein relations among the skein elementsparallel to triangulations.One can as well consider the (upper) cluster algebras with frozen variables not beinginvertible, in which case they coincide with S A sl , Σ . It amounts to consider a partialcompactification of the cluster K -variety.1.1. Comparison of the skein and cluster algebras for sl . Our goal is to find higher-rank analogues of the Muller’s result for semisimple Lie algebras g other than sl . Indeed,for the simply-connected algebraic group G with Lie algebra g , the moduli space A G, Σ hasalso a canonical K -structure, which is encoded in a seed pattern s ( g , Σ) ([FG06a] for sl n ;[Le19] for classical Lie algebras; [GS19] for general). On the other hand, the higher-rank KEIN AND CLUSTER ALGEBRAS FOR sl analogues of the skein theory has been studied by Kuperberg [Kup96] for rank two Liealgebras, Murakami-Ohtsuki-Yamada [MOY98], Sikora [Sik05] and Morrison [Mor07] for sl n .Our aim in this paper is to establish the sl -case. We first define a skein algebra S A sl , Σ spanned by certain sl -webs on a marked surface Σ without punctures, subject to certainboundary skein relations as well as the usual sl -skein relations (see Definition 3.1 andDefinition 3.2). In this case, however, a difficulty arises from the fact that the mutationsdo not necessarily correspond to geometric operations such as flips in general. To isolatesuch a problem, we first construct quantum seeds whose underlying seeds are associatedwith decorated triangulations and those interpolating their flips (see Section 2.2). Thequantum clusters are realized as specific web clusters in S A sl , Σ , where the latter are de-fined to be collections of elementary webs which A -commute with each other, with theprescribed cardinality (see Definition 3.14 and Definition 3.15). Thanks to the connectiv-ity of decorated triangulations by flips and changes of signs, these quantum seeds definea canonical quantum seed pattern s q ( sl , Σ) . Let A q sl , Σ ⊂ A s q ( sl , Σ) denote the subalgebragenerated by the quantum cluster variables contained in the quantum seeds as above.Then we obtain the following sl -analogue of Theorem 1: Theorem 2 (Comparison of skein and cluster algebras:Theorems 4.4 and 4.7) . We have A q sl , Σ ⊂ S A sl , Σ [ ∂ − ] ⊂ U s q ( sl , Σ) with q = A . Here S A sl , Σ [ ∂ − ] denotes the sl -skein algebra localized at boundary webs (seeDefinition 3.37). Moreover, • the inclusions are M C (Σ) × Out( SL ) -equivariant; • the bar-involution on U s q ( sl , Σ) restricts to the mirror-reflection on S A sl , Σ [ ∂ − ] ; • the ensemble grading on U s q ( sl , Σ) restricts to the endpoint grading on S A sl , Σ [ ∂ − ] . The heart of the first inclusion is the comparison of the quantum exchange and skeinrelations, which has been essentially observed by Fomin–Pylyavskyy [FP16]. The secondinclusion is a consequence of the Laurent expansion results of an arbitrary web in certainweb clusters given in Section 3.4.1 and the quantum Laurent phenomenon [BZ05]. The(partially conjectural) correspondence of some notions are summarized below.Quantum cluster algebra A s q ( sl , Σ) Skein algebra S A sl , Σ clusters web clusterscluser variables elementary websquantum exchange relations skein relations(a Z q -basis) graphical basisbar-involution mirror-reflectionensemble grading endpoint gradingAs in the sl -case, we expect the following: TSUKASA ISHIBASHI AND WATARU YUASA
Conjecture 3. A s q ( sl , Σ) = S A sl , Σ [ ∂ − ] = U s q ( sl , Σ) . In particular, we expect a one-to-one correspondence between the quantum clusters in A s q ( sl , Σ) and the web clusters in S A sl , Σ . Indeed, in the subsequent paper [IY], we willconfirm this conjecture when Σ is a k -gon with ≤ k ≤ , which are exactly the caseswhere the cluster algebra is of finite type, by establishing a bijection between the quantumclusters and the web clusters. This gives a geometric description of the quantum clusters(or equivalently, the cluster charts on the moduli spaces A SL , Σ and X P GL , Σ ) apart fromthose associated with decorated triangulations, including even those not represented byideal webs [Go17]. A comparison with the theory of spectral networks [GMN13] will beextremely interesting, which is believed to be another candidate for a geometric modelfor the cluster charts.1.2. Quantum Laurent positivity of webs.
From the second inclusion in Theorem 2,each web x ∈ S A sl , Σ gives rise to a quantum universally Laurent polynomial , i.e. , it isrepresented as a Laurent polynomial in an arbitrary quantum cluster and the quantumparameter q . Such an element is called a quantum universally positive Laurent polynomial if its Laurent expression in each quantum cluster has positive integral coefficients. Thesearch of webs providing such special polynomials is motivated by the Fock–Goncharovduality conjecture: see Section 1.3 below.The expansion of a web in a quantum cluster in the subalgebra A q sl , Σ are obtained byjust successively applying the sl -skein relations. Most of them have a manifest positivity,while the one (3.4) causes a problem. Indeed, one is forced to use the relation (3.4) insome situation during the process getting the cluster expansion of a web. In order to avoidthe usage of (3.4), we only consider the elevation-preserving webs (Definition 3.31) withrespect to an ideal triangulation. For instance, the n -bracelet (resp. the n -bangle ) along anoriented simple loop γ in Σ , obtained by replacing the embedding of γ with an embeddingof the graph shown in the left (resp. right) in Figure 1.1, is an elevation-preserving webfor any ideal triangulation.An element of U s q ( sl , Σ) is called a quantum GS-universally positive Laurent polynomial (after Goncharov–Shen) if it is represented as a positive Laurent polynomial in the quan-tum cluster associated with any decorated triangulation. The following is our result inthis paper: Theorem 4 (Quantum Laurent positivity of webs: Theorem 4.8) . Any elevation-preservingweb with respect to ∆ is expressed as a positive Laurent polynomial in the quantum clusterassociated with a decorated triangulation ∆ = (∆ , s ∆ ) with the underlying triangulation ∆ . In particular, the bracelets and the bangles along an oriented simple loop in Σ arequantum GS-universally positive Laurent polynomials. Related works.
The theory of cluster ensembles [FG09] produces a pair ( A s , X s ) of positive schemes from a given seed pattern s . Thanks to their positivity nature, we KEIN AND CLUSTER ALGEBRAS FOR sl a simple loop γ } n •••••• the n -bracelet of γ } n •••••• the n -bangle of γ Figure 1.1.
The middle shows a tubular neighborhood of an oriented sim-ple loop γ . The n -bracelet (resp. n -bangle) along γ is obtained by replacingit with the graph shown in the left (right).can form their P -valued points ( A s ( P ) , X s ( P )) for any semifield P . The Fock–Goncharovduality conjecture [FG09, Section 4] asks a construction of duality maps A s ( Z T ) → O ( X s ∨ ) , X s ( Z T ) → O ( A s ∨ ) which satisfy certain axioms, where s ∨ denotes the Langlands dual seed pattern of s (inour situation, s ∨ = s ). In particular, each element in the left-hand side is required to giverise to a universally positive Laurent polynomial. One may ask their quantum aspects,by replacing O ( X s ∨ ) with the Fock–Goncharov’s quantized algebra O q ( X s ∨ ) [FG08], andreplacing O ( A s ∨ ) = U s ∨ with O q ( A s ∨ ) := U s ∨ q whenever the full-rank condition holds,though it involves a choice of a quantum seed pattern s ∨ q .For the seed pattern s = s ( sl , Σ) , such quantum duality maps are constructed via theskein theory. The classical duality maps are constructed by Fock–Goncharov [FG06a]when Σ has empty boundary. In the quantum setting, in the O q ( X s ∨ ) -side, when Σ hasempty boundary, first Bonahon–Wong [BW11] made a progress by defining a quantumtrace map from the “stated” Kauffman bracket skein algebra on Σ to the “square-root”of the quantized algebra O q ( X s ∨ ) . This has been upgraded to a quantum duality map A s ( Z T ) → O q ( X s ∨ ) by Allegretti–Kim [AK17], by composing with a skein realization ofeach integral A -lamination. On the O q ( A s ∨ ) -side, when Σ has no punctures, a quantumduality map X s ( Z T ) → O q ( A s ∨ ) is worked out by Musiker–Schiffler–Williams [MSW11,MSW13], and finally established by Thurston [Thu14] by adding the loop elements tothe Muller’s work we mentioned above. There are also related works in the skein theoryside. Lê [Lê18, Lê19] gave an inclusion of the Muller’s skein algebra ( O q ( A s ∨ ) -side) intothe stated skein algebra ( O q ( X s ∨ ) -side), and obtained the explicit formula of the quantumtrace map for certain simple loops. Costantino–Lê [CL19] proved that the stated skeinalgebra of a biangle is isomorphic to the quantized coordinate ring of SL ( C ) and studiedthe correspondence of some algebraic structures.The duality maps for the seed pattern s = s ( sl , Σ) are recently intensively studied.Douglas–Sun [DS20a, DS20b] developed a theory on the sl - A -laminations in terms of TSUKASA ISHIBASHI AND WATARU YUASA sl -webs (without endpoints on marked points) by defining their tropical A -coordinates,based on the ideas of Xie [Xie13] from the viewpoint of Fock–Goncharov duality. A corre-sponding sl -skein algebra (with certain boundary skein relations) and another coordinatesystems of sl -webs are considered by Frohman–Sikora [FS20]. A stated sl -skein algebrawas also defined by Higgins [Hig20]. Then a quantum duality map A s ( Z T ) → O q ( X s ∨ ) has been established by Kim [Kim20], using the Douglas–Sun coordinates.Now our work in this paper can be regarded as a first step to constructing a dualitymap on the O q ( A s ∨ ) -side. Indeed, we will study tropical X -coordinates of sl -webs (an sl -analogue of the lamination shear coordinates) based on the correspondence given in[IY] in a subsequent paper. Organization of the paper.
In Section 2, we recall the general framework of the quan-tum cluster algebra. Here we partially use the terminology from the theory of clustervarieties [FG09], which is reviewed in Appendix A. We also review the construction of theseed pattern related to the moduli space A SL , Σ .In Section 3, we define the skein algebra S A sl , Σ and investigate its basic structures.Expansion formulae and the positivity results are proved in terms of the skein theory.In Section 4, we construct the quantum seed pattern s q ( sl , Σ) by realizing some of thequantum seeds in the skein algebra S A sl , Σ . Theorems 2 and 4 are proved. Acknowledgements
T. I. is supported by JSPS KAKENHI Grant Number JP20K22304.W. Y. is supported by JSPS KAKENHI Grant Numbers JP19J00252 and JP19K14528.
Notation on marked surfaces and their triangulations. A marked surface (Σ , M ) is a compact oriented surface Σ with boundary equipped with a fixed non-empty finiteset M ⊂ Σ of marked points . When the choice of M is clear from the context, we simplydenote a marked surface by Σ . A marked point is called a puncture if it lies in the interiorof Σ , and a special point otherwise. In this paper we assume that there are no punctures,and hence M ⊂ ∂ Σ . Moreover, assume the following conditions:(1) Each boundary component has at least one marked point.(2) n (Σ) := − χ (Σ) + 2 | M | > , and Σ is not a disk with two special points (abiangle).These conditions ensure that the marked surface Σ has an ideal triangulation, that is, theisotopy class of a collection ∆ of simple arcs connecting marked points whose interiorsare mutually disjoint, which decomposes Σ into triangles. The number n (Σ) gives thenumber of edges of any ideal triangulation ∆ . We call a connected component of thepunctured boundary ∂ × Σ := ∂ Σ \ M a boundary interval , and denote by B the set ofboundary intervals. Each boundary interval belongs to any ideal triangulation ∆ . Wecall an edge of ∆ an interior edge if it is not a boundary interval. Denote the set of edges(resp. interior edges, triangles) of ∆ by e (∆) (resp. e int (∆) , t (∆) ). KEIN AND CLUSTER ALGEBRAS FOR sl kk op Figure 1.2.
A local picture of an sl -triangulation. By convention, a por-tion of ∂ Σ is drawn by a thick line together with a gray region indicatingthe “outer side” of Σ .It is sometimes useful to equip ∆ with two distinguished points on the interior of eachedge and one point in the interior of each triangle: see Figure 1.2. The set of such pointsis denoted by I (∆) = I sl (∆) . We refer to an ideal triangulation equipped with suchcollection of points as an sl -triangulation . Let I edge (∆) (resp. I tri (∆) ) denote the setof points on edges (resp. faces of triangles) so that I (∆) = I edge (∆) ⊔ I tri (∆) , where wehave a canonical identification I tri (∆) = t (∆) . For k ∈ I edge (∆) , let k op denote the otherpoint on the same edge. Let I (∆) f ⊂ I edge (∆) be the subset consisting of the points onthe boundary of Σ , and let I (∆) uf := I (∆) \ I (∆) f .More generally, we can consider an ideal cell decomposition of Σ , which is a decompo-sition of Σ into a union of polygons. When it is obtained from an ideal triangulation byremoving k interior edges, it is said to be of deficiency k . In this paper, we only use anideal cell decomposition of deficiency or . The ideal cell decomposition of deficiency obtained from an ideal triangulation ∆ by removing one interior edge E is denoted by (∆; E ) . 2. Quantum cluster algebras
Quantum cluster algebra.
Here we recall the definition of the quantum clusteralgebra following [BZ05] and related fundamental results. We also recall a grading on the(quantum) cluster algebra which we call the ensemble grading from the viewpoint of thecluster variety [FG09], which has been originally investigated in [GSV03].2.1.1.
The exchange graph and the cluster algebra.
Fix a finite linearly ordered set I ofindices and a field F which is isomorphic to the field of rational functions on I variableswith rational coefficients. We also fix a subset I uf ⊂ I and let I f := I \ I uf . A seed in F is a pair ( B, A ) , where This is the set of vertices of the -triangulation [FG06a] associated with ∆ . The linear ordering here is auxiliary, just in order to distinguish a collection labeled by the set I (e.g., A ( t ) ) from its unlabeled version (e.g. A ( ω ) ) by simpler terms “ordered” and “unordered”. TSUKASA ISHIBASHI AND WATARU YUASA • B = ( b ij ) i,j ∈ I is a skew-symmetric matrix with half-integral entries such that b ij ∈ Z unless ( i, j ) ∈ I f × I f ; • A = ( A i ) i ∈ I is a tuple of algebraically independent elements in F .We call a matrix B satisfying the above conditions an exchange matrix . The elements A i for i ∈ I are called the cluster ( A -)variables , and those for i ∈ I f are called the frozenvariables .It is useful to represent an exchange matrix B = ( b ij ) i,j ∈ I by a quiver Q . Let us definethe quiver exchange matrix ε = ( ε ij ) i,j ∈ I by ε ij := b ji . Then the quiver Q correspondingto B has vertices parametrized by the set I and | ε ij | arrows from i to j (resp. j to i ) if ε ij > (resp. ε ji > ). In figures, we draw n dashed arrows from i to j if ε ij = n/ for n ∈ Z , where a pair of dashed arrows is replaced with a solid arrow.Let T I uf be a regular I uf -valent tree whose edges are labeled by indices in I uf , suchthat the edges incident to a common vertex have distinct labels. An assignment s : T I uf ∋ t s ( t ) = ( B ( t ) , A ( t ) = ( A ( t ) i ) i ∈ I ) of a seed s ( t ) in F to each vertex t ∈ T I uf is called a seed pattern (of geometric type) in F if for each edge t k −−− t ′ of T I uf labeled by k ∈ I uf , the two seeds s ( t ) and s ( t ′ ) are relatedby the following seed mutation at k (write s ( t ′ ) = µ k s ( t ) ): b ( t ′ ) ij = ( − b ( t ) ij if i = k or j = k,b ( t ) ij + [ b ( t ) ik ] + [ b ( t ) kj ] + − [ − b ( t ) ik ] + [ − b ( t ) kj ] + otherwise , (2.1) A ( t ′ ) = ( A ( t ) k ) − Y j ∈ I ( A ( t ) j ) [ b ( t ) jk ] + + Y j ∈ I ( A ( t ) j ) [ − b ( t ) jk ] + ! if i = k,A ( t ) i if i = k. (2.2)Here [ a ] + := max { a, } for a ∈ R . The relation (2.1) is called the matrix mutation , and(2.2) is called the exchange relation . It is not hard to check that the seed mutation isinvolutive: s ( t ) = µ k µ k s ( t ) , and hence such an assignment makes sense. Since the matrixmutation rule inductively determines a seed from a given one, for a vertex t ∈ T I uf and aseed ( B, A ) , there exists a unique seed pattern s = s t ;( B, A ) such that s ( t ) = ( B, A ) . Wecall s t ;( B, A ) the seed pattern with the initial seed ( B, A ) at t .Two seed patterns s , s ′ are isomorphic if there exists an automorphism φ of the graph T I uf such that s ( φ ( t )) = s ( t ) for all t ∈ T I uf . Note that the isomorphism class of s t ;( B, A ) only depends on the equivalence class of the initial exchange matrix B for the mutation-equivalence (or the mutation class for short), which is generated by matrix mutations andpermutations of indices.Given a seed pattern s , we consider a quotient of the tree T I uf as follows. We identifytwo vertices t, t ′ ∈ T I uf if the associated seeds s ( t ) and s ( t ′ ) are related by a permutation This is identified with the Fock–Goncharov’s exchange matrix [FG09]. See Appendix A.
KEIN AND CLUSTER ALGEBRAS FOR sl σ ∈ S I uf × S I f (and write s ( t ′ ) = σ. s ( t ) ): b ( t ′ ) ij = b ( t ) σ − ( i ) ,σ − ( j ) , A ( t ′ ) i = A ( t ) σ − ( i ) . Identify two edges if both of their vertices are respectively identified, so that we have nomultiple edges in the quotient.
Definition 2.1.
The quotient graph
Exch s = T I uf / ∼ obtained in this way is called the exchange graph of s . Let π s : T I uf → Exch s denote the graph projection.To each vertex ω ∈ Exch s , associated are the unordered collection A ( ω ) := { A ( t ) i } i ∈ I of cluster variables called a cluster and the sub-collection A f( ω ) := { A ( t ) i } i ∈ I f of frozenvariables, for some lift t ∈ π − s ( ω ) . Let Z [ A ± ω ) ] := Z [( A ( t ) i ) ± | i ∈ I ] denote the ring ofLaurent polynomials. Definition 2.2.
The cluster algebra associated with a seed pattern s is the subring A s ⊂ F generated by the union of the clusters A ( ω ) and the inverses of the frozen variables in A f( ω ) for ω ∈ Exch s . The upper cluster algebra is defined to be the subring U s := \ ω ∈ Exch s Z [ A ± ω ) ] ⊂ F . The
Laurent phenomenon theorem [FZ02, Theorem 3.1] tells us that each cluster vari-able can be expressed as a Laurent polynomial in any cluster, and hence A s ⊂ U s holds.Note that the isomorphism classes of the exchange graph and the cluster algebra de-pend only on the isomorphism class of a seed pattern. When s = s t ;( B, A ) , we may write A s =: A ( B ) and U s =: U ( B ) as in the literature.2.1.2. The quantum cluster algebra.
We basically follow [BZ05], partially employing thenotation in [GS19, Section 13.3]. Recall that for a skew-symmetric form Π on a lattice L ,the associated based quantum torus is the associative Z q -algebra T Π such that • T Π has a free Z q -basis M α parametrized by α ∈ L , and • the product of these basis elements is given by M α · M β = q Π( α,β ) / M α + β .Let F be a skew-field. A quantum seed in F is a triple ( B, Π , M ) , where • B is an exchange matrix; • Π = ( π ij ) i,j ∈ I is a skew-symmetric matrix with integral entries satisfying the com-patibility relation X k ∈ I b ki π kj = δ ij d j for all i ∈ I uf and j ∈ I , where d i is a positive integer for i ∈ I uf . The matrix Π defines a skew-symmetric form on a lattice ◦ Λ = L i ∈ I f i by Π( f i , f j ) := π ij ; Our notation for the lattice is motivated by the connection to the theory of cluster varieties: seeAppendix A. • M : ◦ Λ → F \ { } is a function such that M ( α ) M ( β ) = q Π( α,β ) / M ( α + β ) for α, β ∈ ◦ Λ , and the Z q -span of M ( ◦ Λ) ⊂ F is the based quantum torus of theform Π whose skew-field of fractions coincides with F .We call Π the compatibility matrix , and M the toric frame of the quantum seed. Thecompatibility relation can be written as ε Π = B T Π = ( D, , (2.3)where D := diag( d i | i ∈ I uf ) and denotes the I uf × I f -zero matrix. By [BZ05, Lemma4.4], a toric frame M is uniquely determined by the values A i := M ( f i ) on the basisvectors f i for i ∈ I . Indeed, for a linear ordering I ∼ = { , . . . , N } , we have M X i ∈ I x i f i ! = q P l Let ( B ( t ) , Π ( t ) , M ( t ) ) be a quantum seed in F , k ∈ I uf , and consider theexchange matrix B ( t ′ ) := E ( t ) k,ǫ B ( t ) F ( t ) k,ǫ and the toric frame M ( t ′ ) determined by (2.5) . Let Π ( t ′ ) = ( π ′ ij ) i,j ∈ I be the skew-symmetric matrix associated with M ( t ′ ) , which is uniquelydetermined by the condition A ( t ′ ) i A ( t ′ ) j = q π ′ ij A ( t ′ ) j A ( t ′ ) i for i, j ∈ I . Then the pair ( B ( t ′ ) , Π ( t ′ ) ) satisfies the compatibility relation. The following “decomposition of mutation” formula is useful in the study of the ensemblegrading in Section 2.1.3. Lemma 2.5 ([BZ05, (4.22)]) . The quantum exchange relation (2.5) is written as M ( t ′ ) = ρ b k ,ǫ ◦ M ( t ) ◦ e µ ∗ k,ǫ , where ρ b k ,ǫ is an automorphism on F uniquely determined by ρ b k ,ǫ ( A ( t ) i ) := ( A ( t ) i if i = k,A ( t ) k + M ( t ) ( f ( t ) k + P j ∈ I ǫb ( t ) jk f ( t ) j ) i = k, and e µ ∗ k : ◦ Λ ( t ′ ) → ◦ Λ ( t ) denote the signed seed mutation (A.1) . Given a quantum seed patten s q as above, we write s ( t ′ ) q = σ. s ( t ) q for a permutation σ ∈ S I uf × S I f if b ( t ′ ) ij = b ( t ) σ − ( i ) ,σ − ( j ) , π ( t ′ ) ij = π ( t ) σ − ( i ) ,σ − ( j ) , A ( t ′ ) i = A ( t ) σ − ( i ) hold. The classical specialization of s q is a seed pattern s such that the exchange matrixassigned to each vertex t ∈ T I uf coincides with B ( t ) . Then it is known that s ( t ′ ) q = σ. s ( t ) q holds whenever s ( t ′ ) = σ. s ( t ) holds in the classical specialization [BZ05, Theorem 6.1].Hence to each vertex ω ∈ Exch s , associated is a quantum torus T ( ω ) := span Z q M ( t ) ( ◦ Λ ( t ) ) ⊂F , where t ∈ π − s ( ω ) and the basis of T ( ω ) is given up to permutations. The unordered basis A ( ω ) := { A ( t ) i } i ∈ I is called a quantum cluster , and we have the collection A f( ω ) := { A ( t ) i } i ∈ I f of frozen variables. Definition 2.6. The quantum cluster algebra associated with a quantum seed pattern s q is the Z q -subalgebra A s q ⊂ F generated by the union of the quantum clusters A ( ω ) and the inverses of frozen variables in A f( ω ) for ω ∈ Exch s . The quantum upper cluster algebra is defined to be U s q := \ ω ∈ Exch s T ( ω ) ⊂ F . For each vertex ω ∈ Exch s , the upper bound at ω is defined to be U s q ( ω ) := T ω ′ T ( ω ′ ) ,where ω ′ ∈ Exch s runs over the vertices adjacent to ω . Theorem 2.7 (Quantum Laurent phenomenon [BZ05, Theorem 5.1]) . For any vertices ω, ω ′ ∈ Exch s , we have U s q ( ω ) = U s q ( ω ′ ) . In particular, we have U s q = U s q ( ω ) for any ω ∈ Exch s . Note that it also implies that A s q ⊂ U s q . Bar-involution . For each ω ∈ Exch s , define a Z -linear involution † : T ( ω ) → T ( ω ) by ( q r/ M ( ω ) ( α )) † := q − r/ M ( ω ) ( α ) for r ∈ Z and α ∈ ◦ Λ ( t ) . Then † preserves the subalgebra U s q ⊂ T ( ω ) , and the inducedinvolution does not depend on the choice of ω [BZ05, Proposition 6.2]. Following [BZ05],we call this anti-involution † : U s q → U s q the bar-involution . Each quantum clustervariable is invariant under the bar-involution.2.1.3. Ensemble grading. We have a natural grading on the (quantum) upper clusteralgebra, which we call the ensemble grading (a.k.a. universal grading [Mul16]). In orderto motivate its definition from the algebro-geometric viewpoint, we borrow some notationsfrom [FG09], for which the reader is referred to Appendix A. Lemma-Definition 2.8 (cf. [GSV03, Lemma 5.3]) . For ω ∈ Exch s , define gr ( A ( ω ) i ) ∈ coker p ∗ to be the image of the basis vector f ( t ) i ∈ ◦ Λ ( t ) under the natural projection α ∗ ( t ) : ◦ Λ ( t ) → coker p ∗ ( t ) for some lift t ∈ π − s ( ω ) . Then gr defines a grading on the ring U s ,which we call the ensemble grading . The ensemble grading on the quantum upper clusteralgebra U s q is defined by the same manner, which makes the latter a graded Z q -algebra. Proof. In order to see that the ensemble grading is well-defined, first we need to show thatfor two vertices t , t ∈ T I uf , the condition A ( t ) i = A ( t ) σ − ( i ) for some σ ∈ S I uf × S I f impliesthat α ∗ ( t ) ( f ( t ) i ) = α ∗ ( t ) ( f ( t ) σ − ( i ) ) under the identification coker p ∗ ( t ) ∼ = coker p ∗ ( t ) induced by asequence of signed seed mutations. For this, let γ : t k −−− t k −−− · · · k h − −−− t h = t be the shortest edge path in T I uf . Since the identification coker p ∗ ( t ) ∼ = coker p ∗ ( t ) doesnot depend on signs, we may consider the associated tropical sign ǫ trop γ = ( ǫ , . . . , ǫ h − ) KEIN AND CLUSTER ALGEBRAS FOR sl [NZ12]. Then the composite ˇ µ ∗ γ, ǫ trop γ := ˇ µ ∗ k ,ǫ . . . ˇ µ ∗ k h − ,ǫ h − : ◦ Λ ( t ) → ◦ Λ ( t ) sends the vector f ( t ) i to ˇ µ ∗ γ, ǫ trop γ ( f ( t ) i ) = X j ∈ I f ( t ) j ( E ( t ) k ,ǫ . . . E ( t h − ) k h − ,ǫ h − ) ji . Here G s ; t t := E ( t ) k ,ǫ . . . E ( t h − ) k h − ,ǫ h − is the G -matrix assigned at t with the initial seed t ([NZ12, Proposition 1.3]), and hence ˇ µ ∗ γ, ǫ trop γ ( f ( t ) i ) = f ( t ) j ( G s ; t t ) ji . On the other hand, bythe separation formula [FZ07, Corollary 6.3], the identity A ( t ) i = A ( t ) σ − ( i ) implies that ( G s ; t t ) ji = δ j,σ − ( i ) and thus ˇ µ ∗ γ, ǫ trop γ ( f ( t ) i ) = f ( t ) σ − ( i ) .Secondly, the ensemble grading is compatible with the exchange relation (2.2). Indeed,the birational automorphism (A.3) does not affect on the grading since gr ( p ∗ X ( t ) k ) = 0 .Thus it is well-defined on the upper cluster algebra U s .For the quantum counterpart U s q , notice that the automorphism ρ b k ,ǫ given in Lemma 2.5preserves the grading. Thus the ensemble grading is compatible with the quantum ex-change relations, hence well-defined on U s q as well. (cid:3) Remark 2.9. (1) The grading gr ( A ( ω ) i ) ∈ coker p ∗ is the weight of the coordinatefunction A ( ω ) i ∈ O ( A s ) = U s for the H A -action on A s (see Appendix A).(2) The proof of Lemma-Definition 2.8 shows that we can identify the lattices ◦ Λ ( t ) and ◦ Λ ( t ′ ) with π s ( t ) = π s ( t ′ ) =: ω via the linear isomorphism e µ ∗ γ, ǫ trop γ . We denotethese lattices by ◦ Λ ( ω ) under this identification, which still has an unordered basiswritten as { f ( ω ) i } i ∈ I . Convention 2.10. When we do not care about the global structure of the exchangegraph, we may always choose a lift of a vertex (or a contractible path) in Exch s so thatthe associated objects get linearly ordered labelings by I . This being reminded, we willconfuse the notation as A ( ω ) i := A ( t ) i , b ( ω ) ij := b ( t ) ij , etc. with a choice of a lift t ∈ π − s ( ω ) non-specified.2.2. The seed pattern related to the moduli space A SL , Σ . Let Σ be a markedsurface. As in Section 1, we assume that Σ has no punctures. A decorated triangulation ∆ = (∆ , s ∆ ) consists of: • An ideal triangulation ∆ of Σ ; • A function s ∆ : t (∆) → { + , −} .Given a decorated triangulation ∆ , we define a quiver Q ∆ with the vertex set I (∆) = I sl (∆) as follows. Let Q + and Q − be the quivers shown in the left and right of Figure 2.1,respectively. These quivers are related by the mutation at the central vertex k . For eachtriangle T ∈ t (∆) , we draw the quiver Q s ∆ ( T ) , and glue them via the amalgamation procedure [FG06b] to get a quiver Q ∆ drawn on Σ . In our situation, opposite half-arrowscancel together, and parallel half-arrows combine to give an usual arrow. Some examplesare shown in Figure 2.2. Q + k µ k Q − k Figure 2.1. Two quivers on a triangle. Figure 2.2. The quivers on a quadrilateral with the signs (+ , +) (left) and (+ , − ) (right).Let B ∆ = ( b ∆ ij ) i,j ∈ I ( ∆ ) denote the exchange matrix determined by the quiver Q ∆ . Theorem 2.11 (Fock–Goncharov [FG06a, Section 10.3]) . The exchange matrices B ∆ associated with decorated triangulations ∆ of a fixed marked surface Σ are mutation-equivalent to each other. For later use, we reproduce the proof here. Proof. Let ∆ , ∆ ′ be two decorated triangulations of a marked surface Σ . Since thequiver Q − is transformed into Q + by a mutation and the amalgamations commute withmutations at the vertices in I tri (∆) , we can assume that both ∆ and ∆ ′ have the positivesign on each triangle. Moreover, since any two ideal triangulations are transformed toeach other by a sequence of flips, it suffices to consider the case where the underlyingtriangulations of ∆ and ∆ ′ are related by the flip along an edge. Some sequences ofmutations which realizes a flip is shown in Figure 2.3. The assertion is proved. (cid:3) KEIN AND CLUSTER ALGEBRAS FOR sl E ∆ ii op ii op j op j E ′ j j op ∆ ′ µ i µ i op µ i op µ i µ j µ j op µ j op µ j Figure 2.3. Some of the sequences of mutations that realize the flip f E :∆ → ∆ ′ . Remark 2.12. Via the Goncharov–Shen’s construction [GS19, Section 5.2], we get acollection A ∆ of regular functions on the moduli space A SL , Σ of the decorated SL -localsystems on Σ associated with a decorated triangulation ∆ . Here the sign + (resp. − )assigned to a triangle corresponds to the reduced word (1 , , (resp. (2 , , ) of thelongest element w ∈ W ( sl ) in the Weyl group of sl , and thanks to the cyclic symmetryof the cluster structure on the moduli space A SL ,T , we do not need to choose a vertex ofeach triangle T . Thus we get a seed ( B ∆ , A ∆ ) in the field F SL , Σ of rational functions on A SL , Σ . These seeds are mutation-equivalent to each other [GS19, Theorem 5.7].In particular, the mutation class of the exchange matrices B ∆ defines a unique isomor-phism class of a seed pattern, which is denoted by s ( sl , Σ) .It is typically hard to understand all the seeds in s ( sl , Σ) in geometric terms. Instead,we are going to study those associated with the decorated triangulations and those alongthe flip sequences. A decorated cell decomposition (of deficiency ≤ ) is an ideal celldecomposition (∆; E ) of deficiency equipped with a sign on each triangle and one ofthe quivers shown in Figure 2.3 on the unique quadrilateral. In particular, a decoratedtriangulation is a decorated cell decomposition. Definition 2.13. Define the surface subgraph to be the subgraph Exch sl , Σ ⊂ Exch s ( sl , Σ) such that • the vertices are the seeds corresponding to the decorated cell decompositions; • the edges are mutations realizing changes of the signs (Figure 2.1) and thoserealizing flips (Figure 2.3).The exchange matrix B ( ω ) for any vertex ω ∈ Exch sl , Σ is determined by the correspondingquiver in Figure 2.3. Here notice that for two ideal triangulations ∆ and ∆ ′ related by asingle flip, the two sets I (∆) and I (∆ ′ ) can be canonically identified, and we use this set I (∆) = I (∆ ′ ) as the index set I ( ω ) for the matrices B ( ω ) associated with the interpolatingdecorated cell decompositions ω ∈ Exch sl , Σ as in Figure 2.3.We call the corresponding subring A sl , Σ ⊂ A s ( sl , Σ) generated by the collection S ω ∈ Exch sl , Σ A ( ω ) of cluster variables the surface subalgebra .The next lemma follows from the Laurent phenomenon theorem [FZ02, Theorem 3.1],the identification (A.5) and [GS18, Proposition 3.17]. Lemma 2.14. We have the inclusions A sl , Σ ⊂ A s ( sl , Σ) ⊂ U s ( sl , Σ) = O ( A s ( sl , Σ) ) ⊂ O ( A SL , Σ ) . The notation O cl ( A SL , Σ ) := O ( A s ( sl , Σ) ) is also used in the literature. Some group actions . On any quantum cluster algebra A s q , we have a natural rightaction of the corresponding cluster modular group Γ s [FG09]. An element φ ∈ Γ s sendseach cluster to another cluster, and commutes with mutations. When s = s ( sl , Σ) , it isknown that the cluster modular group contains the group M C (Σ) × Out( SL ) [GS18].Here M C (Σ) denotes the mapping class group of Σ , which consists of the isotopy classesof orientation-preserving homeomorphisms on Σ that preserve ∂ Σ and M set-wisely; Out( SL ) := Aut( SL ) / Inn( SL ) denotes the outer automorphism group, which is gen-erated by the Dynkin involution ∗ : SL → SL . The actions of these groups on A q sl , Σ are described as follows. See [GS18] for a detail. • Each mapping class φ ∈ M C (Σ) sends the cluster A ∆ associated with a deco-rated triangulation ∆ = (∆ , s ∆ ) to the one A φ − ( ∆ ) associated with φ − ( ∆ ) :=( φ − (∆) , φ ∗ s ∆ ) , where ( φ ∗ s ∆ )( T ) := s ∆ ( φ ( T )) for T ∈ t ( φ − (∆)) . The correspon-dence φ : φ − (∆) ∼ −→ ∆ naturally induces a correspondence I ( φ − (∆)) ∼ −→ I (∆) ofthe sl -triangulations, and hence of the cluster variables. The action on the cluster A ( ω ) for general ω ∈ Exch sl , Σ is uniquely interpolated by mutation-equivariance. • The Dynkin involution ∗ sends the cluster A ∆ to the one A ∆ ∗ associated with ∆ ∗ := (∆ , s ∗ ∆ ) , where s ∗ ∆ ( T ) := − s ∆ ( T ) for T ∈ t (∆) . The action on the cluster A ( ω ) for general ω ∈ Exch sl , Σ is uniquely interpolated by mutation-equivariance.Although the interpolation by mutation-equivariance is rather implicit, we will see thatthese actions are described as certain geometric actions on webs on Σ . KEIN AND CLUSTER ALGEBRAS FOR sl Skein algebras of marked surfaces without punctures for sl A skein algebra of a connected compact oriented surface Σ is the quotient of the algebraof links in the thickened surface Σ × [0 , defined by certain skein relations. Skein relationsare obtained from representations of quantum groups associated with simple Lie algebras.For sl , the skein relation is known as the Kauffman bracket skein relation and the skeinalgebra is called the Kauffman bracket skein algebra. Muller [Mul16] introduced theboundary Kauffman bracket skein relation for tangle diagrams on a marked surface (Σ , M ) without punctures, and defined the Kauffman bracket skein algebra of (Σ , M ) . In thissection, we will introduce a skein algebra S A sl , Σ of a marked surface (Σ , M ) for sl . We willsee that this skein algebra has an Ore localization S A sl , Σ [∆ − ] on each ideal triangulation,and closely related to a quantum cluster algebra quantizing the seed pattern s ( sl , Σ) .Moreover, certain tangled trivalent graphs are expressed as positive Laurent polynomialin “elementary webs” in S A sl , Σ [∆ − ] .3.1. Skein algebras of marked surfaces for sl . Let N be the set of non-negativeintegers and Z A := Z [ A / , A − / ] the Laurent polynomial ring in a variable A / . In thissubsection, there is no need for considering the conditions (1) and (2) for a marked surface (Σ , M ) in Section 1.3.1.1. The boundary sl -skein relation. The skein algebra S A sl , Σ treats tangled trivalentgraphs with endpoints in M , and its skein relations are defined by adding boundary sl -skein relations to the sl -skein relations introduced in Kuperberg [Kup96].A tangled trivalent graph G on (Σ , M ) is an immersion of an oriented uni-trivalent graphinto Σ satisfying the following conditions (1) – (7):(1) the valency of a vertex of the underlying graph is or ,(2) the univalent vertices of G are contained in M ,(3) the trivalent vertices of G are distinct points in Σ \ ∂ Σ ,(4) all intersection points of G in Σ \ ∂ Σ are transverse double points of edges,(5) an intersection point p ∈ Σ \ ∂ Σ of G has over-/under-passing information (wecall such p an internal crossings ),(6) for an intersection point p ∈ M of G , the set of univalent vertices on p has a stricttotal order, which we call the elevation at p ,(7) the orientation of edges incident to a trivalent vertex is a sink or a source.We denote the number of sinks in G by t + ( G ) , and sources by t − ( G ) . The over-/under-passing information is indicated as . Two consecutive ordered univalent vertices v < v , whose half-edges e and e are incident to p ∈ M , are indicated as e e or e e .We define skein relations for the tangled trivalent graphs on (Σ , M ) . Definition 3.1 ( sl -skein relations [Kup96]) . = A + A − , (3.1) = A − + A , (3.2) = + , (3.3) = − ( A + A − ) , (3.4) = ( A + 1 + A − ) = . (3.5) Definition 3.2 (boundary sl -skein relations) . = A = A (3.6) = A = A (3.7) = = (3.8) = 0 = 0 (3.9) = 0 = 0 (3.10)It is easy to see that the sl -skein relations, boundary sl -skein relations, and theboundary fixing isotopy realize the following Reidemeister moves (R0), (R2), (R3), (R4)and (bR). KEIN AND CLUSTER ALGEBRAS FOR sl Definition 3.3 (Reidemeister moves) . (R0)(R2)(R3) , (R4)(bR) Definition 3.4 (the sl -skein algebra of a marked surface without punctures) . The sl -skein algebra S A sl , Σ of a marked surface (Σ , M ) is defined to be the quotient moduleof the free Z A -module spanned by tangled trivalent graphs in (Σ , M ) by the sl -skeinrelations (Definition 3.1), the boundary sl -skein relations (Definition 3.2), and isotopyin Σ relative to ∂ Σ . It is equipped with a multiplication defined by the superposition oftangled trivalent graphs. The product G G of two tangled trivalent graphs G and G in generic position is defined by superposing G on G , that is, G is over-passing G inall intersection points. We call an element in S A sl , Σ an sl -web or simply a web . Remark 3.5. In the right-hand sides of the boundary skein relations at p ∈ M , the signof an exponent of A only depends on the orientation from the arc with higher elevationto the lower one with respect to the orientation of Σ . The absolute value of an exponentdepends on whether two arcs have a parallel direction or anti-parallel.It is useful to slightly extend the definition of the sl -webs, allowing them to haveunivalent vertices with the same elevation. Definition 3.6 (simultaneous crossings) . An sl -web with the simultaneous crossing at p ∈ M is recursively defined by the following skein relations: A − l − k ( k, l ) = ( k, l ) = A l + k ( k, l ) ,A − k − l ( k, l ) = ( k, l ) = A k + l ( k, l ) , (3.11)where the thickened edge labeled by ( k, l ) is a collection of k + l half-edges with simulta-neous crossing whose endpoint grading is ( k, l ) . From the above skein relations, we obtain Figure 3.1. elliptic facesthe boundary twist relations for two adjacent half-edges with a simultaneous crossing: = A , = A , (3.12) = A , = A . (3.13)From the above relations, we can obtain more general formula: ( e, f )( c, d )( a, b ) = A ac + bd − ec − fd A ( ad + bc − ec − fd ) / ( e, f )( c, d )( a, b ) . (3.14)For any tangled trivalent graph G with no interior crossings, the Weyl ordering [ G ] isthe sl -web obtained by replacing all the crossings on M with the simultaneous crossings.One can represent the Weyl ordering [ G ] of G by a flat trivalent graph , i.e. , a uni-trivalentgraph such that its univalent vertices lie in M and the other part is embedded into Σ \ ∂ Σ . Definition 3.7 (basis webs) . Let [ G ] be a flat trivalent graph. A polygon P in Σ \ [ G ] is an elliptic face if P is one of the shaded faces shown in Figure 3.1. A flat trivalentgraph [ G ] is elliptic if Σ \ [ G ] has an elliptic face. A basis web is an sl -web representedby non-elliptic flat trivalent graph [ G ] on (Σ , M ) . We call the set BWeb sl , Σ of basis websthe graphical basis .We will see that BWeb sl , Σ is indeed a free Z A -basis of S A sl , Σ soon below. The followingnotion provides a useful tool to study the webs. Definition 3.8 (cut-paths [Kup96]) . Let G be a flat trivalent graph on (Σ , M ) and p, q ∈ ∂ × Σ distinct points.(1) A cut-path from p to q of G is a properly embedded oriented interval from p to q which transversely intersects with edges of G . An identity move of a cut-path α with respect to G is a deformation of α into another cut-path α ′ such that α and α ′ bound only one biangle which cuts out a subgraph of G consisting of anidentity braid between two edges of the biangle. In a similar way, an H -move isdefined, for which the biangle cuts out an H -web. See Figure 3.2.(2) The weight wt α ( G ) = ( k, l ) ∈ N × N of G for a cut-path α is defined by k := { p ∈ G ∩ α | ( G, α ) p = 1 } and l := { p ∈ G ∩ α | ( G, α ) p = − } , where ( G, α ) p is the local intersection number of G and α at p : see Figure 3.3. Let | wt α ( G ) | := k + l denote the total number of intersection points of α and G . KEIN AND CLUSTER ALGEBRAS FOR sl ··· α α ′ α α ′ Figure 3.2. An identity move (left) and an H -move (right). The otherparts of α and α ′ are identical.(3) A cut-path α from p to q is said to be minimal for G if wt α ( G ) is minimal in theset of cut-paths homotopic to α rel. to endpoints, with respect to the partial orderon the weight lattice of sl given by ( k, l ) (cid:23) ( k + 1 , l − , ( k, l ) (cid:23) ( k − , l + 1) . (4) A cut-path α from p to q of G is non-convex to the left (resp. right) side if wt α ( G ) (cid:22) wt β ( G ) for any cut-path β ⊂ Σ \ α from p to q of G such that α ∪ β bounds a biangle, and β lies in the left side (resp. right side) of α .Kuperberg proved some lemmas about cut-paths described below. Lemma 3.9 (Kuperberg [Kup96, Lemma 6.5, 6.6,]) . Let Σ be a marked surface, p, q ∈ ∂ × Σ distinct points, and G a non-elliptic flat trivalent graph. (1) If α and β are homotopic (rel. to endpoints) cut-paths from p to q of G and α isminimal, then wt α ( G ) (cid:22) wt β ( G ) . If β is also minimal, then α is related to β by afinite sequence of H -moves and identity moves. (2) If a cut-path α from p to q of G is non-convex to the left side (resp. right side),there exists a unique class α L (resp. α R ) of cut-paths from p to q under identitymoves such that any cut-path β from p to q with wt β ( G ) = wt α ( G ) in the left side(resp. right side) of α lies between α L (resp. α R ) and α . We call the above cut-path α L (resp. α R ) the left (resp. right) core of α : see Figure 3.3. Proposition 3.10. The skein algebra S A sl , Σ is a free Z A -module generated by BWeb sl , Σ .Proof. Let us consider a neighborhood of a special point p ∈ M . For a given tangledtrivalent graph G with simultaneous crossings, we can expand univalent vertices at p ∈ M as follows: · · · · · · . The special point p is replaced by an interval I p containing all expanded univalent verticesat p . For any tangled trivalent graph, one can obtain a tangled trivalent graph withsimultaneous crossings by (3.11) and expand it by applying the above deformation. The pqαα L β Figure 3.3. The curves α , α L , and β are homotopic cut-paths of G suchthat wt α ( G ) = (2 , , wt α L ( G ) = (2 , , and wt β ( G ) = (0 , . The cut-path α is non-convex to the left side but convex to the right side, and α L is theleft core of α . The cut-paths α and α L are related by an H -move . Thecut-path β is minimal.boundary skein relations (3.8), (3.9), and (3.10) are described as = , = , (3.15) = 0 , = 0 , (3.16) = 0 , = 0 . (3.17)Let us give an orientation, induced from the orientation of Σ , to the union of the intervals { I p | p ∈ M } . For a map f : M → N × N , we define B irr ( f ) to be the set of boundary-fixingisotopy classes of embeddings of trivalent graphs satisfying the following conditions: • G ∈ B irr ( f ) has distinct univalent vertices lying in ∪ p ∈ M I p and gr( G ) = f , • For each interval I p , outgoing univalent vertices follow incoming univalent verticeswith respect to the orientation on I p , • G is non-elliptic, and I p is a cut-path of G which is non-convex to the left side forall p ∈ M .One can use the confluence theory of Sikora–Westbury [SW07]. It shows that ∪ f B irr ( f ) gives a basis of S A sl , Σ as a Z A -module. To apply [SW07, Theorem 2.3], we consider“reductions” from the left-hand side of the first equation in (3.15), (3.16), and (3.17) totheir right-hand sides. Adding these reductions to the reductions of sl -webs in [SW07,Chapter 5], we conclude that B irr ( f ) gives a basis of S A sl , Σ by shrinking I p to p for all p ∈ M . (cid:3) KEIN AND CLUSTER ALGEBRAS FOR sl From the above relations (3.15), (3.16), and (3.17), one can see that our skein algebrais identified with the one in Frohman–Sikora [FS20] with a = 1 by a scalar modificationof the multiplication, where a is a coefficient related to (3.15). Consequently, we get thefollowing: Theorem 3.11 (Frohman–Sikora [FS20, Theorem 7]) . The skein algebra S A sl , Σ is finitelygenerated. Basic structures on the skein algebra S A sl , Σ . The skein algebra S A sl , Σ has the fol-lowing basic structures, which will be compared with the corresponding structures on thequantum cluster algebra in Section 4. The mirror-reflection . The mirror-reflection G † of a tangled trivalent graph G is de-fined by reversing the ordering of the univalent vertices on each special point and ex-changing the over-/under-passing information at each internal crossing. The mirror-reflection is extended to an anti-involution † : S A sl , Σ → S A sl , Σ by Z -linearly and bysetting ( A ± / ) † := A ∓ / . Some group actions . The group M C (Σ) × Out( SL ) acts on S A sl , Σ from the right asfollows. • Each mapping class φ acts on S A sl , Σ by sending each web [ G ] to [ φ − ( G )] . It iswell-defined since φ preserves the set M and respects the defining relations. • The Dynkin involution ∗ ∈ Out( SL ) acts on S A sl , Σ as an Z A -algebra involutionby reversing the orientation of each edge of a basis web. The endpoint grading . The skein algebra has the following gradings. Definition 3.12 (the endpoint grading) . The endpoint grading gr = (gr p ) p ∈ M : BWeb sl , Σ → ( N × N ) M is defined as follows. For a basis web [ G ] ∈ BWeb sl , Σ and p ∈ M , the first (resp. second)entry of gr p ( G ) ∈ N × N is the number of incoming (resp. outgoing) edges of G incidentto p .We define a non-negative bi-grading by the sum ~ gr( G ) = X p ∈ M gr p ( G ) and an augmentation map ε gr ( G ) = k − l for ~ gr( G ) = ( k, l ) . Note that the skein relations(Definition 3.1) and the boundary skein relations (Definition 3.2) are homogeneous withrespect to gr . Hence the skein algebra S A sl , Σ = L ( k,l ) ∈ N × N (cid:0) S A sl , Σ (cid:1) ( k,l ) is a bi-gradedalgebra with respect to ~ gr . The augmentation map ε gr defines a Z -valued grading. Remark 3.13. (1) The endpoint grading at p ∈ M is the wight to a minimal cut-pathseparating p .(2) We have ε gr ( G ) = 3( t − ( G ) − t + ( G )) for G ∈ BWeb sl , Σ . For q ∈ N , consider the lattice L ( q ) := ker (cid:16) ( Z × Z ) M aug −−→ Z mod q −−−→ Z /q Z (cid:17) . Here aug(( k p , l p )) := P p ∈ M ( k p − l p ) . Then by (2) in the remark above, we have gr( G ) ∈ L (3) for any G ∈ BWeb sl , Σ .3.1.3. Elementary webs and web clusters. Let BWeb sl ,∂ × Σ ⊂ BWeb sl , Σ denote the set of boundary webs on Σ , that is, sl -webs consisting of oriented boundary intervals of Σ . Thefollowing notions are expected to be skein theoretic incarnations of some concepts in thequantum cluster algebra. Definition 3.14 (elementary webs) . A basis web G ∈ BWeb sl , Σ is called an elementaryweb if there are no basis webs G and G such that G = A k G G for some k ∈ Z . Wedenote the set of elementary webs by EWeb sl , Σ ⊂ BWeb sl , Σ . Definition 3.15 (web clusters) . A subset C ⊂ EWeb sl , Σ is called a web cluster if C isa maximal A -commutative subset in EWeb sl , Σ with cardinality I sl (∆) . We denote thecollection of web clusters by CWeb sl , Σ . Definition 3.16. For two elementary webs G , G ∈ EWeb sl , Σ contained in a commonweb cluster, define Π( G , G ) ∈ Z by G G = A Π( G ,G ) G G . The sl -skein algebra for a triangle. Let us consider a triangle T with specialpoints p , p , p and the unique triangulation ∆ T . The boundary web e ij is defined to bethe simple oriented arc from p i to p j . In this case, we have BWeb sl ,∂ × T = { e , e , e , e , e , e } . The sl -web t +123 (resp. t − ) is defined to be the flat trivalent graph with a trivalentsink (resp. source) vertex and univalent vertices p , p , p . Note that ∗ ( t +123 ) = t − . SeeFigure 3.4. Lemma 3.17. The complete list of relations among BWeb sl ,∂ × T ∪ { t +123 , t − } in S A sl ,T isgiven as follows. e e = e e , e e = e e , e e = e e ,e e = A − / [ e e ] , e e = A − [ e e ] ,e e = A − [ e e ] , e e = A − / [ e e ] ,e e = A / [ e e ] , e e = A [ e e ] ,e e = A [ e e ] , e e = A / [ e e ] ,e t +123 = A − / [ t +123 e ] , e t +123 = A / [ t +123 e ] ,e t − = A / [ t − e ] , e t − = A − / [ t − e ] , KEIN AND CLUSTER ALGEBRAS FOR sl e p p p e p p p e p p p e p p p e p p p e p p p t +123 p p p t − p p p Figure 3.4. Elementary webs in the triangle t +123 t − = A / [ e e e ] + A − / [ e e e ] ,t − t +123 = A − / [ e e e ] + A / [ e e e ] . Proof. Straightforward calculation by the skein relation and the boundary skein relation.We remark that we do not need to confirm all relations, thanks to the symmetries givenby the Dynkin involution, the mirror-reflection, and rotations of the triangle. (cid:3) Proposition 3.18. The skein algebra S A sl ,T is generated by BWeb sl ,∂ × T ∪ { t +123 , t − } asa Z A -algebra.Proof. For a flat trivalent graph [ G ] in T , relations (3.15)–(3.17) can be used to eliminate -, -, and -gons in Σ \ [ G ] . Obviously, these eliminations do not change the repre-senting basis web. A diagram of the resulting non-elliptic flat trivalent graph, describedin Figure 3.5, was explicitly given by Kim [Kim07] and Frohman–Sikora [FS20]. Herea strand labeled by a positive integer m means the m -parallelization of the strand; thewhite triangle with three strands labeled by l is a triangle web defined by ll l := ··· ··· ··· ··· · · ·· · · } l It can be seen that the triangle web in the left (resp. right) in Figure 3.5 is equal to A r ( t +123 ) l (resp. A r ( t − ) l ) for some r ∈ Z . Thus S A sl ,T is generated by { e , e , . . . , e , t +123 , t − } . (cid:3) Proposition 3.19. Let C (∆ T ,ǫ ) := BWeb sl ,∂ × T ∪ { t ǫ } for ǫ ∈ { + , −} . Then we have EWeb sl ,T = BWeb sl ,∂ × T ∪ { t +123 , t − } and CWeb sl ,T = { C (∆ T , +) , C (∆ T , − ) } .Proof. By the proof of Proposition 3.18, it suffices to show that t +123 and t − can notbe expressed as a product of basis webs. Otherwise, with a notice that the sum ~ gr of m m m n n n ll l m m m n n n ll l Figure 3.5. sl basis webs of S A sl ,T the endpoint grading is additive with respect to the multiplication in the skein algebra, ~ gr( t +123 ) = (3 , means that t +123 must be decomposed into a product of basis webs G and G with ~ gr( G ) = (2 , and ~ gr( G ) = (1 , . However, there exist no such pair ofbasis webs in EWeb sl ,T . Hence EWeb sl ,T = { t +123 , t − } ∪ BWeb sl ,∂ × T . It is easy to see that( t +123 , t − ) is the only pair which do not A -commute with each other from Lemma 3.17. (cid:3) We will use the following notation. Definition 3.20. For a subset S of BWeb sl , Σ , let h S i alg denote the subalgebra of S A sl , Σ generated by S , and mon( S ) the multiplicatively closed set generated by S ∪ { A ± / } in S A sl , Σ . Theorem 3.21 (Laurent expression in web clusters for a triangle) . For any x ∈ S A sl ,T and ǫ ∈ { + , −} , there exists ( t ǫ ) k ∈ mon( C (∆ T ,ǫ ) ) for some k ∈ N such that ( t ǫ ) k x ∈h C (∆ T ,ǫ ) i alg and it has positive coefficients if x ∈ BWeb sl ,T .Proof. By Propositions 3.18 and 3.19, any web x ∈ S A sl ,T can be written as a polyno-mial on the generators EWeb sl ,T . By Lemma 3.17, t +123 t − is expanded as a polynomialin BWeb sl ,∂ × T with positive coefficients. Moreover, t +123 A -commutes with the webs in BWeb sl ,∂ × T . Hence by multiplying a sufficiently large power ( t +123 ) k to x , we can replace t − ’s in each monomials in x with boundary webs, without changing the signs of the coef-ficients. The second assertion follows since each basis web is a monomial on EWeb sl ,T . (cid:3) We remark that the above propositions say that S A sl ,T is generated by EWeb sl ,T , andgenerated by “Laurent polynomials” in a cluster web C (∆ T ,ǫ ) for ǫ ∈ { + , −} .3.3. The sl -skein algebra for a quadrilateral. Let Q be a quadrilateral with specialpoints p , p , p , p . It has two triangulations ∆ (13) Q and ∆ (24) Q shown in Figure 3.6.In the same way as in the triangle case, we define boundary webs BWeb sl ,∂ × Q = { e , e , e , e , e , e , e , e } KEIN AND CLUSTER ALGEBRAS FOR sl p p p p ∆ (13) Q p p p p ∆ (24) Q Figure 3.6. Two sl -triangulations of a quadrilateral Q and introduce the following 16 sl -webs: e , e , e , e , t +124 , t +231 , t +342 , t +413 ,t − , t − , t − , t − , h , h , h , h . Let us denote the set of these webs by EWeb sl ,Q := { e ij , t ǫ , t ǫ , t ǫ , t ǫ , h i | i, j ∈ { , , , } , i = j, ǫ ∈ { + , −}} . We will show that it is exactly the set of elementary webs for Q in Proposition 3.25 soonbelow. Lemma 3.22. The complete list of relations among EWeb sl ,Q is given as follows : t +124 t +231 = A − [ t +124 t +231 ] , (3.18) t +124 t +342 = [ t +124 t +342 ] , (3.19) t +124 t +413 = A [ t +124 t +413 ] , (3.20) Here the left-hand sides of the above equations mean the multiplication of two elementary webs, thefirst web being located right above the second one. A collection of elementary webs depicted in the samequadrilateral mean the Weyl ordering of these webs. t +124 t − = A − [ e e e ] + A [ e e e ] , (3.21) t +124 t − = A − [ e e e ] + A [ e h ] , (3.22) t +124 t − = [ t +124 t − ] , (3.23) t +124 t − = A [ e e e ] + A [ e h ] , (3.24) t +124 e = A − [ e t +231 ] + A [ e t +413 ] , (3.25) t +124 e = A [ t +124 e ] , (3.26) t +124 e = [ t +124 e ] , (3.27) t +124 e = A − [ t +124 e ] , (3.28) t +124 h = A [ e e t +342 ] + A − [ e e t +231 ] , (3.29) t +124 h = A − [ e e t − ] + A [ e e t +413 ] , (3.30) KEIN AND CLUSTER ALGEBRAS FOR sl t +124 h = A [ t +124 h ] , (3.31) t +124 h = A − [ t +124 h ] , (3.32) e e = A [ e e ] + A − h , (3.33) e e = [ e e ] , (3.34) e e = A − [ e e ] + A h , (3.35) e h = A [ e h ] , (3.36) e h = A − [ e h ] (3.37) e h = A − [ t − t +413 ] + A [ e e e ] , (3.38) e h = A [ t +231 t − ] + A − [ e e e ] , (3.39) h h = A [ e e e e ] + A − [ e t − t +342 ] , (3.40) h h = [ e e e e ] + [ e e e e ] + [ e e e e ]+ A [ e e h ] + [ e e h ] , (3.41) h h = A − [ e e e e ] + A [ e t +231 t − ] . (3.42)Indeed, the relations between BWeb sl ,∂ × Q and the other webs follows from the boundaryskein relations (3.6), (3.7). The remaining relations are obtained by applying the Dynkininvolution and rotations of the quadrilateral to the above relations. Remark 3.23. The Weyl ordering of a sl -web appearing in the above relations can berepresented by a flat trivalent graph obtained by the following operations: ,. This operation is an inverse operation of the arborization in [FP16]. Proposition 3.24. The skein algebra S A sl ,Q is generated by EWeb sl ,Q as a Z A -algebra.Proof. We take a point p on an edge of Q and a point q on the opposite side. For anynon-elliptic flat trivalent graph G representing a basis web in BWeb sl ,Q , fix a minimalcut-path α of G from p to q . We remark that the minimal cut-path is non-convex tothe left and right sides by Definition 3.8, thus there exists left and right cores α L and α R Lemma 3.9 (2). An induction on | wt α ( G ) | will prove the proposition. The cases wt α ( G ) = 0 , are easy. Assume | wt α ( G ) | = n , by Lemma 3.9 (1), there is no cut-pathsrelated to α L (resp. α R ) by H -moves in the left (resp. right) side, and α L is related to α R by H -moves and identity moves. The explicit description of the basis webs on T inFigure 3.5 and the proof of Proposition 3.18 imply that there exists a trivalent graph G ′ of the form in the left of Figure 3.7 such that G = A k G ′ for some k . Here the web B in the biangle bounded by α L and α R is constructed by a concatenation of H -webs, asshown in the center of Figure 3.7. By applying the skein relations (3.6) and (3.7), these H -webs can be replaced by internal crossings up to multiplication by A and modulo webswith weights lower than n . Then the web B can be replaced with A • σ + P x λ x x , where x is a trivalent graph whose minimal cut-path between p and q has intersection pointsless than n , and σ is a non-twisting positive braid between α L and α R , as shown in theright of Figure 3.7. By substituting B = A • σ + P x λ x x into G ′ , we obtain an expression KEIN AND CLUSTER ALGEBRAS FOR sl Bp p p p pqG ′ ··· α L α R pqB ··· α L α R pqσ Figure 3.7. A web in G ′ G ′ = G ′ σ + P x λ x G ′ x and see that G ′ σ is described as a product of webs in EWeb sl ,Q . Theproof is finished by applying the induction hypothesis to G ′ x . By the proof, notice thatthe webs h and h are not needed for the generating set. (cid:3) By using the above proposition, we give the set of elementary webs and the collectionof web clusters for the quadrilateral. Proposition 3.25. EWeb sl ,Q = { e ij , t ǫ , t ǫ , t ǫ , t ǫ , h i | i, j ∈ { , , , } , i = j, ǫ ∈{ + , −}} is the set of elementary for Q .Proof. We use the same argument as in the proof of Proposition 3.19. Recall that if abasis web G is decomposed into a product G = G G , then ~ gr( G ) = ~ gr( G ) + ~ gr( G ) .On the other hand, we know the explicit generators of S A sl ,Q given in Proposition 3.24.Observe that ~ gr( t + ijk ) = (3 , , ~ gr( t − ijk ) = (0 , , ~ gr( e ij ) = (1 , , ~ gr( h j ) = (2 , . Hence except for the last h j ’s, one can easily see that these webs are indecomposable.Therefore we only have to care about the possibility that h j is decomposed into two websin { e , e , e , e } , but it is impossible by Lemma 3.22. Thus each web in EWeb sl ,Q is indecomposable. With a notice that each basis web appearing on the right-hand sideof expansions given in Lemma 3.22 is described as a product of webs in EWeb sl ,Q , weconclude that this set is exactly the set of elementary webs. (cid:3) Remark 3.26. We can also determine the web clusters by Lemma 3.22. In fact, CWeb sl ,Q consists web clusters. We will list CWeb sl ,Q up in the forthcoming paper [IY]. We will consider expansions of any webs in the five web clusters C ν = C ′ ν ∪ BWeb sl ,∂ × Q in CWeb sl ,Q for ν = 0 , , , , , where C ′ ν ’s are given as follows: C ′ = ( t +124 , t +342 , e , e ) , C ′ = ( t +124 , t − , e , e ) C ′ = ( t − , t +342 , e , e ) , C ′ = ( t +124 , t +342 , t +231 , e ) C ′ = ( t − , t +342 , e , t +413 ) . As clarified later in Section 4, the web clusters C , C , C , and C are “adjacent” to C bya mutation. It will mean that by replacing t +342 ∈ C with t − we get C , and they satisfythe relation t +342 t − = A / [ e e e ] + A − / [ e e e ] ∈ h C ∩ C i alg . (3.43)Similarly, t +124 t − = A / [ e e e ] + A − / [ e e e ] ∈ h C ∩ C i alg ,e t +231 = A / [ e t +342 ] + A − / [ e t +124 ] ∈ h C ∩ C i alg ,e t +413 = A / [ e t +124 ] + A − / [ e t +342 ] ∈ h C ∩ C i alg . The following lemma gives the “cluster expansion” of any web in S A sl , Σ in the webclusters C ν for ν = 0 , , , , . Proposition 3.27. For any web x ∈ S A sl ,Q and ν = 0 , , , , , there exists J ν ∈ mon( C ν ) such that xJ ν ∈ h C ν i alg .Proof. For the web clusters C , C , and C , the assertion is already proved in Corol-lary 3.29. An expansion in C gives an expansion on C via an automorphism induced KEIN AND CLUSTER ALGEBRAS FOR sl by a rotation of Q . Therefore we only need to obtain an expansion of elementary webs in S A sl ,Q in the web cluster C . For each elementary web in EWeb sl ,Q \ C = { e , e , e , t +413 , t − , t − , t − , t − , h , h , h , h } , we can expand it as a polynomial in C by the right multiplication of webs in C ′ . Indeed,we have e t +231 ∈ * t +124 , t +342 + alg ⊂ h C i alg , (3.44) t +413 e ∈ * t +124 , t +342 + alg ⊂ h C i alg , (3.45) h t +124 ∈ * t +124 , t +342 + alg ⊂ h C i alg ,h t +342 ∈ * t +124 , t +342 + alg ⊂ h C i alg . Similarly, one can confirm the followings by a straightforward computation: e t +124 e ! ∈ h C i alg , e t +342 e ! ∈ h C i alg ,t − t +124 t +231 ! ∈ h C i alg , t − t +342 t +231 ! ∈ h C i alg ,t − t +124 t +342 ! ∈ h C i alg ,h e t +124 t +124 ! ∈ h C i alg , h e t +342 t +124 ! ∈ h C i alg ,t − t +231 t +124 t +342 e ! ∈ h C i alg . For example, the most complicated one will be an expansion of t − ( t +231 t +124 t +342 e ) . Firstly, t +413 e ∈ h C i alg by eq. (3.44). Since e t +124 is a sum of e e t +231 and e t +413 , we get e ( t +124 e ) ∈ h C i alg by eq. (3.45). In the same way, e ( t +342 e ) ∈ h C i alg . By Lemma 3.17, t − t +231 is expanded as a sum of e e e and e e e . We remark that e e e and t +342 are A -commutative, so are e e e and t +124 . Thus we get t − ( t +231 t +124 t +342 e ) ∈h C i alg . (cid:3) Laurent expressions, positivity and localized skein algebras. Based on theexpansion results in triangles and quadrilaterals obtained in the previous subsections, weare going to give an expansion result in a general marked surface and discuss the positivityof the coefficients.3.4.1. Laurent expression. Let ∆ = (∆ , s ∆ ) be a decorated triangulation of Σ . For eachtriangle T ∈ t (∆) , we can naturally regard the set of elementary webs EWeb sl ,T as asubset of BWeb sl , Σ . Similarly we regard each web cluster C (∆ T , s ∆ ( T )) ∈ CWeb sl ,T as asubset of BWeb sl , Σ .We are going to show that any sl -web x ∈ S A sl , Σ can be expressed as a Laurentpolynomial in the web cluster C ∆ := ∪ T ∈ t (∆) C (∆ T , s ∆ ( T )) ⊂ BWeb sl , Σ . Let mon(∆) := mon( EWeb sl ,∂ × T ) ⊂ S A sl , Σ be the multiplicatively closed set generated by the elementary webs along the edges of ∆ and A / . Theorem 3.28 (Expansions in elementary webs on triangles) . For any web x ∈ S A sl , Σ anda triangulation ∆ , there exists a monomial J ∆ ∈ mon(∆) such that xJ ∆ ∈ h∪ T EWeb sl ,T i alg .Proof. Let [ G ] be a ∆ -transverse flat trivalent graph in (Σ , M ) . Here a flat trivalent graphis said to be ∆ -transverse if it has only finitely many transverse intersection points with ∆ , which lie in Σ \ ∂ Σ . For an edge E ij = p i p j ∈ e (∆) connecting p i and p j , assumewe have G ] ∩ E ij = n and let γ , γ , . . . , γ n be the corresponding subarcs of [ G ] in aneighborhood of E ij . By applying the skein relations, the product γ k [ e ij e ji ] is computedas γ k = A G + k + G k + A − G − k (3.46) KEIN AND CLUSTER ALGEBRAS FOR sl On the other hand, it is easy to see that [ e ij e ji ] is A -commutative with the sl -webs G ǫk appearing in the right-hand side from the boundary sl -skein relation. Hence for any ℓ ≥ , we have ( A G + k + G k + A − G − k )[ e ij e ji ] ℓ = [ e ij e ji ] ℓ ( A r + G + k + A r G k + A r − G − k ) forsome r + , r , and r − . Therefore we can independently apply the above computation to all γ k for k = 1 , , . . . n . Thus the product G [ e ij e ji ] n is expanded into a sum of webs withouttransverse intersection points with E ij .Any sl -web x ∈ S A sl , Σ can be written as a sum x = P G λ G [ G ] of ∆ -transversenon-elliptic flat trivalent graphs. For each E ∈ e (∆) , consider the non-negative inte-ger n E ( x ) := max { G ] ∩ E | λ G = 0 } . By applying the above computation, the product x Y E ij ∈ ∆ ( e ij e ji ) n Eij ( x ) is expanded into a polynomial of webs in triangles of ∆ , which can be further writtenas a polynomial in ∪ T ∈ t (∆) EWeb sl ,T by Proposition 3.18. Thus we get the assertion with J ∆ := Q E ij ∈ ∆ ( e ij e ji ) n Eij ( x ) . (cid:3) Naively, the above theorem tells us that the web x has a Laurent expression x = f ∆ J − with f ∆ ∈ h∪ T EWeb sl ,T i alg . This will be made more precise in Section 3.4.3.Let us further consider the multiplicatively closed set mon( C ∆ ) , which obviously con-tains mon(∆) . Corollary 3.29 (Expansions in the web cluster C ∆ ) . For any web x ∈ S A sl , Σ and adecorated triangulation ∆ , there exists a monomial J ∆ ∈ mon( C ∆ ) such that xJ ∆ ∈h C ∆ i alg .Proof. By Theorem 3.28, there exists a monomial J ′ ∆ ∈ mon(∆) such that xJ ′ ∆ ∈ h∪ T EWeb sl ,T i alg .In the same way as the proof of Theorem 3.21, by multiplying ( e v s ( T ) ) k T to xJ ′ ∆ , we canreplace ∗ ( e v s ( T ) ) for each v ∈ I tri (∆) ∩ T with elemenatary webs along the edges. Here k T is the degree of ∗ ( e v s ( T ) ) in the polynomial xJ ′ ∆ , and note that any monomial contain-ing no ∗ ( e v s ( T ) ) is A -commutative with e v s ( T ) . Thus xJ ′ ∆ Q T ∈ t (∆) ( e v s ( T ) ) k T is contained in h C ∆ i alg , and we get the assertion with J ∆ := J ′ ∆ Q T ∈ t (∆) ( e v s ( T ) ) k T ∈ mon( C ∆ ) . (cid:3) Laurent positivity for elevation-preserving webs. We are going to show that theLaurent expressions of webs of certain kind, which we call the elevation-preserving sl -webs , in S A sl , Σ [∆ − ] have positive coefficients. By arguing as in the proof of Corollary 3.29,it implies that the Laurent expressions in the web cluster C ∆ have also positive coefficients.Elevation-preserving sl -webs include the bracelets and the bangles along an orientedsimple closed curve.For an ideal triangulation ∆ of Σ , let ∆ split be the associated splitting triangulation obtained by replacing each edge of ∆ with doubled edges as shown in Figure 3.8. Theset of connected components of Σ \ ∆ split is divided into two subsets: the set t (∆ split ) of triangles and the set b (∆ split ) of biangles. We can canonically identify t (∆ split ) with t (∆) , and b (∆ split ) with e (∆) . We denote a triangle in t (∆ split ) by the same symbol asthe corresponding triangle in t (∆) , while the biangle corresponding to an edge E ∈ e (∆) is denoted by B E ∈ b (∆ split ) . For an edge E ∈ e (∆) and a triangle T ∈ t (∆) adjacent to E , let E T ∈ e (∆ split ) denote the edge shared by B E and T . Definition 3.30. (1) A fundamental piece in T ∈ t (∆ split ) consists of a superpositionof trivalent graphs with at most one trivalent vertex and distinct endpoints on ∂ × T such that endpoints of the same connected component lies in distinct connectedcomponents of ∂ × T each other. An elevation of a fundamental piece of T is alabeling of its connected components by positive integers. See left and right ofFigure 3.9.(2) Let E ∈ e (∆) be an edge shared by T and T ′ in t (∆) . An elevation-preservingbraid in B E ∈ b (∆ split ) connecting fundamental pieces with elevations in T and T ′ is a braid between E T and E T ′ such that • the braid consists of a superposition of strands connecting endpoints of fun-damental pieces of T and T ′ ; • for any strands α and β of the braid, α ( T ) ≤ β ( T ) if and only if α ( T ′ ) ≤ β ( T ′ ) ; • a strand α passes above another strand β if α ( T ) > β ( T ) or α ( T ′ ) > β ( T ′ ) ;where α ( T ) (resp. β ( T ) ) denotes the elevation on the endpoint of α (resp. β ) in E T induced from the fundamental piece with the elevation in T , and similarly for T ′ . See the middle of Figure 3.9.(3) Let E ∈ B be a boundary interval and T ∈ t (∆) the adjacent triangle. An elevation-preserving braid in B E ∈ b (∆ split ) consists of elevation-preserving arcswith no internal crossings connecting E T to one of the tow special points of B E .We define a certain sl -web which satisfies positivity by concatenating fundamentalpieces with elevations by elevation-preserving braids. Definition 3.31 (elevation-preserving sl -webs) . A tangled trivalent graph in Σ is said tobe elevation-preserving with respect to ∆ if it can be decomposed into fundamental piecesin triangles and elevation-preserving braids in biangles connecting them by cutting along e (∆ split ) . An elevation-preserving sl -web is an sl -web such that it is represented by anelevation-preserving graph with respect to some ∆ . Example 3.32. For a triangulation ∆ , a simple trivalent graph obtained by attachingfundamental pieces to the triangles t (∆ split ) with no internal crossings and connectingthem by identity braids in biangles in b (∆ split ) gives an elevation-preserving trivalenttangle with respect to ∆ . In particular, oriented simple loops and oriented simple arcs areelevation-preserving for any ∆ . For any triangulation ∆ , the n -bracelet along a (non-nullhomotopic) simple loop γ ( Figure 1.1) is obtained from the n -bangle of γ by replacingthe identity n -braid in some biangle by a braid corresponding to a cyclic permutation (12 · · · n ) . See Figure 3.10. Theorem 3.33. For any elevation-preserving web x ∈ S A sl , Σ with respect to a triangula-tion ∆ , there exists J ∆ ∈ mon(∆) such that the expansion of xJ ∆ in h∪ T EWeb sl ,T i alg haspositive coefficients. KEIN AND CLUSTER ALGEBRAS FOR sl ET T ′ ∆ B E E T E T ′ T T ′ ∆ split Figure 3.8. The split triangulation ∆ split associated with ∆ . T B E T ′ Figure 3.9. Fundamental pieces in triangles T and T ′ , and an elevation-preserving braid in the biangle B E connecting them. Elevations are pre-sented by positive integers. Proof. Let x be an elevation-preserving web with respect to ∆ . Our strategy of the proof isthe following. Firstly, we decompose x into webs in triangles of ∆ split as in Theorem 3.28,in order of increasing elevation. Next, expand the remaining part in biangles. Notice thatthe right-hand sides of the sl -skein relations have positive coefficients, except for (3.4),and we can avoid using this relation in the above process. Therefore we can observe thatthe coefficients in these expansions are positive.Let us describe the details of the proof. We focus on a piece G of x in T ∈ t (∆ split ) and represent it by G = G n . . . G G as a superposition of connected components, wheresubscripts indicate their elevations. Here G is the connected component of G of the lowestelevation, and each G i is an arc or a trivalent graph with a single vertex. Let { p , p , p } be the three special points of T , E ij the edge between p i and p j , and B ij := B E ij . We willuse the notation in Section 3.2 for the elementary webs in T . γ γ Figure 3.10. The left-hand side shows a portion of the diagram of anoriented simple loop γ in a quadrilateral in of a triangulation ∆ of Σ . Theright-hand side shows the associated -bracelet.Firstly, expand G by multiplying [ e e e e e e ] ∈ mon(∆) and by using eq. (3.46).We remark that one can omit multiplying one of e ij e ji if G is an arc. In a neighborhoodof each edge, the resulting diagrams in the expansion are decomposed into diagrams inthe three parts: in the biangle part (shwon as a shaded region), on the edge part , and the triangle part ( i.e. , the interior of T ) as follows. For a strand incoming to T , T = A T + T + A − T , where the bottom-half belongs to a biangle part and the top-half does to one of thetriangle parts covering T . The oriented edges betwenn the special points belong to theedge part. For a strand outgoing from T (obtained by applying the Dynkin involution tothe incoming case), T = A T + T + A − T . Then the webs in the expansion of G [ e e e e e e ] in T are obtained by concatenat-ing the pieces in the three sectors T , T , T shown in Figure 3.11, and their coefficientsare one of { , A ± , A ± } . In order to list up the concatenation patterns, let us denote theresulting webs in each sector by X + ij = p i p j T ij , X ij = p i p j T ij , X − ij + T ij . KEIN AND CLUSTER ALGEBRAS FOR sl T T T p p p Figure 3.11. Concatenation of webs in three region X, Y, Z gives websappearing in the expansion on T .If G is an arc connecting the edges E ij and E jk , we just concatenate X ǫij and ∗ X ǫ ′ jk . If G is a trivalent graph, the concatenation of the three pieces X ǫ , X ǫ ′ , X ǫ ′′ produce one newsink or source vertex at the center of T . It is easy to confirm that these concatenationsproduce the following webs. In the case that G is an arc, we get X +12 ⋆ ( ∗ X +23 ) = e , X +12 ⋆ ( ∗ X ) = t +123 , X +12 ⋆ ( ∗ X − ) = e ,X ⋆ ( ∗ X +23 ) = 0 , X ⋆ ( ∗ X ) = [ e e ] , X ⋆ ( ∗ X − ) = t − ,X − ⋆ ( ∗ X +23 ) = 0 , X − ⋆ ( ∗ X ) = 0 , X − ⋆ ( ∗ X − ) = e ,X +12 ⋆ ( ∗ X +31 ) = e , X +12 ⋆ ( ∗ X ) = 0 , X +12 ⋆ ( ∗ X − ) = 0 ,X ⋆ ( ∗ X +31 ) = t +123 , X ⋆ ( ∗ X ) = [ e e ] , X ⋆ ( ∗ X − ) = 0 ,X − ⋆ ( ∗ X +31 ) = e , X − ⋆ ( ∗ X ) = t − , X − ⋆ ( ∗ X − ) = e , where X ⋆ Y means the concatenation of X and Y . In the case that G is a trivalentgraph, we get X +12 ⋆ X +23 ⋆ X +31 = t +123 , X +12 ⋆ X +23 ⋆ X = [ e e ] ,X +12 ⋆ X +23 ⋆ X − = 0 , X +12 ⋆ X ⋆ X = [ e t − ] ,X +12 ⋆ X ⋆ X − = [ e e ] , X +12 ⋆ X − ⋆ X − = 0 ,X ⋆ X ⋆ X = [ t +123 t +123 ] , X ⋆ X ⋆ X − = [ t +123 e ] ,X ⋆ X − ⋆ X − = [ e e ] , X − ⋆ X − ⋆ X − = t +123 , where X ⋆ Y ⋆ Z means the concatenation of X , Y and Z . Here we have applied someskein relations. For example, X +12 ⋆ X ⋆ X = = . The above calculation shows that the triangle part of G [ e e e e e e ] in T ∈ t (∆ split ) is expanded as a polynomial in EWeb sl ,T with positive coefficients. The webs appearingin this expansion A -commute with webs along the edges of T , since the biangle part, edgeapart, and triangle part A -commute with each other. Therefore in the product G [ e e e e e e ] = G n · · · G G [ e e e e e e ] = G n · · · G ( G [ e e e e e e ])[ e e e e e e ] , we can move [ e e e e e e ] to the left beyond G [ e e e e e e ] preserving thepositivity of coefficients in the expansion, and hence the component G can be expandedin the same way. Proceeding in this way, one can decompose G [ e e e e e e ] n intowebs in T ∈ ∆ split and webs outside of T with positive coefficients.Applying this operation to x for all T ∈ ∆ split , we obtain a positive sum of webs suchthat KEIN AND CLUSTER ALGEBRAS FOR sl • their triangle parts (and edge parts) in T ∈ t (∆ split ) are expressed as monomialsin EWeb sl ,T , • concatenations of elevation-preserving braids and biangle parts produced in theexpansion procedure.It remains to show that a web in each biangle becomes a polynomial in EWeb sl ,T , es-pecially EWeb sl ,∂ × , with positive coefficients. In the expansion of G , webs in a bianglepart adjacent to T inherits the elevation from the fundamental piece in T . Hence fromthe elevation-preserving assumption, for adjacent triangles T, T ′ ∈ t (∆ split ) , the biangleparts of T and T ′ can be connected by strands with preserving their elevations. Hence,the concatenation of biangle parts and the elevation-preserving braid is presented as asuperposition of the concatenation of biangle parts, as listed below: p p = e , p p = 0 , p p = 0 , p p = 0 , p p = [ e e ] , p p = 0 , p p = 0 , p p = 0 , p p = e . Consequently, x is decomposed into a sum of monomials in ∪ T ∈ t (∆) EWeb sl ,T such that itscoefficients are positive Laurent polynomial in Z A . (cid:3) Corollary 3.34. Let ∆ = (∆ , s ∆ ) be a decorated triangulation of Σ . Then, for anyelevation-preserving web x ∈ S A sl , Σ with respect to ∆ , there exists J ∆ ∈ mon( C ∆ ) suchthat the expansion of xJ ∆ in h C ∆ i alg has positive coefficients.Proof. By Theorem 3.33, an elevation-preserving web is expanded as a positive polynomialin ∪ T ∈ t (∆) EWeb sl ,T by multiplying an appropriate element in mon(∆) . In a similar wayto Corollary 3.29, we can further expand it as a polynomial in a web cluster C ∆ withpositive coefficients by multiplying appropriate elemenatary webs in triangles. (cid:3) Corollary 3.35. Let γ be an oriented simple loop in Σ . Then for any ∆ and n ∈ N , the n -bracelet and n -bangle are expressed as polynomial with positive coefficients in h C ∆ i alg by multiplying some monomial in mon( C ∆ ) . The ∆ -lacalization of S A sl , Σ . In Theorems 3.28 and 3.33, we expanded any sl -websin S A sl , Σ by multiplying some monomials. We are going to see that these expansion giverise to expressions of sl -webs as Laurent polynomials in suitable localizations of S A sl , Σ . Lemma 3.36. The multiplicatively closed set mon(∆) in S A sl , Σ satisfies the Ore condition. Proof. We first show the right Ore condition that for any web x ∈ S A sl , Σ and monomial J ∈ mon(∆) , there exist a web x ′ ∈ S A sl , Σ and a monomial J ′ ∈ mon(∆) such that xJ ′ = J x ′ . By Theorem 3.28, there exist a monomial J ′′ ∈ mon(∆) such that xJ ′′ = X f λ f e f ∈ h∪ T ∈ t (∆) EWeb sl ,T i alg where f : ∪ T ∈ t (∆) EWeb sl ,T → N and e f := Q i ∈ I (∆) e f ( i ) i . Since any monomial in mon(∆) is A -commutative with e f by Lemma 3.17, we obtain the following: xJ ′′ J = X f λ f e f ! J = X f λ f e f J = X f λ f A n ( f,J ) J e f = J X f A n ( f,J ) λ f e f ! , where n ( f, J ) is some half integer. In other words, xJ ′ = J x ′ holds with J ′ := J ′′ J and x ′ := P f A n ( f,J ) λ f e f . By applying the mirror-reflection † , we see that the left Orecondition also holds. (cid:3) Definition 3.37 (the localized sl -skein algebras for (Σ , M ) ) . The ∆ -localized skein al-gebra S A sl , Σ [∆ − ] is the Ore localization of S A sl , Σ by the Ore set mon(∆) . Similarly, the ∂ -localized skein algebra S A sl , Σ [ ∂ − ] is the Ore localization by mon( EWeb sl ,∂ × Σ ) .The following theorem guarantees the existence of the skew-field of fractions Frac S A sl , Σ and embeddings of the above localizations of S A sl , Σ . Theorem 3.38 ([Yua]) . S A sl , Σ is an Ore domain. Here we give a sketch of the proof. Let us consider a ( N ) e (∆) × N t (∆) -filtration { F ( e ∆ , t ∆ ) S A sl , Σ } such that the first component e ∆ is given by weights of basis webs onedges of ∆ , and the second t ∆ is given by a function of trivalent vertices and edges in atriangle. By considering a certain partial order on this filtration, we can define the as-sociated graded algebra gr ∆ S A sl , Σ := L e ∆ , t ∆ F ( e ∆ , t ∆ ) S A sl , Σ /F − ( e ∆ , t ∆ ) S A sl , Σ . In this gradedalgebra, all basis webs are A -commutative. We can also prove that S A sl , Σ is generated by afinite subset of BWeb sl , Σ . Then we conclude that gr ∆ S A sl , Σ is a quantum affine space, thusa Noetherian domain. A Noetherian domain is known to be Ore [GW04, Cororally 6.7]and a filtered algebra inherit this property from its associated graded algebra. Corollary 3.39. We have inclusions S A sl , Σ ⊂ S A sl , Σ [ ∂ − ] ⊂ S A sl , Σ [∆ − ] ⊂ Frac S A sl , Σ . Quantum cluster algebras and Skein algebras Realization of the quantum surface subalgebra in the skein algebra. In thissection, we construct a quantum seed pattern in the skew-field Frac S A sl , Σ of fractions ofthe skein algebra S A sl , Σ , which quantizes the seed pattern s ( sl , Σ) . In what follows, weidentify the quantum parameters as q = A .For any vertex ω ∈ Exch sl , Σ of the surface subgraph, we are going to define a quan-tum seed ( B ( ω ) , Π ( ω ) , M ( ω ) ) in Frac S A sl , Σ . The exchange matrix B ( ω ) is the one already KEIN AND CLUSTER ALGEBRAS FOR sl k k ik k e k e k e k e k e j Figure 4.1. The neighboring vertices to i ∈ I edge ( ∆ ) (left) and the corre-sponding collection of elementary webs (right). An additional elementaryweb e j is also shown in purple.defined in Section 2.2. In order to define the remaining data, we consider a web cluster(Definition 3.15) C ( ω ) = { e ( ω ) i | i ∈ I ( ω ) } defined as follows. See Figure 4.4. • Suppose ω = ∆ = (∆ , s ∆ ) is a decorated triangulation. If i ∈ I tri (∆) , then e ∆ i isone of the elementary webs on the corresponding triangle T ∈ t (∆) . If s ∆ ( T ) = + (resp. s ∆ ( T ) = − ), then it is defined to be the one with the unique trivalentsink (resp. source). If i ∈ I edge (∆) , then e ∆ i is one of the elementary webs givenby assigning an orientation to the edge on which i is located. The orientation isdetermined so that the terminal point is closer to the vertex i . • For a decorated cell decomposition ω obtained by the mutation µ i for i ∈ I edge (∆) from a decorated triangulation ∆ , we set e ( ω ) j := e ∆ j for j = i . Define e ( ω ) i tobe the trivalent sink with endpoints three of the special points on the uniquequadrilateral, which span a triangle that contains i in its interior. • For a decorated cell decomposition ω obtained by the mutation µ op for i ∈ I edge (∆) from the decorated cell decomposition ω ′ := µ i ( ∆ ) , we set e ( ω ) j := e ( ω ′ ) j for j = i op .Define e ( ω ) i op to be the trivalent sink with endpoints three of the special points onthe unique quadrilateral, which span a triangle that contains i op in its interior.Then define the compatibility matrix Π ( ω ) = ( π ( ω ) ij ) i,j ∈ I ( ω ) by π ( ω ) ij := Π( e ( ω ) i , e ( ω ) j ) . Here recall Definition 3.16. Then Π ( ω ) is evidently skew-symmetric. Proposition 4.1. For any decorated triangulation ∆ = (∆ , s ∆ ) with s ∆ ( T ) = + for all T ∈ t (∆) , the pair ( B ∆ , Π ∆ ) satisfies the compatibility relation ( B ∆ ) T Π ∆ = (6 · Id , . Proof. During the proof, we fix a decorated triangulation ∆ and omit the superscript ∆ .Let ε := B T denote the quiver exchange matrix associated with ∆ . For i ∈ I ( ∆ ) uf and ik k k k k k e k e k e k e k e k e k e i e j Figure 4.2. The neighboring vertices to i ∈ I tri ( ∆ ) (left) and the corre-sponding collection of elementary webs (right). An additional elementaryweb e j is also shown in purple. j ∈ I ( ∆ ) , we are going to compute ( ε Π) ij = P k ∈ I ( ∆ ) ε ik π kj . Let us divide into the cases i ∈ I edge (∆) and i ∈ I tri (∆) . The case i ∈ I edge (∆) : Let Q be the quadrilateral having E as its diagonal. Label theneighboring vertices of the quiver as in Figure 4.1. In this case, we have ( ε Π) ij = X ν =1 ( − ν π k ν j . If j lies on an edge outside of Q , then one can easily see that ( ε Π) ij = 0 . One example ofthe elementary web e j corresponding to such a vertex is shown in the right of Figure 4.1.For this example, we have π k j = π k j = +1 and thus ( ε Π) ij = ( − + ( − = 0 . Thisis also the case for the vertices lying on the left and the bottom edges in Figure 4.1, sinceeach entry of the compatibility matrix is defined as the sum of the contribution from eachend. If j lies on the face of a triangle outside of Q , then we get ( ε Π) ij = 0 by a similarconsideration. The remaining entries are computed as follows: ( ε Π) ik = ( − · (+2) + ( − · (+1) + ( − · (+1 − 2) = 0 and similarly for j = k op1 , k , k op2 ; ( ε Π) ik = ( − · ( − 1) + ( − · ( − − · ( − and similarly for j = k ; ( ε Π) ii op = ( − · ( − 1) + ( − · (+1) + ( − · ( − − · ( − , and finally ( ε Π) ii = ( − · ( − 2) + ( − · (+2) + ( − · (+1 − 2) + ( − · ( − . KEIN AND CLUSTER ALGEBRAS FOR sl The case i ∈ I tri (∆) : Label the neighboring vertices of the quiver as in Figure 4.2. Inthis case, we have ( ε Π) ij = X ν =1 ( − ν π k ν j . Then a similar computation shows that the matrix entries ( ε Π) ij vanishes except for ( ε Π) ii = − · (+1 − 2) + 3 · (+2 − 1) = 6 . Thus ( B ( ω ) , Π ( ω ) ) is a compatible pair when ω = (∆ , s ∆ ) is a decorated triangulation with s T = + for all T ∈ t (∆) . (cid:3) The check of the compatibility relation for a general ω ∈ Exch sl , Σ is postponed untilthe proof of Theorem 4.3 below. For any vertex ω ∈ Exch sl , Σ , define a toric frame M ( ω ) : ◦ Λ ( ω ) → Frac S A sl , Σ by sending the basis vector f ( ω ) i to the corresponding elementary web e ( ω ) i , and extendingby M ( ω ) N X i =1 x i f ( ω ) i ! := [( e ( ω )1 ) x . . . ( e ( ω ) N ) x N ] by using the Weyl ordering (Definition 3.6) for an auxiliary linear ordering I ( ω ) ∼ = { , . . . , N } . Note that this is the same extension rule as (2.4), and hence we get: Lemma 4.2. For any vertex ω ∈ Exch sl , Σ , the pair (Π ( ω ) , M ( ω ) ) satisfies M ( ω ) ( α ) M ( ω ) ( β ) = q Π ( ω ) ( α,β ) / M ( ω ) ( α + β ) for α, β ∈ ◦ Λ ( ω ) with q = A . Theorem 4.3. For any vertex ω ∈ Exch sl , Σ , the triple ( B ( ω ) , Π ( ω ) , M ( ω ) ) is a quantumseed in Frac S A sl , Σ . These quantum seeds are mutation-equivalent to each other.Proof. By Proposition 4.1 and Lemma 4.2, the triple ( B ∆ , Π ∆ , M ∆ ) associated with a dec-orated triangulation ω = ∆ is a quantum seed. Here the condition Frac T ∆ = Frac S A sl , Σ follows from Corollary 3.29. We have also seen that the exchange matrices B ( ω ) are relatedto each other by matrix mutations.We are going to first show that the toric frames M ( ω ) are related to each other by thequantum exchange relations (2.5). By the connectivity of the surface subgraph Exch sl , Σ and symmetry, it suffices to consider the toric frames associated with two vertices ω and ω ′ connected by an edge of the following three types.(1) The first mutation from a decorated triangulation, where ω = (∆ , s ∆ ) is a deco-rated triangulation and ω ′ = µ i ( ω ) for a vertex i ∈ I edge (∆) .(2) The second mutation from a decorated triangulation, where ω = µ i op ( ∆ ) and ω ′ = µ i µ i op ( ∆ ) for some decorated triangulation ∆ = (∆ , s ∆ ) and a vertex i ∈ I edge (∆) . (3) A change of a sign at a triangle T , where ω = (∆ , s ∆ ) and ω ′ = (∆ , s ∆ ′ ) areboth decorated triangulations with the same underlying triangulation but with s ∆ ( T ) = + , s ∆ ′ ( T ) = − and s ∆ ( T ′ ) = s ∆ ′ ( T ′ ) for T ′ ∈ t (∆) \ { T } .During the proof, we simply denote the elementary webs by e j := e ( ω ) j and e ′ j := e ( ω ′ ) j ineach case.In the first case, label the vertices of the quiver as in the left in Figure 4.1. Then weneed to check the quantum exchange relation e ′ i = M ( ω ) ( − f i + f k + f k ) + M ( ω ) ( − f i + f k + f k ) (4.1)holds in Frac S A sl , Σ . Using the relation M ( ω ) ( α + β ) = q Π ( ω ) ( α,β ) / M ( ω ) ( β ) M ( ω ) ( α ) whichfollows from the definition of the toric frame, this is equivalent to e i e ′ i = q − / M ( ω ) ( f k + f k ) + q / M ( ω ) ( f k + f k )= q − / [ e k e k ] + q / [ e k e k ] , This is nothing but the (3.25) (with a suitable change of labelings).In the second case, label the vertices of the quiver as in the left of Figure 4.3. Theexpected quantum exchange relation is: e ′ i = M ( ω ) ( − f i + f k + f k ) + M ( ω ) ( − f i + f k + f k )= q − / e − i M ( ω ) ( f k + f k ) + q / e − i M ( ω ) ( f k + f k )= e − i ( q − / [ e k e k ] + q / [ e k e k ]) . This is again the relation (3.25).In the third case, label the vertices of the quiver on T as in the left of Figure 4.2. Theexpected quantum exchange relation is e ′ i = M ( ω ) ( − f i + f k + f k + f k ) + M ( ω ) ( − f i + f k + f k + f k )= e − i ( q − / M ( f k + f k + f k ) + q / M ( f k + f k + f k ))= e − i ( q − / [ e k e k e k ] + q / [ e k e k e k ]) This is the relation (3.21). Thus the toric frames M ( ω ) are related to each other by thequantum exchange relations.Then it follows from Lemma 4.2 and Lemma 2.4 that the pair ( B ( ω ) , Π ( ω ) ) satisfies thecompatibility relation for all ω ∈ Exch sl , Σ . The assertion is proved. (cid:3) It follows from the above theorem that the quantum seeds ( B ( ω ) , Π ( ω ) , M ( ω ) ) generatesa canonical quantum seed pattern s q ( sl , Σ) , whose quantum cluster algebra A s q ( sl , Σ) isrealized in Frac S A sl , Σ . Let A q sl , Σ ⊂ A s q ( sl , Σ) denote the Z q -subalgebra generated by theunion of the quantum clusters A ( ω ) and the inverses of frozen variables in A f( ω ) associatedwith the vertices ω ∈ Exch sl , Σ of the surface subgraph, which we call the quantum surfacesubalgebra . Now we are ready to prove our comparison result of the skein and clusteralgebras. KEIN AND CLUSTER ALGEBRAS FOR sl ik k k k e k e k e k e k Figure 4.3. The neighboring vertices to i ∈ I edge ( µ i op ( ∆ )) (left) and thecorresponding collection of elementary webs (right). ∆ e ∆ i op e ∆ i ∆ ′ e ∆ ′ j e ∆ ′ j op µ i µ i op µ i op µ i µ j µ j op µ j op µ j Figure 4.4. The web clusters associated with the flip sequence in Fig-ure 2.3. Here the relevant elementary webs are just overwritten, not mean-ing the simultaneous crossing. The elementary webs on the boundary ofthe quadrilateral are omitted. Theorem 4.4. We have an M C (Σ) × Out( SL ) -equivariant inclusions A q sl , Σ ⊂ S A sl , Σ ( A )[ ∂ − ] ⊂ U s q ( sl , Σ) with q = A . Moreover, the bar-involution on U s q ( sl , Σ) restricts to the mirror-reflection on S A sl , Σ [ ∂ − ] .Proof. Since each cluster variable in A q sl , Σ is realized as an elementary web M ( ω ) ( f ( ω ) i ) ∈ S A sl , Σ for some ω ∈ Exch sl , Σ by Theorem 2.11, we get the first inclusion. Here note thatthe inverses of the frozen variables are contained in A q sl , Σ .In order to prove the second inclusion, thanks to Theorem 2.7, it suffices to check theinclusion S A sl , Σ ⊂ U s q ( sl , Σ) ( ∆ ) for a decorated triangulation ∆ . Moreover, note that allthe vertices of Exch s ( sl , Σ) adjacent to ∆ are decorated cell decompositions. Therefore itsuffices to see that for any x ∈ S A sl , Σ and a decorated cell decomposition ω , there existsa monomial J ( ω ) of elementary webs in the web cluster C ( ω ) such that xJ ( ω ) ∈ h C ( ω ) i alg .When ω = ∆ is a decorated triangulation, it is exactly Corollary 3.29. Indeed, wehave seen that by multiplying a product of elementary webs e ∆ i for i ∈ I edge(∆) to x , wecan decompose it into a sum of webs in triangles ( i.e. , a product of e ∆ j ’s and ∗ ( e ∆ j ) ’sfor j ∈ I tri (∆) ), and these webs are further decomposed into a polynomial in C ∆ bymultiplying some product of ∗ ( e ∆ j ) ’s.When ω is a decorated cell decomposition of deficiency , the assertion follows fromProposition 3.27. Indeed, we can similarly decompose x ∈ S A sl , Σ into a sum of webs intriangles and the unique quadrilateral, and the latter webs can be expanded in the webcluster associated with ω . Thus we get the inclusion S A sl , Σ [ ∂ − ] ⊂ U s q ( sl , Σ) ( ∆ ) = U s q ( sl , Σ) as desired.The equivariance is easily seen by comparing the descriptions of the group actions givenin Section 2.2 and Section 3.1.2. To compare the bar-involution and the mirror-reflection,note that they coincide on the elementary webs associated with a decorated triangulation.Then the assertion follows from the cluster expansion (Corollary 3.29). (cid:3) We move on to the comparison of the two notions of gradings. Recall the lattice L (3) := ker(( Z × Z ) M aug −−→ Z mod −−−→ Z ) , where aug(( k p , l p ) p ∈ M ) := P p ∈ M ( k p − l p ) . For any ω ∈ Exch sl , Σ , take the associated webcluster ( e ( ω ) i ) i ∈ I ( ω ) and consider the map end ( ω ) : ◦ Λ ( ω ) → L (3) , f ( ω ) i gr( e ( ω ) i ) . Lemma 4.5. For an edge ω k −−− ω ′ in Exch sl , Σ , we have end ( ω ′ ) = end ( ω ) ◦ µ ∗ k,ǫ for each ǫ ∈ { + , −} .Proof. It suffices to check the relation end ( ω ′ ) ( f ( ω ′ ) k ) = end ( ω ) ( µ ∗ k,ǫ ( f ( ω ) k )) , which is equivalentto gr( e ( ω ′ ) k ) = gr ( e ( ω ) k ) − Y j ∈ I ( ω ) ( A ( ω ) j ) [ ǫb ( ω ) jk ] + , or KEIN AND CLUSTER ALGEBRAS FOR sl gr( e ( ω ) k e ( ω ′ ) k ) = gr Y j ∈ I ( ω ) ( A ( ω ) j ) [ ǫb ( ω ) jk ] + . On the other hand, the comparison of the skein and quantum exchange relations obtainedin the proof of Theorem 4.3 tells us that the monomial appearing in the right-hand side isexactly one of the two terms in the corresponding skein relation. Since the skein relationsare homogeneous with respect to the endpoint grading, we get the desired assertion. (cid:3) Lemma 4.6. For any ω ∈ Exch sl , Σ , we have an isomorphism end ( ω ) : coker p ∗ ( ω ) ∼ −→ L (3) of lattices which fits into the following diagram: ◦ Λ ( ω ) L (3) . coker p ∗ ( ω ) end ( ω ) α ∗ ( ω ) end ( ω ) Here α ∗ ( ω ) : ◦ Λ ( ω ) → coker p ∗ ( ω ) denotes the natural projection. Together with the mutation-invariance (Lemma 4.5), we get a canonical isomorphism end : coker p ∗ ∼ −→ L (3) .Proof. Thanks to the mutation-invariance (Lemma 4.5), it suffices to prove the statementfor a decorated triangulation ω = ∆ . For the well-definedness of end ∆ , we need to checkthat end ∆ ( p ∗ ∆ e ∆ i ) = end ∆ X j ∈ I (∆) ε ∆ ij f ∆ j = gr Y j ∈ I (∆) ( e ∆ j ) ε ∆ ij = 0 for all i ∈ I ( ∆ ) uf . Let us write e j := e ∆ j ∈ S A sl , Σ for simplicity.If i ∈ I edge (∆) , then label the neighboring vertices as in the left of Figure 4.1. Thenone can see gr( e j e j /e j e j ) = 0 by inspection into the right of Figure 4.1. If i ∈ I tri (∆) ,then label the neighboring vertices as in the left of Figure 4.2. Then one can see gr( e j e j e j /e j e j e j ) = 0 by inspection into the right of Figure 4.2. Thus the map end ∆ is well-defined.For the surjectivity, first note that the lattice L (3) is generated by the following gradingvectors(1) gr( e ∆ i ) for i ∈ I tri ( ∆ ) ;(2) gr( e ( ω ) i ) for ω ∈ Exch sl , Σ and i ∈ I edge ( ω ) .Indeed, given any vector in L (3) , by adding a suitable number of vectors of the the form(1), we can translate it so that aug = P p ∈ M k p − P p ∈ M l p = 0 . Such a grading vector canbe written as a sum of the grading vectors of the oriented arcs, which are of the form (2).Since each web cluster C ( ω ) is mutation-equivalent to C ∆ , each grading vector in (1)(2)can be written as a sum of the vectors gr( e ∆ i ) for i ∈ I ( ∆ ) . Thus end ∆ is surjective.Since L (3) is a full-rank sub-lattice of ( Z × Z ) M , we have rank L (3) = 2 | M | . On theother hand, we know from the geometric realizations of the cluster varieties as certain moduli spaces that rank coker p ∗ = dim A SL , Σ − dim X P GL , Σ = 2 | M | [FG06a, Lemma2.4]. It follows that end ∆ is an isomorphism of lattices as a surjective morphism betweentwo lattices of the same rank. (cid:3) Theorem 4.7. The ensemble grading on U s q ( sl , Σ) restricts to the endpoint grading on S A sl , Σ [ ∂ − ] . More precisely, we have end( gr ( A ( ω ) i )) = gr( e ( ω ) i ) for ω ∈ Exch sl , Σ and i ∈ I ( ω ) .Proof. From Lemma 4.6, we get end( gr ( A ( ω ) i )) = end ( ω ) ( α ∗ ( ω ) ( f ( ω ) i )) = end ( ω ) ( f ( ω ) i ) = gr( e ( ω ) i ) as desired. (cid:3) Quantum Laurent positivity of sl -webs. As a quantum counterpart of (A.6),we have the semiring L + s q := \ ω ∈ Exch s Z + q [ A ± ω ) ] ⊂ U s q of quantum universally Laurent polynomials , where Z + q [ A ± ω ) ] ⊂ Z q [ A ± ω ) ] denote the semir-ing of quantum Laurent polynomials in the variables A ( ω ) i for i ∈ I and q / with non-negative coefficients. When s q = s q ( sl , Σ) , one may ask the existence of the followingdiagram: BWeb sl , Σ U s q ( sl , Σ) L + s q ( sl , Σ) . ? In order to state our partial result, consider a larger semiring e L + sl , Σ := \ ∆ Z + q [ A ± ∆ ] ⊂ U s q , where ∆ runs over all the decorated triangulations of Σ . An element of e L + sl , Σ is calleda quantum GS-universally positive Laurent polynomial . The following is a rephrasing ofCorollaries 3.34 and 3.35: Theorem 4.8 (Quantum Laurent positivity of webs) . Any elevation-preserving web withrespect to ∆ is contained in Z + q [ A ± ∆ ] . In particular, the n -bracelet or the n -bangle alongan oriented simple loop in Σ for any n are contained in the semiring e L + sl , Σ . KEIN AND CLUSTER ALGEBRAS FOR sl Appendix A. Relation to the cluster varieties Here we recall some relations between the theory of cluster algebras [FZ02] and thatof cluster varieties [FG09]. Although we mainly deal with (the quantum aspects of) theformer in the text, we borrow some notations from the latter to indicate connections torelevant geometries. For a comparison of their quantizations given by [BZ05] and [FG08],see [GS19, Section 13.3].Given a seed pattern s (see Section 2.1.1), the corresponding Fock–Goncharov seedpattern is an assignment s FG : T I uf ∋ t (Λ ( t ) , ( , ) ( t ) , ( e ( t ) i ) i ∈ I ) , where Λ ( t ) = L i ∈ I Z e ( t ) i is a lattice with a fixed basis, ( , ) ( t ) : Λ ( t ) × Λ ( t ) → Q is a bilinearform given by ( e ( t ) i , e ( t ) j ) ( t ) := ε ( t ) ij = b ( t ) ji . For an edge t k −−− t ′ , the lattices Λ ( t ) and Λ ( t ′ ) are related by two linear isomorphisms µ ∗ k,ǫ : Λ ( t ′ ) ∼ −→ Λ ( t ) , e ( t ′ ) i X j ∈ I e ( t ) j ( F ( t ) k,ǫ ) ji for ǫ ∈ { + , −} , which we call the signed seed mutations . The dual lattices ◦ Λ ( t ) :=Hom(Λ ( t ) , Z ) are related by ˇ µ ∗ k,ǫ := (( µ ∗ k,ǫ ) T ) − : ◦ Λ ( t ′ ) ∼ −→ ◦ Λ ( t ) , f ( t ′ ) i X j ∈ I f ( t ) j ( E ( t ) k,ǫ ) ji (A.1)with the dual basis ( f ( t ) i ) i ∈ I of ( e ( t ) i ) i ∈ I . The bilinear form ( , ) ( t ) induces the ensemblemap p ∗ ( t ) : Λ ( t )uf → ◦ Λ ( t ) , e ( t ) i ( e ( t ) i , ) ( t ) = X j ∈ I ε ( t ) ij f ( t ) j , where Λ ( t )uf := L i ∈ I uf Z e ( t ) i . For each t k −−− t ′ and ǫ ∈ { + , −} , the following diagramcommutes: Λ ( t ′ )uf ◦ Λ ( t ′ ) Λ ( t )uf ◦ Λ ( t ) . p ∗ ( t ′ ) µ ∗ k,ǫ ˇ µ ∗ k,ǫ p ∗ ( t ) (A.2)The lattice ◦ Λ ( t ) parametrizes the character on the algebraic torus A ( t ) := Hom( ◦ Λ ( t ) , G m ) ,where G m := Spec Z [ t, t − ] denotes the multiplicative algebraic group . The correspon-dence is given by the natural isomorphism ch • : ◦ Λ ( t ) ∼ −→ O ( A ( t ) ) , ch m ( ψ ) := ψ ( m ) for m ∈ ◦ Λ ( t ) and ψ ∈ A ( t ) . Let A ( t ) i := ch f ( t ) i for i ∈ I , and similarly consider the characters X ( t ) i := ch e ( t ) i on the algebraic torus X uf( t ) := Hom(Λ ( t )uf , G m ) . The above mentioned linear A reader not familier with such a notion may substitute any field k to get G m ( k ) = k ∗ , and A ( t ) ( k ) ∼ =( k ∗ ) I . This amounts to consider schemes over k , making their function rings k -algebras in the sequel. maps between the lattices of the form f ∗ naturally induce monomial morphisms f betweenthe corresponding tori.For t k −−− t ′ , the cluster A -transformation is the birational map µ ak := ˇ µ k,ǫ ◦ µ ,ak,ǫ : A ( t ) A ( t ′ ) , where µ ,ak,ǫ is the birational automorphism on A ( t ) given by ( µ ,ak,ǫ ) ∗ A ( t ) i := A ( t ) i (1 + ( p ∗ X ( t ) i ) ǫ ) − δ ik . (A.3)The cluster K -variety (or the cluster A -variety ) is defined by gluing the tori A ( t ) viathese cluster A -transformations: A s := [ t ∈ T I uf A ( t ) . From the commutative diagram (A.2), the signed seed mutation induces a linear isomor-phism ˇ µ ∗ k,ǫ : coker p ∗ ( t ′ ) ∼ −→ coker p ∗ ( t ) for each t k −−− t ′ , which does not depend on the sign ǫ . Via these linear isomorphisms, we identify the lattices coker p ∗ ( t ) for any vertex t ∈ T I uf and simply denote it by coker p ∗ . Let H A := Hom(coker p ∗ , G m ) denote the correspondingsplit algebraic torus. The projections α ∗ ( t ) : ◦ Λ ( t ) → coker p ∗ ( t ) combine to give an action[FG09, Lemma 2.10(a)] α : H A × A s → A s . (A.4) Reconstruction of the (upper) cluster algebra. For t k −−− t ′ , one can easily see that ( µ ak ) ∗ A ( t ′ ) i coincides with the right-hand side of the exchange relation (2.2).Fixing an “initial” vertex t ∈ T I uf , consider the rational functions A ( t ; t ) i := ( µ aγ ) ∗ A ( t ) i for i ∈ I and t ∈ T I uf on A ( t ) , where γ is the unique shortest edge path from t to t and µ aγ denotes the composite of the cluster A -transformations associated with theedges γ traverses. Then the pair s ( t ) = ( B ( t ) , A ( t ; t ) := ( A ( t ; t ) i ) i ∈ I ) is a seed in the field F ( t ) of rational functions on A ( t ) . Thus the original seed pattern s is recovered via anisomorphism F ∼ = F ( t ) . The cluster algebra A s is the ring generated by these rationalfunctions A ( t ; t ) . The upper cluster algebra is the ring of global functions on A s : O ( A s ) = \ t ∈ T I uf Z [( A ( t ) i ) ± | i ∈ I ] ∼ = \ t ∈ T I uf Z [( A ( t ; t ) i ) ± | i ∈ I ] = U s . (A.5)Here we use the fact that A ( t ′ ; t ) i and A ( t ; t ) σ − ( i ) are the same function for i ∈ I if and onlyif s ( t ′ ) = σ. s ( t ) . 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Westbury, Confluence theory for graphs , Algebr. Geom. Topol. (2007),439–478.[Thu14] D. P. Thurston, Positive basis for surface skein algebras , Proc. Natl. Acad. Sci. USA, (2014), no. 27, 9725–9732.[Xie13] D. Xie, Higher laminations, webs and N = 2 line operators , arXiv:1304.2390 (hep-th).[Yua] W. Yuasa, Filtered and graded sl -skein algebras of marked surface without punctures , in prepa-ration. KEIN AND CLUSTER ALGEBRAS FOR sl Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan Email address : [email protected] URL : https://sites.google.com/view/tsukasa-ishibashi/home Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan Email address : [email protected] URL ::