Cluster algebra structures on module categories over quantum affine algebras
aa r X i v : . [ m a t h . QA ] A p r CLUSTER ALGEBRA STRUCTURES ON MODULE CATEGORIESOVER QUANTUM AFFINE ALGEBRAS
MASAKI KASHIWARA, MYUNGHO KIM, SE-JIN OH, AND EUIYONG PARK
Abstract.
We study monoidal categorifications of certain monoidal subcategories C J offinite-dimensional modules over quantum affine algebras, whose cluster algebra structurescoincide and arise from the category of finite-dimensional modules over quiver Heckealgebra of type A ∞ . In particular, when the quantum affine algebra is of type A or B , the subcategory coincides with the monoidal category C g introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are realsimple modules over quantum affine algebras. Contents
Introduction 11. Quantum groups, quantum coordinate rings and quantum affine algebras 52. Quantum cluster algebras and monoidal categorification 143. Quiver Hecke algebras 194. Cluster structure on R A ∞ -gmod 275. The category T N and its cluster structure 396. Main result 50References 63 Introduction
The quiver Hecke algebras (or
Khovanov-Lauda-Rouquier algebras ), introduced inde-pendently by Khovanov-Lauda ([37, 38]) and Rouquier ([47]), provide the categorification
Date : April 2, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Quantum affine algebra, Quiver Hecke algebra, Quantum group, Quantumcluster algebra.The research of M. Kashiwara was supported by Grant-in-Aid for Scientific Research (B) 15H03608,Japan Society for the Promotion of Science.The research of M. Kim was supported by the National Research Foundation of Korea(NRF) Grantfunded by the Korea government(MSIP) (NRF-2017R1C1B2007824).The research of S.-j. Oh was supported by the National Research Foundation of Korea(NRF) Grantfunded by the Korea government(MSIP) (NRF-2016R1C1B2013135).The research of E. P. was supported by the National Research Foundation of Korea(NRF) Grantfunded by the Korea Government(MSIP)(NRF-2017R1A1A1A05001058). of a half of quantum group U q ( g ). Since then, quiver Hecke algebras have been studiedactively and various new features were discovered in the viewpoint of categorification.Studying quantum groups via the quiver Hecke algebras has become one of the mainresearch themes on quantum groups in aspect of categorification. In particular, the stud-ies on representations of a quantum affine algebra U ′ q ( g ) via the generalized quantumSchur-Weyl duality functors ([25]) and the quantum cluster algebra structures of quan-tum unipotent coordinate algebras A q ( n ( w )) via the monoidal categorifications ([29]) drewbig attention of researchers on various areas.The generalized quantum Schur-Weyl duality functor was developed in [25], which is avast generalization of quantum affine Schur-Weyl duality. Let U ′ q ( g ) be a quantum affinealgebra of arbitrary type over a base field k , and let { V j } j ∈ J be a family of quasi-good U ′ q ( g )-modules. The generalized quantum Schur-Weyl duality provides a procedure tomake a symmetric quiver Hecke algebra R J from the R-matrices among { V j } j ∈ J and toconstruct a monoidal functor F : R J -gmod → C g enjoying good properties. Here R J -gmod(resp. C g ) denotes the monoidal category of graded R J -modules (resp. integrable U ′ q ( g )-modules) which are finite-dimensional over k .Let us recall briefly the results of [25]. Let N ∈ Z > and J = Z , and consider the family { V ( ̟ ) q k } k ∈ Z of the quantum affine algebra U ′ q ( g ) of type A (1) N − . Then the correspondingquiver Hecke algebra R J is of type A ∞ and the generalized quantum Schur-Weyl dualityassociated with { V ( ̟ ) q k } k ∈ Z gives a monoidal functor F : R J -gmod → C J ⊂ C g . Here C g is the smallest Serre subcategory of C g which is stable by taking tensor products andcontains a sufficiently large family of fundamental representations, and C J is a certainsubcategory of C g determined by the functor F (see Section 1.7 and 6.1). Let S N be thesmallest Serre subcategory of A := R J -gmod such that(i) S N contains L [ a, a + N ] for any a ∈ J ,(ii) X ◦ Y, Y ◦ X ∈ S N for all X ∈ A and Y ∈ S N ,where L [ a, a + N ] is the 1-dimensional R J ( ǫ a − ǫ a + N +1 )-module (see Section 4.2 for thedefinition of L [ a, b ]). Then they constructed the monoidal category T N with the modifiedconvolution product ⋆ by localizing the quotient category A / S N at the commuting familycoming from the objects L a = L [ a, a + N − T N is rigid,the functor F : A → C J factors through the canonical functor Ω N : A → T N , and theresulting functor T N → C J induces an isomorphism between the Grothendieck rings of C J and T N | q =1 . The category T N also plays the same role in the studies on generalizedquantum Schur-Weyl duality for C A (2) N − ([28]) and C B (1) N/ (for N ∈ Z ) ([32]).The monoidal categorification of a quantum unipotent coordinate algebra A q ( n ( w ))using a certain monoidal subcategory C w (see Definition 3.19) of R -gmod for symmetric quiver Hecke algebras was provided in [29], which gives a monoidal categorical explanationon the quantum cluster algebra structure of A q ( n ( w )) given in Geiß–Leclerc–Schr¨oer [13].For each reduced expression e w of a Weyl group element w , the initial quantum monoidalseed in C w is given by the determinantial modules M e w ( s,
0) which correspond to certain
LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 3 unipotent quantum minors of A q ( n ( w )). It was shown in [29] that every cluster monomialsis a member of the upper global basis of A q ( n ( w )) by using the monomial categorificationof A q ( n ( w )) arising from C w .The cluster algebra structures also appear in certain monoidal subcategories of the cate-gory C g of finite-dimensional integrable representations of a quantum affine algebra U ′ q ( g ).The Grothendieck ring of the subcategory C ℓ ( ℓ ∈ Z > ) of C g introduced in Hernandez–Leclerc [16, Section 3.8] was studied by using cluster algebra structures ([16]), and analgorithm for calculating q -character of Kirillov-Reshetikhin modules for any untwistedquantum affine algebras was described in [17] by studying the cluster algebra A which isisomorphic to the Grothendieck ring of the subcategory C − g of C g (see Section 2.3). It wasconjectured in [17] that all cluster monomials of A correspond to the classes of certainsimple objects of C − g (see Conjecture 2.7).In a viewpoint of the categorification using quiver Hecke algebras, it is natural to studya cluster algebra structure on monoidal subcategories of quantum affine algebras using themonoidal categorification of quantum unipotent coordinate algebras via the generalizedquantum Schur-Weyl duality. In fact, it is what we perform in this paper. We study aquantum cluster algebra structure of the category A := R J -gmod of type A ∞ and showthat this quantum cluster algebra structure is compatible with the functor A → T N forany N ∈ Z > (see Theorem 5.24).We further provide the condition (6.1) on quasi-good modules of a quantum affinealgebra U ′ q ( g ) to make the generalized Schur-Weyl duality functor F : A → C J factorthrough the canonical functor Ω N : A → T N so that we have the exact monoidal functor e F : T N → C J by using the framework given in [32] (see Theorem 6.7). In the cases of types A ( t ) N − ( t = 1 , B (1) N/ (for N ∈ Z ), C (1) N − (for N > D ( t ) N (for ( N ≥ t = 1 ,
2) or( N = 4 and t = 3)), we give explicit quasi-good modules satisfying the condition, whichprovide the cluster algebra structure on the Grothendieck ring K ( C J ) induced from T N .Moreover, all cluster monomials of K ( C J ) correspond to the classes of real simple modulesin C J (Theorem 6.10). Note that the families for the types A and B appeared in [25, 28],and [32], but the ones for types C and D are new. Since C J = C g in types A and B , thereexists a cluster algebra structure on K ( C g ) coming from the quantum cluster algebrastructure of T N . In particular, we prove that the conjecture given in [17] (see Theorem6.15) is true when g is of type A (1) N − : all cluster monomials of A correspond to the classesof certain real simple objects of C − A (1) N − . Remark that for the case of the categories C ℓ , thecorresponding property was proved in [46].Let us explain our results more precisely. Let J = Z and let A J be the Cartan matrixof type A ∞ . Let R J be the symmetric quiver Hecke algebra of type A ∞ . We first choosea special infinite sequence e w of simple reflections s j ( j ∈ J ) of the Weyl group W J oftype A ∞ (see (4.2)). It is shown in Proposition 4.1 that every p -prefix e w ≤ p of e w is areduced expression in W J for each p ∈ Z ≥ and, if p = t (2 t + 1) for t ∈ Z ≥ , then e w ≤ p can be viewed as a reduced expression of the longest element of the parabolic subgroup M. KASHIWARA, M. KIM, S.-J. OH, AND E. PARK of W J generated by s j for j ∈ [ −⌊ t − ⌋ , ⌊ t ⌋ ] (see Remark 4.2). In this sense, the category R J -gmod can be understood as a limit of the subcategories C e w ≤ p . We then consider thedeterminantial modules M e w ( p,
0) and the cuspidal modules M e w ( p + , p ) associated with e w .Let c : Z ≥ → Z ≥ × Z ≥ be the bijection given in Definition 4.3 and let L [ a, b ] be the 1-dimensional R -module for a, b ∈ Z defined in Section 4.2. For p ∈ Z ≥ with c ( p ) = ( ℓ, m ),we prove that M e w ( p,
0) is isomorphic to the head W ( ℓ ) m, j p of a certain convolution productof m -many R -modules L [ a, b ] with the length ℓ = | b − a | + 1 (Proposition 4.10), anddescribe also the cuspidal module M e w ( p + , p ) in terms of L [ a, b ] (Corollary 4.13). We nextdescribe the quiver Q associated with the initial seed given by e w , which can be viewed asthe square product of the bipartite Dynkin quiver Q A ∞ (see (4.15)). Theorem 4.18 tellsthat the category R J -gmod is a monoidal categorification of the quantum cluster algebras A q / ( ∞ ) with the quantum seed [ S ∞ ] arising from the determinantial modules M e w ( p, Q .We next consider the monoidal categorification structure of R J -gmod which we con-structed. In Proposition 5.7, we describe the mutation M e w ( p, ′ of M e w ( p,
0) as follows:M e w ( p, ′ ≃ ( W ( ℓ ) m, j p +1 if ℓ + m ≡ , W ( ℓ ) m, j p − if ℓ + m ≡ . We then find sequences of mutations µ Σ + and µ Σ − such that µ Σ + ( Q ) and µ Σ − ( Q ) are thesame as Q as quivers and µ Σ + (M e w ( p, ≃ W ( ℓ ) m, j p +1 and µ Σ − (M e w ( p, ≃ W ( ℓ ) m, j p − for p ∈ K with c ( p ) = ( ℓ, m ) (Proposition 5.9). In this viewpoint, since M e w ( p,
0) isisomorphic to W ( ℓ ) m, j p , applying µ Σ + (resp. µ Σ − ) to Q is understood as shifting the indices j p of W ( ℓ ) m, j p at all vertices of Q by 1 (resp. − N ∈ Z ≥ , we define thesubquiver Q N of Q as in (5.6) and investigate compatibility with µ Σ + , µ Σ − and theprocedure from A to T N . Then we can conclude that the category T N is a monoidalcategorification of the quantum cluster algebras A q / ( N ) with the quantum seed [ S N ]arising from { Ω N (M e w ( p, } c ( p ) ∈ K ex N and the quiver Q N (Theorem 5.24).For the construction of generalized quantum Schur-Weyl duality functor T N → C g ,we provide the condition (6.1) on quasi-good modules of the quantum affine algebra U ′ q ( g ). Let { V a } a ∈ J be a family of quasi-good modules in C g satisfying the condition(6.1). Note that, to define Schur-Weyl duality functor, we have to choose duality coef-ficients P i,j ( u, v ) ( i, j ∈ J ) which are elements in k [[ u, v ]] satisfying certain conditionsdetermined by { V a } a ∈ J . In Lemma 6.1, we prove that any Schur-Weyl duality functorarising from { V a } a ∈ J sends L [ a, b ] to 0 for any a, b ∈ Z with b − a ≥ N , which implies thatit factors through A → A / S N . We prove that there exists a suitable duality coefficients { P i,j ( u, v ) } i,j ∈ J such that the corresponding Schur-Weyl duality functor F factors throughthe canonical functor Ω N : A → T N by following the framework given in [32]. Theorem LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 5 e F : T N → C J such that the following diagram quasi-commutes : A Ω N / / F & & ◆◆◆◆◆◆◆◆◆◆◆◆◆ T N e F (cid:15) (cid:15) C J . Then, the functor e F induces an isomorphism K ( T N ) | q =1 ∼−→ K ( C J ) and the category C J gives a monoidal categorification of the cluster algebra A ( N ) := A q / ( N ) | q =1 via thefunctor e F . Therefore, every cluster monomial in A ( N ) corresponds to the isomorphismclass of a real simple object in C J (Theorem 6.10). For types A ( t ) N − ( t = 1 , B (1) N/ (for N ∈ Z ), C (1) N − (for N > D ( t ) N (for ( N ≥ t = 1 ,
2) or ( N = 4 and t = 3)), we giveexplicit quasi-good modules satisfying the condition (6.1) (see Section 6.2). For types A and B , C J is equal to C g , which means that there exists a cluster algebra structure on K ( C g ) coming from the quantum cluster algebra structure of T N . For the other types C and D , C J is a proper subcategory of C g . As for type A (1) N − , we obtain an additional resultusing our monoidal categorification. In [17], for an untwisted quantum affine algebras,Hernandez and Leclerc studied the cluster algebra A with initial quiver G − of infiniterank, which is isomorphic to the Grothendieck ring of a half C − g of C g . It was conjecturedin [17] that all cluster monomials of A correspond to the classes of certain simple objectsof C − g . When A is of type A (1) N − , we prove that this conjecture is true using our monoidalcategorification C g (see Theorem 6.15).This paper is organized as follows. In Section 1, we review quantum groups and quan-tum affine algebras. In Section 2, we recall quantum cluster algebras and monoidal cat-egorification. In Section 3, we review quiver Hecke algebras for monoidal categorifica-tion and generalized quantum Schur-Weyl duality. In Section 4, we study the monoidalcategorification of R J -gmod associated with the infinite sequence e w . In Section 5, weinvestigate the cluster algebra structure of R J -gmod and prove that T N has a monoidalcategorification structure induced from R J -gmod via Ω N : R J -gmod → T N . In Section6, we provide the condition on quasi-good U ′ q ( g )-modules to make the generalized Schur-Weyl duality functor F : A → C J factor through the functor Ω N : A → T N , and give anexplicit quasi-good modules satisfying the condition for various types. Acknowledgments
The second, third and fourth authors gratefully acknowledge for the hospitality of RIMS(Kyoto University) during their visits in 2018 and 2019.1.
Quantum groups, quantum coordinate rings and quantum affinealgebras
In this section, we shortly recall the basic materials on quantum groups, quantumcoordinate rings and quantum affine algebras. We refer to [1, 22, 23, 24, 29] for details.
M. KASHIWARA, M. KIM, S.-J. OH, AND E. PARK
For simplicity, we use the following convention:For a statement P , δ ( P ) is 1 if P is true and 0 if P is false.1.1. Quantum groups.
Let I be an index set. A Cartan datum is a quintuple ( A , P , Π , P ∨ , Π ∨ )consisting of (i) a symmetrizable generalized Cartan matrix A = ( a i,j ) i,j ∈ I , (ii) a freeabelian group P , called the weight lattice , (iii) Π = { α i ∈ P | i ∈ I } , called the set of sim-ple roots , (iv) P ∨ := Hom Z ( P , Z ), called the co-weight lattice , (v) Π ∨ = { h i | i ∈ I } ⊂ P ∨ ,called the set of simple coroots , satisfying (1) h h i , α j i = a i,j for all i, j ∈ I , (2) Π islinearly independent, (3) for each i ∈ I there exists a Λ i ∈ P such that h h j , Λ i i = δ i,j forall j ∈ I . We call Λ i the fundamental weights . Note that there exists a diagonal matrix D = diag( d i | i ∈ I ) such that d i ∈ Z > and DA is symmetric.We denote by Q := L i ∈ I Z α i the root lattice , Q + := L i ∈ I Z ≥ α i the positive root lattice and Q − := L i ∈ I Z ≤ α i the negative root lattice . For β = P i ∈ I m i α i ∈ Q + , we set | β | = P i ∈ I m i .Set h = Q ⊗ Z P ∨ . Then there exists a symmetric bilinear form ( , ) on h ∗ such that( α i , α j ) = d i a i,j ( i, j ∈ I ) and h h i , λ i = 2( α i , λ )( α i , α i ) for any λ ∈ h ∗ and i ∈ I. Let g be the Kac-Moody algebra associated with a Cartan datum ( A , P , Π , P ∨ , Π ∨ ).The Weyl group W of g is the subgroup of GL( h ∗ ) generated by the set of reflections S = { s i | i ∈ I } , where s i ( λ ) := λ − h h i , λ i α i for λ ∈ h ∗ .Let q be an indeterminate and set q i := q d i for each i ∈ I . We denote by U q ( g ) the quantum group associated to g , which is a Q ( q )-algebra generated by e i , f i ( i ∈ I ) and q h ( h ∈ P ∨ ). We set U + q ( g ) (resp. U − q ( g )) the subalgebra of U q ( g ) generated by e i ’s(resp. f i ’s).Recall that U q ( g ) admits the weight space decomposition U q ( g ) = L β ∈ Q U q ( g ) β , where U q ( g ) β := { x ∈ U q ( g ) | q h xq − h = q h h,β i for any h ∈ P ∨ } . For x ∈ U q ( g ) β , we set wt( x ) = β .Set A = Z [ q ± ]. Let us denote by U − A ( g ) the A -subalgebra of U q ( g ) generated by f ( n ) i := f ni / [ n ] i !, and by U + A ( g ) the A -subalgebra of U q ( g ) generated by e ( n ) i := e ni / [ n ] i !( i ∈ I , n ∈ Z ≥ ), where [ n ] i = q ni − q − ni q i − q − i and [ n ] i ! = n Y k =1 [ k ] i .There is a Q ( q )-algebra anti-automorphism ϕ of U q ( g ) given as follows : ϕ ( e i ) = f i , ϕ ( f i ) = e i , ϕ ( q h ) = q h . We say that a U q ( g )-module M is called integrable if the actions of e i and f i on M are locally nilpotent for all i ∈ I . We denote by O int ( g ) the category of integrable left U q ( g )-module M satisfying(i) M = L η ∈ P M η where M η = { m ∈ M | q h m = q h h,η i m } , dim M η < ∞ ,(ii) there exist finitely many weights λ , . . . , λ m such that wt( M ) ⊂ ∪ j ( λ j + Q − ), wherewt( M ) = { η ∈ P | dim M η = 0 } . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 7
It is well-known that O int ( g ) is a semisimple category such that every simple object isisomorphic to an irreducible highest weight module V (Λ) with the highest weight vector u Λ of highest weight Λ. Here Λ is an element of the set P + of dominant integral weights : P + := { µ ∈ P | h h i , µ i ≥ i ∈ I } . Unipotent quantum coordinate rings and unipotent quantum minors.
Notethat U + q ( g ) ⊗ U + q ( g ) has an algebra structure defined by( x ⊗ x ) · ( y ⊗ y ) = q − (wt( x ) , wt( y )) ( x y ⊗ x y ) , for homogeneous elements x , x , y and y . Then we have the algebra homomorphism∆ n : U + q ( g ) → U + q ( g ) ⊗ U + q ( g ) given by∆ n ( e i ) = e i ⊗ ⊗ e i . Definition 1.1.
We define the unipotent quantum coordinate ring A q ( n ) as follows : A q ( n ) = L β ∈ Q − A q ( n ) β where A q ( n ) β := Hom Q ( q ) ( U + q ( g ) − β , Q ( q )) . Note that A q ( n ) has a ring structure given as follows :( ψ · θ )( x ) := θ ( x (1) ) ψ ( x (2) ) where ∆ n ( x ) = x (1) ⊗ x (2) . The A -form of A q ( n ) is defined by A A ( n ) := { ψ ∈ A q ( n ) | h ψ, U + A ( g ) i ⊂ A } .Recall that the algebra A q ( n ) has the upper global basis ([23]) B up ( A q ( n )) := { G up ( b ) | b ∈ B ( A q ( n )) } , where B ( A q ( n )) denotes the crystal of A q ( n ).Let ( , ) Λ be the non-degenerate symmetric bilinear form on V (Λ) such that ( u Λ , u Λ ) Λ =1 and ( xu, v ) Λ = ( u, ϕ ( x ) v ) Λ for u, v ∈ V (Λ) and x ∈ U q ( g ).For µ, ζ ∈ W Λ, the unipotent quantum minor D ( µ, ζ ) is an element in A q ( n ) given by D ( µ, ζ )( x ) = ( xu µ , u ζ ) Λ for x ∈ U + q ( g ), where u µ and u ζ are the extremal weight vectors in V (Λ) of weight µ and ζ , respectively. Lemma 1.2 ([29, Lemma 9.1.1]) . D( η, ζ ) is either contained in B up ( A q ( n )) or zero. For η, ζ ∈ W Λ, we write η (cid:22) ζ if there exists a sequence { β k } ≤ k ≤ l of positive real rootssuch that we have ( β k , λ k − ) ≥
0, where λ := ζ and λ k := s β k λ k − for 1 ≤ k ≤ l . Notethat η (cid:22) ζ implies η − ζ ∈ Q − .By [29, Lemma 9.1.4], D( η, ζ ) = 0 if and only if η (cid:22) ζ , for Λ ∈ P + and η, ζ ∈ W Λ.The behaviors of multiplications among unipotent quantum minors were investigatedintensively (see [4, 13, 29]):
Proposition 1.3.
Let λ, µ ∈ P + . M. KASHIWARA, M. KIM, S.-J. OH, AND E. PARK (i)
For u, v ∈ W such that u ≥ v , we have D( uλ, vλ )D( uµ, vµ ) = q − ( vλ,vµ − uµ ) D( u ( λ + µ ) , v ( λ + µ )) . (ii) For s, t, s ′ , t ′ ∈ W satisfying • ℓ ( s ′ s ) = ℓ ( s ′ ) + ℓ ( s ) and ℓ ( t ′ t ) = ℓ ( t ′ ) + ℓ ( t ) , • s ′ sλ (cid:22) t ′ λ and s ′ µ (cid:22) t ′ tµ ,we have D( s ′ sλ, t ′ λ )D( s ′ µ, t ′ tµ ) = q ( s ′ sλ + t ′ λ, s ′ µ − t ′ tµ ) D( s ′ µ, t ′ tµ )D( s ′ sλ, t ′ λ ) . The subalgebra A q ( n ( w )) of A q ( n ) . In this subsection, we assume that the gener-alized Cartan matrix A is symmetric.Let w be a sequence of the set S := { s i | i ∈ I } of reflections of W : w = s i s i . . . s i ℓ . (1.1)We set • w ≤ k := s i · · · s i k ∈ W for 0 ≤ k ≤ ℓ , • λ k := w ≤ k Λ i k ∈ P for 1 ≤ k ≤ ℓ .(1.2)For a given sequence w of S , p ∈ { , . . . , ℓ } and j ∈ I , we set p + := min( { k | p < k ≤ ℓ, i k = i p } ∪ { ℓ + 1 } ) ,p − := max( { k | ≤ k < p, i k = i p } ∪ { } ) ,p + ( j ) := min( { k | p < k ≤ ℓ, i k = j } ∪ { ℓ + 1 } ) ,p − ( j ) := max( { k | ≤ k < p, i k = j } ∪ { } ) . For a reduced expression e w = s i s i · · · s i ℓ of w ∈ W and 0 ≤ t ≤ s ≤ ℓ , we setD e w ( s, t ) := D( λ s , λ t ) if t > i s = i t , D( λ s , Λ i s ) if 0 = t < s ≤ ℓ, otherwise . (1.3)The Q ( q )-subalgebra of A q ( n ) generated by D e w ( i, i − ) (1 ≤ i ≤ ℓ ), is independent of thechoice of e w . We denote it by A q ( n ( w )). Then every D e w ( s, t ) is contained in A q ( n ( w )) [13,Corollary 12.4]. The set B up ( A q ( n ( w ))) := B up ( A q ( n )) ∩ A q ( n ( w )) forms a Q ( q )-basis of A q ( n ( w )) [39, Theorem 4.25]. We call B up ( A q ( n ( w ))) the upper global basis of A q ( n ( w )).The A -module A A ( n ( w )) generated by B up ( A q ( n ( w ))) is an A -subalgebra of A q ( n ) ([39,Theorem 4.27]).1.4. Quantum affine algebras.
In this subsection, we briefly review the representationtheory of finite-dimensional integrable modules over quantum affine algebras by follow-ing [1, 24].When concerned with quantum affine algebras, we always take the algebraic closure of C ( q ) in S m> C (( q /m )) as the base field k . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 9
Let I be an index set and let A = ( a ij ) i,j ∈ I be a generalized Cartan matrix of affinetype. We choose 0 ∈ I as the leftmost vertices in the tables in [21, pages 54, 55] except A (2)2 n -case where we take the longest simple root as α . Set I = I \ { } .We normalize the Q -valued symmetric bilinear form ( • , • ) on P by( δ , λ ) = h c , λ i for any λ ∈ P , where δ denotes the null root and c = P i ∈ I c i h i denotes the center . We denote by γ thesmallest positive integer such that γ ( α i , α i )2 ∈ Z for all i ∈ I .Let us denote by U q ( g ) the quantum group over Q ( q /γ ) associated with the affineCartan datum ( A , P , Π , P ∨ , Π ∨ ). We denote by U ′ q ( g ) the subalgebra of U q ( g ) generatedby e i , f i , t ± i := q ± ( αi,αi )2 h i for i ∈ I and call it the quantum affine algebra .We use the comultiplication ∆ of U ′ q ( g ) given by∆( e i ) = e i ⊗ t − i + 1 ⊗ e i , ∆( f i ) = f i ⊗ t i ⊗ f i , ∆( q h ) = q h ⊗ q h . Let us denote by ¯ the involution of U ′ q ( g ) defined as follows: e i e i , f i f i , t i t − i , q /γ → q − /γ . We denote by C g the category of finite-dimensional integrable U ′ q ( g )-modules. For M ∈ C g , we denote by M aff the affinization of M which is k [ z, z − ] ⊗ M as a vector spaceendowed with the U ′ q ( g )-module structure given by e i ( u z ) = z δ i, ( e i u ) z , f i ( u z ) = z − δ i, ( f i u ) z , t i ( u z ) = ( t i u ) z . Here u z denote the element ⊗ u ∈ M aff for u ∈ M . We also write M z instead of M aff .A simple module M in C g contains a non-zero vector u of weight λ ∈ P cl := P / Z δ such that (1) h h i , λ i ≥ i ∈ I , (2) all the weight of M are contained in λ − P i ∈ I Z ≥ cl( α i ), where cl : P → P cl denotes the canonical projection. Such a λ is uniqueand u is unique up to a constant multiple. We call λ the dominant extremal weight of M and u the dominant extremal weight vector of M .For x ∈ k × and M ∈ C g , we define M x := M aff / ( z M − x ) M aff , where z M denotes the U ′ q ( g )-module automorphism of M aff of weight δ . We call x the spectral parameter .For each i ∈ I , we set ̟ i := gcd( c , c i ) − cl( c Λ i − c i Λ ) ∈ P cl . Then there exists a unique simple U ′ q ( g )-module V ( ̟ i ) in C g , called the fundamentalmodule of ( level weight ̟ i , satisfying the certain conditions (see [24, § U ′ q ( g )-module M , we denote by the U ′ q ( g )-module M = { ¯ u | u ∈ M } whosemodule structure is given as x ¯ u := xu for x ∈ U ′ q ( g ). Then we have M a ≃ ( M ) a , M ⊗ N ≃ N ⊗ M . (1.4)In particular, V ( ̟ i ) ≃ V ( ̟ i ) (see [1, Appendix A]). For a module M in C g , let us denote the right and the left dual of M by ∗ M and M ∗ ,respectively. That is, we have isomorphismsHom U ′ q ( g ) ( M ⊗ X, Y ) ≃ Hom U ′ q ( g ) ( X, ∗ M ⊗ Y ) , Hom U ′ q ( g ) ( X ⊗ ∗ M, Y ) ≃ Hom U ′ q ( g ) ( X, Y ⊗ M ) , Hom U ′ q ( g ) ( M ∗ ⊗ X, Y ) ≃ Hom U ′ q ( g ) ( X, M ⊗ Y ) , Hom U ′ q ( g ) ( X ⊗ M, Y ) ≃ Hom U ′ q ( g ) ( X, Y ⊗ M ∗ ) , which are functorial in U ′ q ( g )-modules X and Y . In particular, the module V ( ̟ i ) x ( x ∈ k × ) has the left dual and right dual as follows: (cid:0) V ( ̟ i ) x (cid:1) ∗ ≃ V ( ̟ i ∗ ) x ( p ∗ ) − , ∗ (cid:0) V ( ̟ i ) x (cid:1) ≃ V ( ̟ i ∗ ) xp ∗ where p ∗ is an element in k × depending only on U ′ q ( g ) (see [1, Appendix A]), and i i ∗ denotes the involution on I given by α i = − w α i ∗ . Here w is the longest element of W = h s i | i ∈ I i ⊂ W .We say that a U ′ q ( g )-module M is good if it has a bar involution , a crystal basis with simple crystal graph , and a global basis (see [24] for the precise definition). For instance,every fundamental module V ( ̟ i ) for i ∈ I is a good module. Note that every goodmodule is a simple U ′ q ( g )-module. Moreover the tensor product of good modules is againgood. Hence any good module M is real simple, i.e., M ⊗ M is simple. Definition 1.4.
We call a U ′ q ( g ) module M quasi-good if M ≃ V c for some good module V and c ∈ k × .1.5. R-matrices.
In this subsection, we briefly review the notion of R -matrices for quan-tum affine algebras following [24, § M, N ∈ C g , there is a morphism of k [[ z N /z M ]] ⊗ k [ z N /z M ] k [ z ± M , z ± N ] ⊗ U ′ q ( g )-modules,denoted by R univ M zM ,N zN and called the universal R -matrix : R univ M zM ,N zN : k [[ z N /z M ]] ⊗ k [ z N /z M ] ( M z M ⊗ N z N ) → k [[ z N /z M ]] ⊗ k [ z N /z M ] ( N z N ⊗ M z M ) . We say that R univ M zM ,N zN is rationally renormalizable if there exist a ∈ k (( z N /z M )) and a k [ z ± M , z ± N ] ⊗ U ′ q ( g )-module homomorphism R ren M zM ,N zN : M z M ⊗ N z N → N z N ⊗ M z M such that R ren M zM ,N zN = aR univ M zM ,N zN . Then we can choose R ren M zM ,N zN so that for any a M , a N ∈ k × , the specialization of R ren M zM ,N zN at z M = a M , z N = a N , R ren M zM ,N zN | z M = a M ,z N = a N : M a M ⊗ N a N → N a N ⊗ M a M does not vanish provided that M and N are non-zero U ′ q ( g )-modules in C g . It is called a renormalized R -matrix .We denote by r M,N := R ren M zM ,N zN | z M =1 ,z N =1 : M ⊗ N → N ⊗ M and call it the R -matrix . By the definition r M,N never vanishes.
LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 11
For simple U ′ q ( g )-modules M and N in C g , the universal R -matrix R univ M zM ,N zN is rationallyrenormalizable. Then, for dominant extremal weight vectors u M and u N of M and N ,there exists a M,N ( z N /z M ) ∈ k [[ z N /z M ]] × such that R univ M zM ,N zN (cid:0) ( u M ) z M ⊗ ( u N ) z N ) (cid:1) = a M,N ( z N /z M ) (cid:0) ( u N ) z N ⊗ ( u M ) z M ) (cid:1) . (1.5)Then R norm M zM ,N zN := a M,N ( z N /z M ) − R univ M zM ,N zN is a unique k ( z M , z N ) ⊗ k [ z N /z M ] k [ z ± M , z ± N ] ⊗ U ′ q ( g )-module homomorphism sending (cid:0) ( u M ) z M ⊗ ( u N ) z N (cid:1) to (cid:0) ( u N ) z N ⊗ ( u M ) z M (cid:1) .It is known that k ( z M , z N ) ⊗ k [ z ± M ,z ± N ] ( M z M ⊗ N z N ) is a simple k ( z M , z N ) ⊗ k [ z ± M ,z ± N ] U ′ q ( g )-module ([24, Proposition 9.5]). We call R norm M zM ,N zN the normalized R -matrix .Let us denote by d M,N ( u ) ∈ k [ u ] a monic polynomial of the smallest degree suchthat the image of d M,N ( z N /z M ) R norm M zM ,N zN is contained in N z N ⊗ M z M . We call d M,N the denominator of R norm M zM ,N zN . Then, d M,N ( z N /z M ) R norm M zM ,N zN : M z M ⊗ N z N → N z N ⊗ M z M is a renormalized R -matrix, and the R -matrix r M,N : M ⊗ N → N ⊗ M is equal to d M,N ( z N /z M ) R norm M zM ,N zN (cid:12)(cid:12) z M =1 ,z N =1 up to a constant multiple.1.6. Denominators of normalized R -matrices. The denominators of the normalized R -matrices d k,l ( z ) := d V ( ̟ k ) ,V ( ̟ l ) ( z ) between V ( ̟ k ) and V ( ̟ l ) were calculated in [1, 6, 26,44] for classical affine types and in [45] for exceptional affine types (see also [10, 30, 34, 51]).In this subsection, we recall d k,l ( z ) for quantum affine algebras of type A and B . In Table 1,we list the Dynkin diagrams with an enumeration of simple roots and the correspondingfundamental weights for types A and B . Remark 1.5. (a) Note that the convention for Dynkin diagram of type A (2)2 n is different from the onein [21, page 54, 55]. However, for each i ∈ I the corresponding fundamental modules V ( ̟ i ) are isomorphic to each other, since the corresponding fundamental weights areconjugate to each other under the Weyl group action (see [24, § B (1)2 and A (2)3 in Table 1 are denoted by C (1)2 and D (2)3 in [21, page 54, 55], respectively.(c) Our conventions on quantum affine algebras are different from [17, 18]. To compare,we refer to [18, Remark 3.28]. Theorem 1.6 ([6, 44]) . We have the following denominator formulas. (a)
For g = A (1) n − ( n ≥ , ≤ k, l ≤ n − , we have d k,l ( z ) = min( k,l,n − k,n − l ) Y s =1 (cid:0) z − ( − q ) | k − l | +2 s (cid:1) . Type Dynkin diagram Fundamental weights A (1)1 ◦ k s α + ◦ α ̟ = cl(Λ − Λ ) A (1) n ( n ≥ ◦ α ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ ◦ α ◦ α ◦ α ◦ α n − ◦ α n ◦ ̟ i = cl(Λ i − Λ )(1 ≤ i ≤ n ) A (2)2 ❝ ❝ > α α ̟ = cl(2Λ − Λ ) A (2)3 ◦ k s α ◦ α + α ◦ ̟ = cl(Λ − Λ ), ̟ = cl(Λ − ) A (2)2 n − ( n ≥ ◦ α ◦ α ◦ α ◦ α ◦ α n − ◦ k s α n ◦ ̟ i = cl(Λ i − Λ )( i = 1 , n ), ̟ i = cl(Λ i − )(2 ≤ i ≤ n − A (2)2 n ( n ≥ ◦ α + ◦ α ◦ α ◦ α n − ◦ α n + ◦ ̟ i = cl(Λ i − Λ )( i = 1 , . . . , n − ̟ n = cl(2Λ n − Λ ) B (1) n ( n ≥ ◦ ❍❍❍❍❍❍❍ ◦ ◦ n − ◦ n + ◦◦ ✈✈✈✈✈✈✈ ̟ = cl(Λ − Λ ), ̟ i = cl(Λ i − )(2 ≤ i ≤ N − ̟ n = cl(Λ n − Λ ) B (1)2 ◦ α + ◦ k s α α ◦ ̟ = cl(Λ − Λ ), ̟ = cl(Λ − Λ ) Table 1.
Dynkin diagrams and fundamental weights(b)
For g = A (2) n − ( n ≥ , ≤ k, l ≤ ⌊ n/ ⌋ , we have d k,l ( z ) = min( k,l ) Y s =1 ( z − ( − q ) | k − l | +2 s )( z + q n ( − q ) − k − l +2 s ) . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 13 (c)
For g = B (1) n ( n ≥
2) 1 ≤ k, l ≤ n − , we have d k,l ( z ) = min( k,l ) Y s =1 ( z − ( − q ) | k − l | +2 s )( z + ( − q ) n − k − l − s ) . (1.6) For ≤ k ≤ n , we have d k,n ( z ) = k Y s =1 ( z − ( − n + k q n − k − ss ) if ≤ k ≤ n − , n Y s =1 ( z − ( q s ) s − ) . if k = n, (1.7) where q s := q n = q / . Hernandez-Leclerc category.
For each quantum affine algebra U ′ q ( g ), we definea quiver σ ( g ) as follows:(i) Take the set of equivalence classes ˆ I g := ( I × k × ) / ∼ as the set of vertices, wherethe equivalence relation is given by ( i, x ) ∼ ( j, y ) if and only if V ( ̟ i ) x ≃ V ( ̟ j ) y .(ii) Put d -many arrows from ( i, x ) to ( j, y ), where d denotes the order of zero of d i,j ( z j /z i )at z j /z i = y/x .Note that ( i, x ) and ( j, y ) are connected by at least one arrow in σ ( g ) if and only if V ( ̟ i ) x ⊗ V ( ̟ j ) y is reducible ([1, Corollary 2.4]).Let σ ( g ) be a connected component of σ ( g ). Note that a connected component of σ ( g )is unique up to a spectral parameter shift and hence σ ( g ) is uniquely determined up toa quiver isomorphism. For types A and B , one can take σ ( A (1) n ) := { ( i, ( − q ) p ) ∈ I × k × | p ≡ i + 1 mod 2 } , σ ( A (2)2 n − ) := { ( i, ± ( − q ) p ) ∈ I × k × | i ∈ I , p ≡ i + 1 mod 2 } , σ ( A (2)2 n ) := { ( i, ( − q ) p ) ∈ I × k × | p ∈ Z } , σ ( B (1) n ) := { ( i, ( − i − q κ q m ) , ( n, q m ) | ≤ i ≤ n − , m ∈ Z } , (1.8)where q κ in (1.8) is defined as follows: q κ := ( − n +1 q n +1 s . We remark here that(i) V ( ̟ n ) x ≃ V ( ̟ n ) − x in the A (2)2 n − -case,(ii) σ ( A (1)2 n − ) ≃ σ ( A (2)2 n − ) as quivers ([28, (2.7)]).Let us denote by C g the smallest abelian subcategory of C g such that(a) C g contains { V ( ̟ i ) x | ( i, x ) ∈ σ ( g ) } ,(b) it is stable under taking submodules, quotients, extensions and tensor products. The category C g for symmetric affine type U ′ q ( g ) was introduced in [16]. Note that everysimple module in C g is a tensor product of certain parameter shifts of some simple modulesin C g [16, § K ( C g ) of C g is the polynomial ring generated bythe classes of modules in { V ( ̟ i ) x | ( i, x ) ∈ σ ( g ) } [9].2. Quantum cluster algebras and monoidal categorification
In this section, we recall the definition of quantum cluster algebras introduced in [5, 7].Then we review the monoidal categorification of a quantum cluster algebra developedin [29] (see also [16]).2.1.
Quantum cluster algebras.
Fix a countable index set K = K ex ⊔ K fr which isdecomposed into the subset K ex of exchangeable indices and the subset K fr of frozenindices. Let L = ( λ ij ) i,j ∈ K be a skew-symmetric integer-valued K × K -matrix.Let A be a Z [ q ± / ]-algebra. We say that a family { x i } i ∈ K of elements in A is L -commuting if it satisfies x i x j = q λ ij x j x i for any i, j ∈ K. We say that an L -commuting family { x i } i ∈ K is algebraically independent if the family { x i · · · x i ℓ | ℓ ∈ Z ≥ , i , . . . , i ℓ ∈ K, i ≤ · · · ≤ i ℓ } is linearly independent over Z [ q ± / ]. Here ≤ is a total order on K .Let e B = ( b ij ) ( i,j ) ∈ K × K ex be an integer-valued matrix such that(a) for each j ∈ K ex , there exist finitely many i ∈ K such that b ij = 0,(b) the principal part B := ( b ij ) i,j ∈ K ex is skew-symmetric.(2.1)We extend the definition of b ij for ( i, j ) ∈ K × K by: b ij = − b ji if i ∈ K ex and j ∈ K and b ij = 0 for i, j ∈ K ex .To the matrix e B , we associate the quiver Q e B such that the set of vertices is K and thenumber of arrows from i ∈ K to j ∈ K is max(0 , b ij ). Then, Q e B satisfies that(a) the set of vertices of Q e B are labeled by K ,(b) Q e B does not have loops, 2-cycle and arrows between frozen vertices,(c) each exchangeable vertex v of Q e B has finite degree ; that is, the numberof arrows incident with v is finite.(2.2)Conversely, for a given quiver satisfying (2.2), we can associate a matrix e B by b ij := (the number of arrows from i to j ) − (the number of arrows from j to i ) . (2.3)Then e B satisfies (2.1).We say that the pair ( L, e B ) is compatible with a positive integer d , if X k ∈ K λ ik b kj = δ i,j d for each i ∈ K and j ∈ K ex . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 15
Definition 2.1.
For a Z [ q ± / ]-algebra A , a triple S = ( { x i } i ∈ K , L, e B ) consisting of(i) a compatible pair ( L, e B ),(ii) an L -commuting algebraically independent family { x i } i ∈ K is called a quantum seed in A . We also call(a) { x i } i ∈ K the cluster of S and elements in { x i } i ∈ K the cluster variables ,(b) x i ( i ∈ K ex ) the exchangeable variables and x i ( i ∈ K fr ) the frozen variables ,(c) x a (cid:0) a ∈ Z ⊕ K ≥ (cid:1) the quantum cluster monomials ,where x a := q / P s>t a is a it λ is,it x a i i · · · x a ir i r , for a = ( a i ) i ∈ K , { i ∈ K | a i = 0 } ⊂ { i , . . . , i r } . Note that x a does not depend on thechoice of { i , . . . , i r } .For k ∈ K ex , the mutation µ k ( L, e B ) := ( µ k ( L ) , µ k ( e B )) of a compatible pair ( L, e B ) indirection k is a pair ( µ k ( L ) , µ k ( e B )) consisting of a K × K -matrix µ k ( L ) and a K × K ex -matrix µ k ( e B ) defined as follows :(a) µ k ( L ) ij = − λ ij + X t ∈ K max(0 , − b tk ) λ tj if i = k, j = k, − λ ij + X t ∈ K max(0 , − b tk ) λ it if i = k, j = k,λ ij otherwise.(b) µ k ( e B ) ij = ( − b ij if i = k or j = k,b ij + ( − δ ( b ik < max( b ik b kj ,
0) otherwise.(2.4)Then one can check that µ k ( e B ) satisfies (2.1) and the pair ( µ k ( L ) , µ k ( e B )) is compatiblewith the same integer d as in [5].We define a ′ i = ( − i = k, max(0 , b ik ) if i = k, a ′′ i = ( − i = k, max(0 , − b ik ) if i = k. and set a ′ := ( a ′ i ) i ∈ K and a ′′ := ( a ′′ i ) i ∈ K which are contained in Z ⊕ K .Let A be a Z [ q ± / ]-algebra contained in a skew-field F . Let S = ( { x i } i ∈ K , L, e B ) bea quantum seed in A . For k ∈ K ex , we define the elements µ k ( x ) i of K as follows : µ k ( x ) i = ( x a ′ + x a ′′ , if i = k,x i if i = k. Then { µ k ( x ) i } i ∈ K is a µ k ( L )-commuting algebraically independent family. We call µ k ( S ) := (cid:0) { µ k ( x ) i } i ∈ K , µ k ( L ) , µ k ( e B ) (cid:1) the mutation of S in direction k . Definition 2.2.
Let S = ( { x i } i ∈ K , L, e B ) be a quantum seed in A . The quantum clusteralgebra A q / ( S ) associated to the quantum seed S is the Z [ q ± / ]-subalgebra of the skewfield F generated by all the quantum cluster variables in the quantum seeds obtainedfrom S by any sequence of mutations.We call S the initial quantum seed of the quantum cluster algebra A q / ( S ).2.2. Quantum cluster algebras A q / ( n ( w )) . In this subsection, we assume that thegeneralized Cartan matrix A is symmetric. Let e w = s i · · · s i r be a reduced expression of w ∈ W . By Proposition 1.3, D e w ( i,
0) and D e w ( j, q -commute ; i.e., there exists λ ij ∈ Z satisfying D e w ( i, e w ( j,
0) = q λ ij D e w ( j, e w ( i, . Hence we have an integer-valued skew-symmetric matrix L = ( λ ij ) ≤ i,j ≤ r .Set K = { , . . . , r } , K fr = { k ∈ K | k + = r + 1 } , and K ex := K \ K fr .(2.5) Definition 2.3 ([13]) . We define the quiver Q with the set of vertices Q and the set ofarrows Q which is associated to e w as follows:( Q ) Q = K = { , . . . , r } .( Q ) There are two types of arrows: • ordinary arrows : s | a is,it | −−−−−→ t if 1 ≤ s < t < s + < t + ≤ r + 1, • horizontal arrows : s −−−−−→ s − if 1 ≤ s − < s ≤ r .Let e B = ( b ij ) be the integer-valued K × K ex -matrix associated to the quiver Q by (2.3). Proposition 2.4 ([13, Proposition 10.1]) . The pair ( L, e B ) is compatible with d = 2 . Theorem 2.5 ([13, Theorem 12.3], [29, Corollary 11.2.8]) . Let A q / ( S ) be the quantumcluster algebra associated to the initial quantum seed S := ( { q − ( d s ,d s ) / D e w ( s, } ≤ s ≤ r , L, e B ) , where d s := wt (cid:0) D e w ( s, (cid:1) . Then we have Z [ q ± / ] -algebra isomorphism A q / ( S ) ≃ A q / ( n ( w )) := Z [ q ± / ] ⊗ Z [ q ± ] A A ( n ( w )) . Cluster algebras K ( C − g ) . In [17], Hernandez and Leclerc introduced a proper sub-category C − g of C g for untwisted quantum affine algebras which contains all simple objectsof C g up to parameter shifts. The definitions of C − g for types A (1) n and B (1) n can be takenas follows. Take σ − ( g ) the subset of I × k × as σ − ( A (1) n ):= { ( i, ( − q ) p ) | p ≤ p ≡ i mod 2 } , σ − ( B (1) n ):= { ( i, ( − i − q κ q m ) ∈ σ ( B (1) n ) | n + 12 + m ∈ Z ≤ }∪ { ( n, q m ) ∈ σ ( B (1) n ) | m ∈ Z ≤ } . The category C − g is the smallest abelian full subcategory of C g such that LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 17 (a) C − g contains { V ( ̟ i ) x | ( i, x ) ∈ σ − ( g ) } ,(b) it is stable under taking submodules, quotients, extensions and tensor products.Also, they defined the quiver G − g of infinite rank, whose vertices are labeled by a certainsubset { ( i, x ) } of I × Z ≤ (see [17] for details). Let z = { z i,x } be the indeterminateslabeled by the { ( i, x ) } . Theorem 2.6 ([17, Theorem 5.1]) . There exists an isomorphism between the clusteralgebra A associated with the initial seed ( z , G − g ) and the Grothendieck ring K ( C − g ) of C − g . Conjecture 2.7 ([16, Conjecture 13.2], [17, Conjecture 5.2]) . The cluster monomialsof A can be identified with the real simple modules of C − g under the isomorphism inTheorem 2.6.We will give a proof of Conjecture 2.7 for C − A (1) N − in Section 6.2.1.2.4. Monoidal categorification of quantum cluster algebras.
In this subsection,we fix a base field k and a free abelian group Q equipped with a symmetric bilinear form( , ) : Q × Q such that ( β, β ) ∈ Z for all β ∈ Q .Let C be a k -linear abelian monoidal category (see [25, Appendix A.1]) in the sensethat it is abelian and the tensor functor ⊗ is k -bilinear and exact. A simple object M in C is called real if M ⊗ M is simple.We assume that C satisfies the following conditions :(i) Any object of C is of a finite length.(ii) k ∼−→ Hom C ( M, M ) for any simple object M of C .(iii) C admits a direct sum decomposition C = L β ∈ Q C β such that the tensor functor ⊗ sends C β × C γ → C β + γ for every β, γ ∈ Q .(iv) There exists an object Q ∈ C satisfying(a) there is an isomorphism R Q ( X ) : Q ⊗ X ∼−→ X ⊗ Q functorial in X ∈ C that Q ⊗ X ⊗ Y R Q ( X ) / / R Q ( X ⊗ Y ) , , X ⊗ Q ⊗ Y R Q ( Y ) / / X ⊗ Y ⊗ Q commutes for any X, Y ∈ C ,(b) the functor X Q ⊗ X is an equivalence of categories.(v) For any M , N ∈ C , we have Hom C ( M, Q ⊗ n ⊗ N ) = 0 except finitely many integers n (see Remark 2.8 below). Remark 2.8.
Note that, since the functor X Q ⊗ X is an equivalence of categories,there exists an object Q − ∈ C such that Q ⊗ Q − ≃ Q − ⊗ Q ≃ , and the functor X Q − ⊗ X give an equivalence of categories also (see [25, Appendix A.1]). Thus, foreach n ∈ Z , we define Q ⊗ n := Q ⊗ Q ⊗ · · · ⊗ Q | {z } n -times if n ≥ Q − ⊗ Q − ⊗ · · · ⊗ Q − | {z } − n -times if n < We denote by q the auto-equivalence Q ⊗ • , and call it the grading shift functor . Withthe grading shift functor, the Grothendieck ring K ( C ) becomes a Q -graded Z [ q ± ]-algebra.For M ∈ C β , we write β = wt( M ) and call it the weight of M . Similarly, for x ∈ Z [ q ± / ] ⊗ Z [ q ± ] K ( C β ), we write β = wt( x ) and call it the weight of x . Definition 2.9.
A pair ( { M i } i ∈ K , e B ) consisting of(1) a family of real simple objects { M i } i ∈ K of C , where M i ∈ C d i for some d i ∈ Q ,(2) an integer valued K × K ex -matrix e B is called a quantum monoidal seed if it satisfies the following properties:(i) for all i, j ∈ K , there exists an integer λ ij satisfying M i ⊗ M j ≃ q λ ij M j ⊗ M i , (ii) M i ⊗ · · · ⊗ M i t is simple for any finite sequence ( i , . . . , i t ) in K ,(iii) the integer valued K × K ex -matrix e B satisfies (2.1),(iv) ( L, e B ) is compatible with d = 2, where L = ( λ ij ) i,j ∈ K ,(v) λ ij − ( d i , d j ) ∈ Z for all i, j ∈ K , where D := = { d i ∈ Q | M i ∈ C d i } i ∈ K ⊂ Q .(vi) P i ∈ K b ik d i = 0 for all k ∈ K ex .Note that for a quantum monoidal seed ( { M i } i ∈ K , e B ), the matrix L = ( λ ij ) i,j ∈ K andthe family D := { d i } i ∈ K is determined by the family { M i } i ∈ K .Let S = ( { M i } i ∈ K ) , e B ) be a quantum monoidal seed in C . For X ∈ C β and Y ∈ C γ such that X ⊗ Y ≃ q c Y ⊗ X and c + ( β, γ ) ∈ Z , we define e λ ( X, Y ) := 12 (cid:0) − c + ( β, γ ) (cid:1) ∈ Z and X J Y := q e λ ( X,Y ) X ⊗ Y ≃ q e λ ( Y,X ) Y ⊗ X. Then we have X J Y ≃ Y J X .For any finite sequence ( i , . . . , , i ℓ ) in K , we define ℓ J k =1 M i k := ( · · · ( M i J M i ) J · · · ) J M i ℓ − ) J M i ℓ . When the L -commuting family { [ M i ] } i ∈ K of elements in Z [ q ± / ] ⊗ Z [ q ± ] K ( C ) is alge-braically independent, we define a quantum seed [ S ] in Z [ q ± / ] ⊗ Z [ q ± ] K ( C ) by[ S ] = (cid:0) { q − ( d i ,d i ) / [ M i ] } i ∈ K , L, e B (cid:1) . For a given k ∈ K ex , we define the mutation µ k ( D ) ∈ Q of D in direction k with respectto e B by µ k ( D ) i = d i ( i = k ) , µ k ( D ) k = − d k + X b ik > b ik d i . Note that, for any k ∈ K ex , we have µ k ( µ k ( D )) = D , and ( µ k ( L ) , µ k ( e B ) , µ k ( D )) satisfies(v) and (vi) in Definition 2.9. LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 19
For k ∈ K ex , set m k := 12 ( d k , ζ ) + 12 X b ik < λ ki b ik and m ′ k := 12 ( d k , ζ ) + 12 X b ik > λ ki b ik , (2.6)where ζ = − d k + P b ik > b ik d i . Definition 2.10.
We say that a quantum monoidal seed S in C admits a mutation indirection k ∈ K ex if there exists a real simple object M ′ k ∈ C µ k ( D ) k such that(i) there exist exact sequences in C → q J b ik > M ⊙ b ik i → q m k M k ⊗ M ′ k → J b ik < M ⊙ ( − b ik ) i → , → q J b ik < M ⊙ ( − b ik ) i → q m ′ k M ′ k ⊗ M k → J b ik > M ⊙ b ik i → , where m k and m ′ k are given in (2.6).(ii) µ k ( S ) := (cid:0) { M i } i = k ⊔ { M ′ k } , µ k ( e B ) (cid:1) is a quantum monoidal seed in C .We call µ k ( S ) the mutation of S in direction k . Definition 2.11.
Assume that a k -linear abelian monoidal category C satisfies the con-ditions (i)–(v) in the beginning of this subsection. The category C is called a monoidalcategorification of a quantum cluster algebra A over Z [ q ± / ] if(i) Z [ q ± / ] ⊗ Z [ q ± ] K ( C ) is isomorphic to A ,(ii) there exists a quantum monoidal seed S = ( { M i } i ∈ K , e B ) in C such that [ S ] :=( { q − ( d i ,d i ) / [ M i ] } i ∈ K , L, e B ) is a quantum seed of A ,(iii) S admits successive mutations in all the directions.3. Quiver Hecke algebras
Quiver Hecke algebras.
Now we briefly recall the definition of quiver Hecke alge-bra associated to a symmetrizable Cartan datum ( A , P , Π , P ∨ , Π ∨ ) (see [37, 47] for moredetail).Let k be a base field. We take a family of polynomials ( Q i,j ) i,j ∈ I in k [ u, v ] satisfying Q i,j ( u, v ) = δ ( i = j ) × X ( p,q ) ∈ Z ≥ ( α i ,α i ) p +( α j ,α j ) q = − α i ,α j ) t i,j ; p,q u p v q , (3.1)where t i,j ; p,q = t j,i ; q,p and t i,j : − a ij , ∈ k × . Then one can check that Q i,j ( u, v ) = Q j,i ( v, u ).For n ∈ Z ≥ and β ∈ Q + such that | β | = n , we set I β = { ν = ( ν , . . . , ν n ) ∈ I n | α ν + · · · + α ν n = β } . We denote by S n = h s , . . . , s n − i the symmetric group of degree n , where s i := ( i, i + 1)is the transposition of i and i + 1. Then S n acts on I β by place permutations. Definition 3.1.
For β ∈ Q + with | β | = n , the quiver Hecke algebra R ( β ) at β associatedwith a symmetrizable Cartan datum ( A , P , Π , P ∨ , Π ∨ ) and a matrix ( Q i,j ) i,j ∈ I is the k -algebra generated by the elements { e ( ν ) } ν ∈ I β , { x k } ≤ k ≤ n and { τ m } ≤ m ≤ n − satisfying thefollowing defining relations : e ( ν ) e ( ν ′ ) = δ ν,ν ′ e ( ν ) , X ν ∈ I β e ( ν ) = 1 , x k x m = x m x k , x k e ( ν ) = e ( ν ) x k ,τ m e ( ν ) = e ( s m ( ν )) τ m , τ k τ m = τ m τ k if | k − m | > , τ k e ( ν ) = Q ν k ,ν k +1 ( x k , x k +1 ) e ( ν ) , ( τ k x m − x s k ( m ) τ k ) e ( ν ) = − e ( ν ) if m = k, ν k = ν k +1 ,e ( ν ) if m = k + 1 , ν k = ν k +1 , , ( τ k +1 τ k τ k +1 − τ k τ k +1 τ k ) e ( ν )= Q ν k ,ν k +1 ( x k , x k +1 ) − Q ν k ,ν k +1 ( x k +2 , x k +1 ) x k − x k +2 e ( ν ) if ν k = ν k +2 , . The above relations are homogeneous withdeg e ( ν ) = 0 , deg x k e ( ν ) = ( α ν k , α ν k ) , deg τ l e ( ν ) = − ( α ν l , α ν l +1 ) , and R ( β ) is endowed with a Z -graded algebra structure.For a graded R ( β )-module M = L k ∈ Z M k , we define qM = L k ∈ Z ( qM ) k , where( qM ) k = M k − ( k ∈ Z ) . We call q the grading shift functor on the category of graded R ( β )-modules.For graded R ( β )-modules M and N , Hom R ( β ) ( M, N ) denotes the space of degree pre-serving module homomorphisms. We set deg( f ) := k for f ∈ Hom R ( β ) ( q k M, N ).For an R ( β )-module M , we set wt( M ) := − β ∈ Q − and call it the weight of M .Let us denote by R ( β )-gmod the category of graded R ( β )-modules which are finite-dimensional over k . We set R -gmod = L β ∈ Q + R ( β )-gmod . For β, γ ∈ Q + with | β | = m , | γ | = n , we define an idempotent e ( β, γ ) as follows: e ( β, γ ) := X ν ∈ I β + γ , ( ν ,...,ν m ) ∈ I β e ( ν ) ∈ R ( β + γ ) . Then we have an injective ring homomorphism R ( β ) ⊗ R ( γ ) / / / / e ( β, γ ) R ( β + γ ) e ( β, γ ) . For an R ( β )-module M and an R ( γ )-module N , LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 21 • the convolution product M ◦ N is an R ( β + γ )-module defined by M ◦ N = R ( β + γ ) e ( β, γ ) ⊗ R ( β ) ⊗ R ( γ ) ( M ⊗ N ) , • the dual space M ∗ := Hom k ( M, k ) admits an R ( β )-module structure via( r · f )( u ) := f ( ψ ( r ) u ) ( r ∈ R ( β ) , u ∈ M ) , where ψ denotes the k -algebra anti-involution on R ( β ) fixing the generators.We denote by u ⊠ v the image of u ⊗ v in M ◦ N .A simple module M in R -gmod is called self-dual if M ∗ ≃ M . Every simple module isisomorphic to a grading shift of a self-dual simple module ([37, § R -gmod has a monoidal category structure with ◦ as a tensor product. Letus denote by K ( R -gmod) the Grothendieck ring of R -gmod which is an algebra over A = Z [ q ± ] with the multiplication induced by the convolution product and the A -actioninduced by the grading shift functor q .In [37, 38, 47], it is shown that a quiver Hecke algebra categorifies the correspondingunipotent quantum coordinate ring. More precisely, we have the following theorem. Theorem 3.2 ([37, 38, 47]) . For a given symmetrizable Cartan datum ( A , P , Π , P ∨ , Π ∨ ) ,we take a parameter matrix ( Q ij ) i,j ∈ I satisfying the conditions in (3.1) , and let A q ( n ) and R ( β ) be the associated unipotent quantum coordinate ring and quiver Hecke algebra,respectively. Then there exists an A -algebra isomorphism ch : K ( R - gmod) ∼−→ A A ( n ) . (3.2) Definition 3.3.
We say that a quiver Hecke algebra R is symmetric if Q i,j ( u, v ) is apolynomial in u − v for all i, j ∈ I .In particular, the corresponding generalized Cartan matrix A is symmetric. In sym-metric case, we assume d i = 1 for all i ∈ I . Theorem 3.4 ([48, 50]) . Assume that the quiver Hecke algebra R is symmetric and thebase field k is of characteristic . Then, under the isomorphism (3.2) in Theorem 3.2 ,the upper global basis B up ( A q ( n )) corresponds to the set of the isomorphism classes ofself-dual simple R -modules. R-matrices.
For β, γ ∈ Q + with | β | = m and | γ | = n , let M be an R ( β )-moduleand N an R ( γ )-module. Then, by [25, Lemma 1.5], there exists an R ( β + γ )-modulehomomorphism (up to a grading shift) R M,N : M ◦ N −−→ N ◦ M, (3.3)which is defined by intertwiners ϕ k ∈ R ( β + γ ) (1 ≤ k ≤ m + n −
1) (see [25, § For the rest of this paper, we assume that quiver Hecke algebra is symmetric and d i = 1 for all i ∈ I . We also work always in the category of graded R -modules . Then each R ( β )-module M admits an affinization M z ≃ k [ z ] ⊗ k M with the action of R ( β ) twisted by the algebra homomorphism ψ z : R ( β ) → k [ z ] ⊗ R ( β )(see [25, § x k to x k + z , where z is an indeterminate of homogeneous degree2. By [25, Proposition 1.10], the R ( β + γ )-module homomorphism (up to a grading shift) R M z ,N : M z ◦ N −−→ N ◦ M z induces an R ( β + γ )-module homomorphism r M,N : M ◦ N → q − Λ( M,N ) N ◦ M, which also satisfies the Yang-Baxter equation and is non-zero provided that M and N arenon-zero. Here Λ( M, N ) denotes the degree of r M,N and we call r M,N the R -matrix . Definition 3.5 ([29]) . For non-zero R -modules M and N in R -gmod, we define integers d ( M, N ) and e Λ( M, N ) as follows :(a) d ( M, N ) = 12 (Λ(
M, N ) + Λ(
N, M )) ∈ Z ≥ ,(b) e Λ( M, N ) = 12 (cid:0) Λ( M, N ) + (cid:0) wt( M ) , wt( N ) (cid:1)(cid:1) ∈ Z ≥ .A simple module M ∈ R -gmod is called real if M ◦ M is simple. Lemma 3.6 ([27]) . Let M and N be simple modules in R - gmod , and assume that one ofthem is real. Then (i) M ◦ N and N ◦ M have simple socles and simple heads. (ii) Im( r M,N ) is equal to the head of M ◦ N and the socle of N ◦ M . For R -modules M and N , we denote by M ∇ N the head of M ◦ N and by M ∆ N thesocle of M ◦ N . Proposition 3.7 ([27, Corollary 3.7]) . For β, γ ∈ Q + , let M be a real simple R ( β ) -module. Then the map N M ∇ N is injective from the set of the isomorphism classesof simple objects N of R ( γ ) - gmod to the set of the isomorphism classes of simple objectsof R ( β + γ ) - gmod . Lemma 3.8 ([29, Lemma 3.1.4]) . Let M and N be self-dual simple modules. If one ofthem is real, then q e Λ( M,N ) M ∇ N is a self-dual simple module . Thus e Λ( M, N ) indicates the degree shift that makes M ∇ N self-dual. Definition 3.9.
For non-zero R -modules M and M , we set M J M := q e Λ( M ,M ) M ◦ M . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 23
The non-negative integer d ( M, N ) measures the degree of complexity of M ◦ N as seenin the following lemma. Lemma 3.10 ([27, 29]) . Let M and N be simple modules in R - gmod , and assume thatone of them is real. (i) M ◦ N is simple if and only if d ( M, N ) = 0 . (ii) If d ( M, N ) = 1 , then M ◦ N has length , and there exists an exact sequence → M ∆ N → M ◦ N → M ∇ N → . Definition 3.11.
For simple R -modules M and N , we say that(i) M and N strongly commute if M ◦ N is simple,(ii) M and N are simply-linked if d ( M, N ) = 1.3.3.
Determinantial modules.
The set of self-dual simple R -modules corresponds tothe upper global basis of A q ( n ) by Theorem 3.4.Recall the A -algebra isomorphism ch in (3.2). Definition 3.12 ([29, § § . For Λ ∈ P + and η, ζ ∈ W Λ suchthat η (cid:22) ζ , let M( η, ζ ) be the self-dual simple R ( ζ − η )-module such that ch (cid:0) M( η, ζ ) (cid:1) =D( η, ζ ) . By Proposition 1.3 (i), the module M( η, ζ ) is real. We call M( η, ζ ) the determinantialmodule . We also write M e w ( s, t ) (see (1.3)) which is the determinantial module such thatD e w ( s, t ) = ch(M e w ( s, t )) . (3.4) Theorem 3.13 ([29, Theorem 10.3.1], [33, Proposition 4.6]) . For Λ ∈ P + and η , η , η ∈ W Λ with η (cid:22) η (cid:22) η , we have M( η , η ) ∇ M( η , η ) ≃ M( η , η ) . Now we recall the definition of several functors on R -modules. Definition 3.14.
Let β ∈ Q + .(i) For i ∈ I and 1 ≤ a ≤ | β | , set e a ( i ) = X ν ∈ I β ,ν a = i e ( ν ) and e ∗ a ( i ) = X ν ∈ I β ,ν | β | +1 − a = i e ( ν ) ∈ R ( β ) . (ii) For M ∈ R ( β )-gmod, E i M := e ( i ) M and E ∗ i M = e ∗ ( i ) M, which are functors from R ( β )-gmod to R ( β − α i )-gmod.(iii) Let L ( i ) be the 1-dimensional R ( α i )-module R ( α i ) /R ( α i ) x . For a simple R -module M , we set ε i ( M ) = max { n ∈ Z ≥ | E ni M = 0 } ,ε ∗ i ( M ) = max { n ∈ Z ≥ | E ∗ ni M = 0 } , e F i M = q ε i ( M ) L ( i ) ∇ M, e F ∗ i M = q ε ∗ i ( M ) M ∇ L ( i ) , e E i M = q − ε i ( M ) soc( E i M ) , e E ∗ i M = q − ε ∗ i ( M ) soc( E ∗ i M ) . Here, for an R -module N , soc( N ) denotes the socle of N and hd( N ) denotes thehead of N .(iv) For i ∈ I and n ∈ Z ≥ , we set L ( i n ) = q n ( n − / L ( i ) ◦ n which is a self-dual realsimple R ( nα i )-module. Proposition 3.15 ([29, Proposition 10.2.3]) . Let Λ ∈ P + , η, ζ ∈ W Λ such that η (cid:22) ζ and i ∈ I . (i) If n := h h i , η i ≥ , then ε i (M( η, ζ )) = 0 and M( s i η, ζ ) ≃ e F ni M( η, ζ ) ≃ L ( i n ) ∇ M( η, ζ ) in R - gmod . (ii) If h h i , η i ≤ and s i η (cid:22) ζ , then we have ε i (M( η, ζ )) = −h h i , η i and M( s i η, ζ ) ≃ e E −h h i ,η i i M( η, ζ ) . (iii) If m := −h h i , ζ i ≥ , then ε ∗ i (M( η, ζ )) = 0 and M( η, s i ζ ) ≃ e F ∗ mi M( η, ζ ) ≃ M( η, ζ ) ∇ L ( i m ) in R - gmod . (iv) If h h i , ζ i ≥ and η (cid:22) s i ζ , then ε ∗ i (M( η, ζ )) = h h i , ζ i and M( η, s i ζ ) ≃ e E h h i ,ζ i i M( η, ζ ) . Admissible pair and A q / ( n ( w )) . In this subsection, we review the results in [29]on the monoidal categorification by using graded modules over quiver Hecke algebras.Throughout this subsection, we focus on a category C which is a full subcategory of R -gmod which is stable under taking convolution products, subquotients, extensions, andgrading shift. Then we have C = L β ∈ Q − C β where C β := C ∩ R ( − β )-gmod . Definition 3.16 (cf. Definition 2.9) . A pair ( { M i } i ∈ K , e B ) consisting of(i) a family of self-dual real simple objects { M i } i ∈ K of C strongly commuting witheach other,(ii) an integer valued K × K ex -matrix e B satisfying (2.1),is called admissible if, for each k ∈ K ex , there exists an object M ′ k of C such that(a) there is an exact sequence in C → q J b ik > M ⊙ b ik i → q e Λ( M k ,M ′ k ) M k ◦ M ′ k → J b ik < M ⊙ ( − b ik ) i → , (3.5)(b) M ′ k is self-dual simple and strongly commutes with M i for any i = k .For an admissible pair ( { M i } i ∈ K , e B ), we can take • − Λ = ( − Λ i,j ) i,j ∈ K the skew-symmetric integer-valued matrix byΛ i,j := Λ( M i , M j ) , • D = { d i } the family of elements in Q − by d i = wt( M i ), LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 25 as in Section 2.4.
Proposition 3.17 ([29, Proposition 7.1.2]) . For an admissible pair ( { M i } i ∈ K , e B ) , wehave the following properties :(i) S := ( { M i } i ∈ K , e B ) is a quantum monoidal seed in C . (ii) The self-dual simple object M ′ k is real for every k ∈ K ex . (iii) The quantum monoidal seed S admits a mutation in each direction k ∈ K ex . (iv) M k and M ′ k is simply-linked for any k ∈ K ex ( i.e., d ( M k , M ′ k ) = 1) . For an admissible pair ( { M i } i ∈ K , e B ), let us denote by[ S ] := ( { q − (wt( M i ) , wt( M i )) [ M i ] } i ∈ K , − Λ , e B ) . Theorem 3.18 ([29, Theorem 7.1.3, Corollary 7.1.4]) . For an admissible pair ( { M i } i ∈ K , e B ) ,we assume that Z [ q ± / ] ⊗ Z [ q ± ] K ( C ) is isomorphic to the quantum cluster algebra A q / ([ S ]) . Then C is a monoidal categorification of the quantum cluster algebra A q / ([ S ]) .In particular, the following statements holds :(i) Each cluster monomial in A q / ([ S ]) corresponds to the isomorphism class of areal simple object in C up to a power of q / . (ii) Each cluster monomial in A q / ([ S ]) is a Laurent polynomial of the initial clustervariables with coefficient in Z ≥ [ q ± / ] . Definition 3.19.
For w ∈ W , let C w be the smallest full subcategory of R -gmod satisfyingthe following properties:(i) C w is stable by convolution, taking subquotients, extensions, and grading shifts,(ii) C w contains M e w ( s, s − ) (1 ≤ s ≤ ℓ ( w )), where e w is a reduced expression of w .By [13], we have an A -algebra isomorphism K ( C w ) ≃ A A ( n ( w )) . Let e B be the integer-valued K × K ex -matrix associated to e w (see Definition 2.3). Theorem 3.20 ([29, Theorem 11.2.2, Theorem 11.2.3]) . The pair ( { M e w ( s, } ≤ s ≤ r , e B ) is admissible. Thus C w is a monoidal categorification of the quantum cluster algebra A q / ( n ( w )) , with the quantum monoidal seed ( { M e w ( s, } ≤ s ≤ r , e B ) .Furthermore, the self-dual real simple R -module M e w ( k, ′ for k ∈ K ex in (3.5) is givenas follows : ( up to a grade shift )M e w ( k, ′ ≃ M e w ( k + , k ) ∇ (cid:16) J t In this subsection,we recall the generalized quantum affine Schur-Weyl duality functor in [25].Let us assume that we are given an index set J and a family { V j } j ∈ J of quasi-good U ′ q ( g )-modules.We define a quiver Γ J associated with the pair ( J, { V j } j ∈ J ) as follows:(a) we take J as the set of vertices,(b) we put d ij many arrows from i to j , where d ij denotes the order of zero of d V i ,V j ( u ) at u = 1.(3.7)Note that we have d ij d ji = 0 for i, j ∈ J (see [25, Theorem 2.2]).We define a symmetric Cartan matrix A J = ( a Jij ) i,j ∈ J by a Jij = 2 if i = j and a Jij = − d ij − d ji otherwise. Then we choose a family of polynomial { Q i,j ( u, v ) } i,j ∈ J satisfying (3.1) withthe form, Q i,j ( u, v ) = ± ( u − v ) − a Jij for i = j for some choices of sign ± .Now we take a family { P i,j ( u, v ) } i,j ∈ J of elements in k [[ u, v ]] satisfying the followingconditions :(a) P i,i ( u, v ) = 1 for i ∈ J .(b) The homomorphism P ij ( u, v ) R norm V i ,V j ( z V i , z V j ) has neither pole nor zero at u = v = 0, where k [ z ± V i , z ± V j ] → k [[ u, v ]] is given by z V i u and z V j v .(c) Q i,j ( u, v ) = P i,j ( u, v ) P j,i ( v, u ) for i = j .(3.8)We call such a family { P i,j ( u, v ) } i,j ∈ J a duality coefficient .Let { α Ji | i ∈ J } be the set of simple roots associated to A J and Q + J = P i ∈ J Z ≥ α Ji be the corresponding positive root lattice. We define the symmetric bilinear form ( , )satisfying ( α Ji , α Jj ) = a Jij .Let us denote by R J ( β ) ( β ∈ Q + J ) the symmetric quiver Hecke algebra associated with A J and { Q i,j ( u, v ) } i,j ∈ J .In [25, § generalized quantumaffine Schur-Weyl duality functor F : L β ∈ Q + J R J ( β )-gmod → C g , (3.9)which is a monoidal functor in the following sense: There exist canonical U ′ q ( g )-isomorphisms F ( R J (0)) ≃ k , F ( M ◦ M ) ≃ F ( M ) ⊗ F ( M )for any M , M ∈ R J -gmod such that the diagrams in [25, (A.2)] are commutative. Proposition 3.21 ([25, Proposition 3.2.2]) . The monoidal functor F is a unique ( up toan isomorphism ) functor which satisfies the following properties: LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 27 (i) For any i ∈ J , we have F ( L ( i ) z ) ≃ k [[ z ]] ⊗ k [ z ± Vi ] ( V i ) aff , where k [ z ± V i ] → k [[ z ]] is given by z V i z . In particular, we have F ( L ( i )) ≃ V i for i ∈ J. (ii) For i, j ∈ J , let R L ( i ) z ,L ( j ) z ′ : L ( i ) z ◦ L ( j ) z ′ → L ( j ) z ′ ◦ L ( i ) z be the R J ( α Ji + α Jj ) -module homomorphism in (3.3) . Then we have F ( R L ( i ) z ,L ( j ) z ′ ) = P i,j ( z, z ′ ) R norm V i ,V j ( z V i , z V j ) . (3.10)Note that the functor F depends on the choice of { P i,j ( u, v ) } i,j ∈ J as seen in (3.10). Lemma 3.22 ([32, Lemma 1.7.8]) . Let M , N ∈ R J - gmod be simple modules, and assumethat one of them is real. Assume that (a) the functor F in (3.9) is exact, (b) d ( M, N ) ≤ , (c) F ( M ) , F ( N ) = 0 and F ( M ) ⊗ F ( N ) is not simple.Then we have (i) F ( r M,N ) = r F ( M ) , F ( N ) up to a non-zero constant multiple, (ii) F ( M ∇ N ) ≃ F ( M ) ∇ F ( N ) which is simple. Theorem 3.23 ([25, Theorem 3.8]) . If the Cartan matrix A J associated with Γ J is oftype A n ( n ≥ , D n ( n ≥ or E n ( n = 6 , , , then the functor F is exact. Cluster structure on R A ∞ - gmodIn this section, we study the cluster structure on the monoidal category R A ∞ -gmod.Here Dynkin diagram of type A ∞ is depicted as follows: ◦ − ◦ ◦ ◦ ◦ We first introduce an infinite sequence e w of simple reflections whose first p -parts is reducedfor every p ∈ Z ≥ . Then we explicitly compute determinantial modules of type A ∞ whichare associated to e w . In the last part of this section, we will construct a certain quantummonoidal seed S ∞ and prove that the category R J -gmod of type A ∞ gives a monoidalcategorification of the quantum cluster algebra A A ( n ) of type A ∞ whose initial quantumseed is [ S ∞ ].4.1. Sequence of simple reflections of length ∞ . Let J = Z be an index set. We alsotake the weight lattice P J = L i ∈ Z Z Λ i with (Λ i , Λ j ) = −| i − j | / 2. We set ǫ a = Λ a − Λ a − and α j = ǫ j − ǫ j +1 . We have ( ǫ a , ǫ b ) = δ a,b and (Λ i , α j ) = δ i,j . The matrix A J := (cid:0) ( α i , α j ) (cid:1) i,j ∈ J is a Cartan matrix of type A ∞ . We write the root lattice Q J = L j ∈ J Z α j ⊂ P J . Let W J be the Weyl group of type A ∞ generated by the set of simple reflections S J := { s j | j ∈ J } . Note that, for all i, j ∈ J , we have s i ( ǫ j ) = ǫ s i ( j ) and s i (Λ j ) = ( Λ j +1 − ǫ j = Λ j − + ǫ j +1 if i = j, Λ j otherwise . (4.1)For k ∈ Z > , we sometimes use the notation k to denote − k ∈ Z < ⊂ J .A pair of integers [ a, b ] with a ≤ b is called a segment . The length of [ a, b ] is defined tobe the positive integer b − a + 1. For a segment [ a, b ], we denote by W [ a,b ] the subgroupof W J generated by s k for a ≤ k ≤ b .Recall the convention in (1.1) and (1.2) on sequences of simple reflections. For each t ∈ Z ≥ , let w ( t ) be the sequence in S J of length 4 t + 3 defined by : w ( t ) = s t s t +1 · · · s s s · · · s t − s t s t +1 s t s t − · · · s s s s · · · s t +1 s t . We set w ( s,t ) := ( w ( s ) ∗ w ( s +1) ∗ · · · w ( t − ∗ w ( t ) if 0 ≤ s ≤ t, id otherwise,where ∗ denotes the concatenation of sequences.Finally, we define an infinite sequence e w of S J by :(4.2) e w := lim t →∞ w (0 ,t ) = w (0) ∗ w (1) ∗ w (2) ∗ · · · = s s s s s s s s s s s s s s s s s s s s s · · · · · · . We define j p ∈ J ( p ∈ Z ≥ ) by e w = s j s j s j · · · · · · . For t ∈ Z , we set a ( t ) := t (2 t + 1) ∈ Z . Note that, for t ∈ Z ≥ , a ( t ) coincides with the length of w (0 ,t − .We have for t ∈ Z ≥ j p = p − a ( t ) − t − p − a ( t + 1 / 2) + t if a ( t ) + 1 ≤ p ≤ a ( t + 1 / 2) + 1, t + 2 + a ( t + 1 / − p = − t + a ( t + 1) − p if a ( t + 1 / 2) + 1 ≤ p ≤ a ( t + 1) + 1.(4.3)For each p ∈ Z ≥ , we will denote the element in the Weyl group W J defined by thesequence e w ≤ p by the same symbol. Proposition 4.1. The sequence e w has the following properties : LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 29 (i) For t ∈ Z ≥ , we have e w ≤ p − ( α j p ) = ǫ − t − ǫ t +2 − p + a ( t ) = ǫ − t − ǫ − t + a ( t +1 / − p if a ( t ) + 1 ≤ p ≤ a ( t + 1 / , ǫ − t − p − a ( t +1 / − ǫ t +2 = ǫ t +1+ p − a ( t +1) − ǫ t +2 if a ( t + 1 / 2) + 1 ≤ p ≤ a ( t + 1) . (ii) For any p ∈ Z ≥ , e w ≤ p is reduced.Proof. Let us first show (i). Note that, for each t ∈ Z ≥ , we have(a) j a ( t )+1 = − t and j p > − t for all p ≤ a ( t ),(b) j a ( t + )+1 = t + 1 and j p < t + 1 for all p < a ( t + ) + 1.Assume that a ( t ) + 1 ≤ p ≤ a ( t + ). Then we have e w ≤ p − ( α j p ) = e w ≤ a ( t ) s − t · · · s j p − ( ǫ j p − ǫ j p +1 ) = e w ≤ a ( t ) ( ǫ − t − ǫ j p +1 ) . Note that e w ≤ a ( t ) is contained in W [ − t +1 ,t ] as a Weyl group element. Thus, for all a ( t ) + 1 ≤ p ≤ a ( t + ), we have e w ≤ p − ( α j p ) = ǫ − t − ǫ e w ≤ a ( t ) ( j p +1) = ǫ − t − ǫ e w ≤ a ( t ) ( − t + p − a ( t )) . Now we claim that e w ≤ a ( t ) ( − t + p − a ( t )) = t + 2 − ( p − a ( t )) . Note that • e w ≤ a ( t ) = w (0 ,t − = w (0 ,t − ∗ w ( t − , • w ( t − ( − t + 1) = t + 1 and w ( t − ( t + 1) = − t + 1, • w ( t − ( k ) = k for all k ∈ Z \ { t + 1 , − t + 1 } .Thus the claim follows from an induction on t . Hence we have e w ≤ p − ( α j p ) = ǫ − t − ǫ t +2 − p + a ( t ) .The case when a ( t + ) + 1 ≤ p ≤ a ( t + 1) can be proved in a similar way.(ii) follows from (i) since e w ≤ p − α j p is a positive root for any p ∈ Z ≥ . (cid:3) Remark 4.2. (a) For each k ∈ Z ≥ , a ( k ) coincides with the length of the longest element of Weylgroup of type A k . Thus e w ≤ a ( k ) is the longest element of ( W [ − k +1 ,k ] if k ∈ Z ≥ ,W [ − k + ,k − ] if k ∈ + Z ≥ .(b) For t > 0, the reduced expressions w (0 ,t ) are not adapted ([2]) in the sense that thereexists no Dynkin quiver Q of type A t +2 satisfying i k is a sink of s i k − s i k − · · · s i ( Q ) for all k. Here s i ( Q ) denotes the quiver obtained by reversing all arrows incident with i . Definition 4.3. For each p ∈ Z ≥ , we assign a pair c ( p ) = ( ℓ, m ) of positive integers inthe following way :( ℓ, m ) := (cid:0) p − a ( t ) , a ( t + ) − p + 1 (cid:1) if a ( t ) < p ≤ a ( t + ) , (2 t + 2 , 1) if p = a ( t + ) + 1 , (cid:0) a ( t + 1) + 1 − p, p − a ( t + ) (cid:1) if a ( t + ) + 2 ≤ p ≤ a ( t + 1) , (4.4)where t is a unique non-negative integer satisfying a ( t ) < p ≤ a ( t + 1) . (4.5) Remark 4.4. The map c : Z ≥ → Z ≥ × Z ≥ in (4.4) satisfies the following properties : • ℓ + m = ( t + 2 ≡ a ( t ) < p ≤ a ( t + ) , t + 3 ≡ a ( t + ) < p ≤ a ( t + 1) . • c is a bijective map whose inverse is given as follows : c − ( ℓ, m ) = ( a ( ℓ + m − ) + ℓ if ℓ + m ≡ ,a ( ℓ + m − ) + m if ℓ + m ≡ . (4.6) • The integer t in (4.5) is equal to ⌊ ( ℓ + m − / ⌋ = ⌈ ( ℓ + m − / ⌉ .Using the lattice point Z ≥ × Z ≥ , the pairs ( p, j p ) corresponding to c ( p ) = ( ℓ, m ) canbe exhibited as follows: ... ... , − · · · , − 1) (12 , − · · · , − 1) (9 , 0) (13 , · · · , 0) (5 , 0) (8 , 1) (14 , · · · , 0) (2 , 1) (6 , , (7 , 2) (15 , · · · m/ℓ · · · Lemma 4.5. For p ∈ Z ≥ with c ( p ) = ( ℓ, m ) , the index j p is given as follows : j p = ⌊ ( ℓ − m + 1) / ⌋ = ⌈ ( ℓ − m ) / ⌉ = ( ( ℓ − m ) / if ℓ + m ≡ , ( ℓ − m + 1) / if ℓ + m ≡ .Proof. It is enough to show that ℓ − m − j p is 0 or − 1. Assume that a ( t ) < p ≤ a ( t + 1 / t ∈ Z ≥ . Then by (4.3) and (4.4), we have ℓ − m − j p = (cid:0) p − a ( t ) (cid:1) − (cid:0) a ( t + 1 / − p + 1 (cid:1) − (cid:0) p − a ( t ) − t − (cid:1) − (cid:0) p − a ( t + 1 / 2) + t (cid:1) = 0 . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 31 If a ( t + 1 / 2) + 2 ≥ p ≤ a ( t + 1), then we have ℓ − m − j p = (cid:0) a ( t + 1) + 1 − p (cid:1) − (cid:0) p − a ( t + 1 / (cid:1) − (cid:0) t + 2 + a ( t + 1 / − p (cid:1) − (cid:0) − t + a ( t + 1) − p (cid:1) = − . If p = a ( t + 1 / 2) + 1, then we have ℓ − m − j p = (cid:0) t + 2 (cid:1) − − (cid:0) t + 2 + a ( t + 1 / − p (cid:1) = − . (cid:3) Determinantial modules of type A ∞ . We keep the notations in the previoussubsection. For i, j ∈ J , let us set Q Ji,j ( u, v ) = ± ( u − v ) if j = i ± , i = j, . We denote by R J the quiver Hecke algebra of type A ∞ which is associated to ( Q Ji,j ) i,j ∈ J . In the sequel, we sometimes drop the subscript J , if there is no danger of confusion. A multisegment is a finite sequence of segments. We assign a total order on the set ofsegments as follows :[ a , b ] > [ a , b ] if a > a , or a = a and b > b . If a multisegment (cid:0) [ a , b ] , . . . , [ a r , b r ] (cid:1) satisfies [ a k , b k ] ≥ [ a k +1 , b k +1 ] for each k , we callit an ordered multisegment .For j ∈ J and a pair of positive integer ( ℓ, m ) ∈ Z ≥ × Z ≥ , we define an orderedmultisegment J ℓ, m K ( j ) as follows : J ℓ, m K ( j ) := (cid:0) [ j − ℓ + m, j + m − , . . . , [ j − ℓ + 2 , j + 1] , [ j − ℓ + 1 , j ] | {z } m -times (cid:1) . For each segment [ a, b ] of length ℓ , there exists a graded 1-dimensional R ( ǫ a − ǫ b +1 )-module L [ a, b ] = k u [ a, b ] which is generated by a vector u [ a, b ] of degree 0. The action of R ( ǫ a − ǫ b +1 ) is given as follows (see [25, Lemma 1.16]) : x k u [ a, b ] = 0 , τ m u [ a, b ] = 0 , e ( ν ) u [ a, b ] = ( u [ a, b ] if ν = ( a, a + 1 , . . . , b ) , . Then one can check that L [ a, b ] ≃ e F a e F a +1 · · · e F b · ≃ e F ∗ b · · · e F ∗ a +1 e F ∗ a · , where is the trivial R (0)-module. Proposition 4.6 ( [41, Theorem 7.2 and Section 8.4]) . (i) Let M be a simple module in R J ( β ) - gmod with β ∈ Q + J . Then there exists a uniquepair of an ordered multisegment (cid:0) [ a , b ] , . . . , [ a t , b t ] (cid:1) and c ∈ Z such that M ≃ q c hd (cid:0) L [ a , b ] ◦ · · · ◦ L [ a t , b t ] (cid:1) , where hd denotes the head. (ii) For an ordered multisegment (cid:0) [ a , b ] , . . . , [ a t , b t ] (cid:1) , hd (cid:0) L [ a , b ] ◦ · · · ◦ L [ a t , b t ] (cid:1) is a simple R J ( β ) -module, where β = P tk =1 ( ǫ a k − ǫ b k +1 ) . We call the ordered multisegment (cid:0) [ a , b ] , . . . , [ a t , b t ] (cid:1) in Proposition 4.6 (i) the multi-segment associated with M . It is uniquely determined by M .For j ∈ J and a pair of positive integer ( ℓ, m ) ∈ Z ≥ × Z ≥ , we define a self-dual simple R J -module W ( ℓ ) m,j associated to J ℓ, m K ( j ) as follows : (up to a grading shift)W ( ℓ ) m,j ≃ hd( L [ j − ℓ + m, j + m − ◦ · · · ◦ L [ j − ℓ + 2 , j + 1] ◦ L [ j − ℓ + 1 , j ] | {z } m -times ) . (4.7) Remark 4.7. The R J -module W ( ℓ ) m,j is known to be homogeneous in the sense that itsgrading is concentrated in a single degree ([40]).Since e w ≤ p is reduced for any p ∈ Z ≥ , we can define M e w ( p , p ) for 0 ≤ p ≤ p (see(1.3) and (3.4)): M e w ( p , p ) := M e w ≤ p ( p , p ) . Note that, for any w ∈ W J and Λ j ( j ∈ J ), we have − ≤ h h k , w Λ j i ≤ k ∈ J . Let us find the multisegment corresponding to M e w ( p, j ∈ J and ( ℓ, m ) ∈ Z ≥ × Z ≥ , we shall define a sequence e w J ℓ,m K ( j ) in S J which isreduced (as seen in Lemma 4.9 below), and arises from the ordered multisegment J ℓ, m K ( j ) as follows : • e w [ a,b ] := s a s a +1 · · · s b − s b for a ≤ b . • e w J ℓ,m K ( j ) = e w [ j − ℓ + m,j + m − · · · · · · e w [ j − ℓ +2 ,j +1] e w [ j − ℓ +1 ,j ] | {z } m -times . Proposition 4.8. For every p ∈ Z ≥ , e w ≤ p Λ j p = e w J ℓ,m K ( j p ) Λ j p , where c ( p ) = ( ℓ, m ) . Proof. We shall use the induction on ℓ + m . Let t be a unique non-negative integer in (4.5).(a) Assume that a ( t ) < p ≤ a ( t + ). In this case, we have • ℓ + m ≡ • j p + t +1 = ℓ since j a ( t )+1 = − t and j a ( t )+ u = − t + u − ≤ u ≤ a ( t + ) − a ( t )+1.Then we have e w ≤ p = w (0 ,t − s t s t +1 · · · s j p − s j p = w (0 ,t − ∗ e w [ − t, j p ] . Using (4.1), we have(4.8) e w [ − t, j p ] Λ j p = s t s t +1 · · · s j p − s j p Λ j p = Λ − t − + j p +1 X k = − t +1 ǫ k = Λ j p +1 − ǫ − t . LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 33 Note that s k (Λ j p +1 − ǫ − t ) = Λ j p +1 − ǫ − t for − t + 1 ≤ k ≤ j p . Thus we have w (0 ,t − (Λ j p +1 − ǫ − t ) = e w ≤ p − ( j p +1) (Λ j p +1 − ǫ − t ) , where c ( p − ( j p + 1)) = ( ℓ, m − 1) by Lemma 4.5. (If m = 1, i.e. p = a ( t + 1 / p − ( j p + 1) = 0 and e w ≤ p − ( j p +1) = 1.)Note that, w (0 ,t − ∈ W [ − t +1 ,t ] and W [ − t +1 ,t ] fixes ǫ − t . Hence we have e w ≤ p Λ j p = e w ≤ p − ( j p +1) Λ j p +1 − ǫ − t . By the induction hypothesis, we have e w ≤ p − ( j p +1) Λ j p +1 = e w J ℓ,m − K ( j p +1) Λ j p +1 , and one can check that the following two reduced expressions coincide e w J ℓ,m − K ( j p +1) ∗ e w [ − t, j p ] = e w J ℓ,m K ( j p ) , where ∗ denotes the concatenation. Note that e w ≤ p − ( j p +1) ∈ W [ − t +1 ,t ] also fixes ǫ − t . Thenour assertion follows from (4.8).(b) Assume that a ( t + ) < p ≤ a ( t + 1). Then one can check that(4.9) e w ≤ p Λ j p = w (0 ,t − s t s t +1 · · · s j p − s j p · · · s t s t +1 s t · · · s j p +1 s j p Λ j p = w (0 ,t − s t s t +1 · · · s j p − s j p · · · s t (cid:0) Λ j p − + ǫ t +2 (cid:1) = w (0 ,t − s t s t +1 · · · s j p − (cid:0) Λ j p − + ǫ t +2 (cid:1) = e w ≤ p − ( j p − (cid:0) Λ j p − + ǫ t +2 (cid:1) , as in the previous case. Note that • ℓ + m ≡ • t + 2 − j p = m and c ( p − ( j p − ℓ − , m ), by Lemma 4.5, • e w ≤ p − ( j p − ∈ W [ t, − t ] ,By the induction hypothesis, we have e w ≤ p − ( j p − Λ j p − = e w J ℓ − ,m K ( j p − Λ j p − , and one can check that the following two reduced expressions are in the same equivalenceclass e w J ℓ − ,m K ( j p − ∗ s t +1 s t · · · s j p +1 s j p ≡ e w J ℓ,m K ( j p ) with respect to the commutation relation s i s j ≡ s j s i ( | i − j | > (cid:3) Lemma 4.9. For j ∈ J and ( ℓ, m ) ∈ Z ≥ × Z ≥ , we have (i) e w J ℓ,m K ( j ) is reduced, (ii) e w J ℓ,m K ( j ) Λ j = Λ j + m − P mk =1 ǫ j − ℓ + k , (iii) M( e w J ℓ,m K ( j ) Λ j , Λ j ) ≃ W ( ℓ ) m,j . Proof. Let us prove (i) and (ii) by induction on m . Write e w := e w J ℓ,m K ( j p ) = s i s i · · · s i r .Now we shall prove that h h i s , e w ≤ s − Λ j p i = 1 for all 1 ≤ s ≤ r. (4.10)By the induction hypothesis, we have e w J ℓ,m − K ( j ) (Λ j ) = Λ j + m − − m − X k =1 ǫ j − ℓ + k . Hence, we have e w J ℓ,m K ( j ) (Λ j ) = s j − ℓ + m · · · s j + m − e w J ℓ,m − K ( j ) (Λ j )= s j − ℓ + m · · · s j + m − (Λ j + m − − m − X k =1 ǫ j − ℓ + k ) . Since s j − ℓ + m · · · s j + m − (Λ j + m − ) = s j − ℓ + m · · · s j + m − (Λ j + m − ǫ j + m − ) = Λ j + m − ǫ j − ℓ + m and h h j + m − k , s j + m − k +1 · · · s j + m − e w J ℓ,m − K ( j ) Λ j i = 1 for 1 ≤ k ≤ ℓ ,(ii) and (4.10) follow. (i) follows from (4.10).(iii) Let us prove (iii) by induction on m . By Theorem 3.13, we haveM( e w J ℓ,m K ( j ) Λ j , Λ j ) ≃ M( e w J ℓ,m K ( j ) Λ j , e w J ℓ,m − K ( j ) Λ j ) ∇ M( e w J ℓ,m − K ( j ) Λ j , Λ j ) ≃ M( e w J ℓ,m K ( j ) Λ j , e w J ℓ,m − K ( j ) Λ j ) ∇ W ( ℓ ) m − ,j . Hence it is enough to showM( e w J ℓ,m K ( j ) Λ j , e w J ℓ,m − K ( j ) Λ j ) ≃ L [ j − ℓ + m, j + m − . Then Proposition 3.15 impliesM( e w J ℓ,m K ( j ) Λ j , e w J ℓ,m − K ( j ) Λ j ) ≃ e F j − ℓ + m · · · e F j + m − ≃ L [ j − ℓ + m, j + m − . (cid:3) Theorem 4.10. For p ∈ Z ≥ with c ( p ) = ( ℓ, m ) , we have M e w ( p, 0) = W ( ℓ ) m, j p . Proof. The assertion immediately follows from Proposition 4.8 and Lemma 4.9. (cid:3) LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 35 Using the lattice points Z ≥ × Z ≥ , we can exhibit { M e w ( p, ≃ W ( ℓ ) m,i p } as follows : ... ... ... ... ... ... W (1)3 , − W (2)3 , W (3)3 , W (4)3 , W (5)3 , · · · W (1)2 , W (2)2 , W (3)2 , W (4)2 , W (5)2 , · · · W (1)1 , W (2)1 , W (3)1 , W (4)1 , W (5)1 , · · · = ... ... ... ... ... ... M e w (4 , 0) M e w (9 , 0) M e w (13 , 0) M e w (18 , 0) M e w (27 , · · · M e w (3 , 0) M e w (5 , 0) M e w (8 , 0) M e w (14 , 0) M e w (17 , · · · M e w (1 , 0) M e w (2 , 0) M e w (6 , 0) M e w (7 , 0) M e w (15 , · · · Note that we have { M e w ( p, } p ∈ Z ≥ = { hd (cid:0) L [ a, b ] ◦ L [ a − , b − · · · ◦ L [ − b, − a ] (cid:1) | a ≤ b, a + b ≥ }⊔{ hd (cid:0) L [ a, b ] ◦ L [ a − , b − · · · ◦ L [1 − b, − a ] (cid:1) | a ≤ b, a + b ≥ } . Here we confuse M and the isomorphic class of M . Corollary 4.11. For p ∈ Z ≥ with c ( p ) = ( ℓ, m ), we haveM e w ( p, 0) = M (cid:16) Λ j p + m − m X k =1 ǫ j p − ℓ + k , Λ j p (cid:17) and wt(M e w ( p, m X k =1 ( ǫ j p + k − ǫ j p − ℓ + k ) . For a pair ( a, b ) of integers with a > b , we denote also by L r ev [ a, b ] = e F a e F a − · · · e F b · ≃ e F ∗ b · · · e F ∗ a +1 e F ∗ a · the graded 1-dimensional R ( ǫ b − ǫ a +1 )-module. Proposition 4.12. (a) For p, p ′ ∈ Z ≥ with c ( p ) = ( ℓ, m ) and with c ( p ′ ) = ( ℓ, m + 1) , we have ( L [ j p − ℓ + m + 1 , j p + m ] ◦ M e w ( p, ։ M e w ( p ′ , if ℓ + m ≡ , M e w ( p, ◦ L [ j p − ℓ, j p − ։ M e w ( p ′ , if ℓ + m ≡ . (b) For each p ∈ Z ≥ with c ( p ) = ( ℓ, m ) , we have ( L [ j p − ℓ + m + 1 , j p + m ] ∇ M e w ( p, ≃ M e w ( p + , if ℓ + m ≡ ,L r ev [ j p − ℓ + m − , j p − ℓ ] ∇ M e w ( p, ≃ M e w ( p + , if ℓ + m ≡ . Proof. (a) is a consequence of Theorem 4.10, since we have j p ′ = ( j p if ℓ + m ≡ , j p − ℓ + m ≡ , by Lemma 4.5.For (b), if ℓ + m ≡ j p ′ = j p + and hence it is the case of (a). Now let usconsider when ℓ + m ≡ c ( p + ) = ( ℓ + 1 , m ) by Lemma 4.5. Then wehave • M e w ( p, ≃ hd (cid:0) L [ j p − ℓ + m, j p + m − ◦ · · · ◦ L [ j p − ℓ + 2 , j p + 1] ◦ L [ j p − ℓ + 1 , j p ] (cid:1) , • M e w ( p + , ≃ hd (cid:0) L [ j p − ℓ + m − , j p + m − ◦ · · · ◦ L [ j p − ℓ + 1 , j p + 1] ◦ L [ j p − ℓ, j p ] (cid:1) . Note that w J ℓ +1 ,m K ( j p ) ≡ s j p − ℓ + m − · · · s j p − ℓ +1 s j p − ℓ ∗ w J ℓ,m K ( j p ) with respect to the commutation relation. By Theorem 3.13, we haveM( e w J ℓ +1 ,m K ( j p ) Λ j p , Λ j p ) ≃ M( e w J ℓ +1 ,m K ( j p ) Λ j p , e w J ℓ,m K ( j p ) Λ j p ) ∇ M( e w J ℓ,m K ( j p ) Λ j p , Λ j p ) ≃ M( e w J ℓ +1 ,m K ( j p ) Λ j p , e w J ℓ,m K ( j p ) Λ j p ) ∇ W ( ℓ ) m, j p . As in the proof of Lemma 4.9 (iii), we haveM( e w J ℓ +1 ,m K ( j p ) Λ j p , e w J ℓ,m K ( j p ) Λ j p ) ≃ e F j p − ℓ + m − · · · e F j p − ℓ +1 e F j p − ℓ · . Hence our assertion follows. (cid:3) Corollary 4.13. For each p ∈ Z ≥ with c ( p ) = ( ℓ, m ), we haveM e w ( p + , p ) ≃ ( L [ j p − ℓ + m + 1 , j p + m ] if ℓ + m ≡ ,L r ev [ j p − ℓ + m − , j p − ℓ ] if ℓ + m ≡ . Proof. By Theorem 3.13 and Proposition 4.12,M e w ( p + , p ) ∇ M e w ( p, ≃ M e w ( p + , ( L [ j p − ℓ + m + 1 , j p + m ] ∇ M e w ( p, ≃ M e w ( p + , 0) if ℓ + m ≡ ,L r ev [ j p − ℓ + m − , j p − ℓ ] ∇ M e w ( p, ≃ M e w ( p + , 0) if ℓ + m ≡ . Then our assertion follows from Proposition 3.7. (cid:3) Quantum monoidal seed S ∞ . Since Proposition 4.1 tells that e w ≤ p is reduced forevery p ∈ Z ≥ , we can consider the quiver Q associated to e w by taking p → ∞ and usingDefinition 2.3. The set of vertices of Q is K := Z ≥ . The set of the frozen vertices of K is empty. Note that there is a bijection c : K → Z ≥ × Z ≥ . Proposition 4.14. Each vertex of the quiver Q is of finite degree. Hence the associatedmatrix e B satisfies the conditions in (2.1) .Proof. Note that for p, p ′ ∈ Z ≥ with c ( p ) = ( ℓ, m ) and c ( p ′ ) = ( ℓ − , m + 1), ( p = p ′ + 1 if ℓ + m ≡ ,p = p ′ − ℓ + m ≡ . By Lemma 4.5, for p ∈ Z ≥ with c ( p ) = ( ℓ, m ), we have(4.11) c ( p − ) = ( ( ℓ − , m ) if it exists and ℓ + m ≡ , ( ℓ, m − 1) if it exists and ℓ + m ≡ ,c ( p + ) = ( ( ℓ, m + 1) if ℓ + m ≡ , ( ℓ + 1 , m ) if ℓ + m ≡ , LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 37 (4.12) c ( p + ( j p − ( ( ℓ − δ ( ℓ = 1) , m + 2) if ℓ + m ≡ , ( ℓ − δ ( ℓ = 1) , m + 1) if ℓ + m ≡ ,c ( p + ( j p + 1)) = ( ( ℓ + 1 , m − δ ( m = 1)) if ℓ + m ≡ , ( ℓ + 2 , m − δ ( m = 1)) if ℓ + m ≡ . Combining (4.11) and (4.12), we have c ( p + ( j p − + ) = ( ( ℓ, m + 2 + δ ( ℓ = 1)) if ℓ + m ≡ , ( ℓ, m + 1 + δ ( ℓ = 1)) if ℓ + m ≡ ,c ( p + ( j p + 1) + ) = ( ( ℓ + 1 + δ ( m = 1) , m ) if ℓ + m ≡ , ( ℓ + 2 + δ ( m = 1) , m ) if ℓ + m ≡ . Hence we have the followings for c ( p ) = ( ℓ, m ):(i) If ℓ + m ≡ m = 1, then p < p + ( j p + 1) < p + ( j p + 1) + < p + < p + ( j p + 1) ++ and p < p + < p + ( j p − . (ii) If ℓ + m ≡ m = 1, then p < p + ( j p + 1) < p + < p + ( j p + 1) + and p < p + < p + ( j p − . (iii) If ℓ + m ≡ ℓ = 1, then p < p + ( j p − < p + ( j p − + < p + < p + ( j p − ++ and p < p + < p + ( j p + 1) . (iv) If ℓ + m ≡ ℓ = 1, then p < p + ( j p − < p + < p + ( j p − + and p < p + < p + ( j p + 1) . Hence, Definition 2.3 tells that there is only one ordinary arrow with source p , whichis given by p → p + ( j p + 1) + or p → p + ( j p + 1) if ℓ + m ≡ p → p + ( j p − + or p → p + ( j p − 1) if ℓ + m ≡ p ∈ J with c ( p ) = ( ℓ, m )has two incoming arrows and two outgoing arrow unless min( ℓ, m ) = 1 : p + (cid:15) (cid:15) p − p o o / / p + ( j p +1) + p − ( j p +1) O O p + ( j p − + p − ( j p − / / p (cid:15) (cid:15) O O p + o o p − (4.13) if ℓ + m ≡ ℓ + m ≡ . In particular, when (i) ℓ + m ≡ m = 1, or (ii) ℓ + m ≡ ℓ = 1,the arrows incident with vertex p can be described as follows:(i) p + (cid:15) (cid:15) p − p o o / / p + ( j p +1) (ii) p + ( j p − p O O (cid:15) (cid:15) p + o o p − (4.14)By replacing p = c − ( ℓ, m ) with W ( ℓ ) m, j p in the diagrams above, the quiver Q can beexhibited as follows: ... ... (cid:15) (cid:15) ... ... (cid:15) (cid:15) ... ... (cid:15) (cid:15) . . . (1)3 , − / / W (2)3 , O O (cid:15) (cid:15) W (3)3 , / / o o W (4)3 , O O (cid:15) (cid:15) W (5)3 , / / o o · · · (1)2 , O O (cid:15) (cid:15) W (2)2 , / / o o W (3)2 , O O (cid:15) (cid:15) W (4)2 , / / o o W (5)2 , O O (cid:15) (cid:15) · · · (1)1 , / / W (2)1 , O O W (3)1 , / / o o W (4)1 , O O W (5)1 , / / o o · · · m/ℓ · · · (4.15)Hence our assertion follows. (cid:3) Note that the arrows of Q oriented right or above are arrows of horizontal type, andarrows of Q oriented left or below are arrows of ordinary type. Also the quiver Q isknown as the square product Q A ∞ (cid:3) Q A ∞ of the bipartite Dynkin quiver Q A ∞ of type A ∞ ,which is related to the periodicity conjecture (see [8, 19, 20, 35, 36, 49, 52]). Here Q A ∞ is the quiver ◦ / / ◦ o o ◦ / / ◦ . Remark 4.15. Take k ∈ Z ≥ . When we restrict the full subquiver Q ( k ) of Q consistingof vertices ( ℓ, m ) with ℓ + m ≤ k + 1, the Q ( k ) is mutation equivalent to the well-knownquiver Q BF Zk of the coordinate ring C [ N ] of the unipotent group N of type A k (see [3]and [35, Theorem 4.5]). For example k = 3, Q (3) is given as follows: •• (cid:15) (cid:15) O O • o o • / / • O O • o o Note that Q BF Zk is isomorphic to the quiver associated to some adapted reduced ex-pression of the longest element of the Weyl group of A k (see Definition 2.3). LUSTER ALGEBRA STRUCTURES OVER QUANTUM AFFINE ALGEBRAS 39 Now we take the skew-symmetric integer-valued Z ≥ × Z ≥ -matrix Λ as follows:Λ = (cid:0) Λ (cid:0) M e w ( i, , M e w ( j, (cid:1)(cid:1) i,j ∈ Z ≥ . Note that for each p ∈ Z ≥ , we can take sufficiently large t such that p is an exchangeableindex of e w ≤ t (see (2.5)). Thus the following corollary follows from Theorem 3.18 for e w ≤ t : Corollary 4.16. (a) For any finite sequence Σ in K , µ Σ ( − Λ , e B ) is compatible with d = 2, where e B is thematrix associated with Q .(b) For every pair of positive integers p = c − ( ℓ, m ) and p ′ = c − ( ℓ ′ , m ′ ), W ( ℓ ) m, j p and W ( ℓ ′ ) m ′ , j p ′ strongly commute.(c) For each p ∈ Z ≥ , there exists M e w ( p, ′ in R J -gmod satisfying (3.5).Let us denote by S ∞ = ( { M e w ( p, } p ∈ K , e B ) . Then S ∞ becomes a quantum monoidal seed. Furthermore, we have • the pair ( { M e w ( p, } p ∈ K , e B ) is admissible, • S ∞ admits successive mutations in R J -gmod for all the directions.Let A q / ( ∞ ) be the quantum cluster algebra associated with the quantum seed[ S ∞ ] := ( { q − ( d p ,d p ) / [M e w ( p, } i ∈ K , − Λ , e B )without frozen variables. Remark 4.17. Note that the quantum cluster algebras associated with a quiver of infiniterank is the Z [ q ± / ]-subalgebra of skew field F generated by all elements obtained frominitial cluster variables by finite sequences of mutations. We refer to [12, 17] for the detailsof the definition of (quantum) cluster algebra of infinite rank. Theorem 4.18. The category R J - gmod is a monoidal categorification of the quantumcluster algebra A q / ( ∞ ) .Proof. Theorem 3.20 implies that any cluster monomial of A q / ( ∞ ) is contained in K ( C e w ≤ t ) ⊂ K ( R J -gmod) for sufficiently large t ∈ Z ≥ . On the other hand, let M bea simple R J ( β )-module. We write β = P pk = − p a k α k . Then by Remark 4.4 (a), [ M ] is con-tained in K ( C e w ≤ a ( p +1) ). Thus we conclude that K ( R J -gmod) = A q / ( ∞ ), which completesthe proof. (cid:3) The category T N and its cluster structure Category T N . We keep the notations in § 4. In this subsection we briefly recallthe quotient category and the localizations of R -gmod introduced in [25, § § F factors thorough the category T N defined below. Set A β := R J ( β )-gmod and A := L β ∈ Q + J A β . Let S N be the smallest Serre subcategory of A such that(i) S N contains L [ a, a + N ] for any a ∈ J ,(ii) X ◦ Y, Y ◦ X ∈ S N for all X ∈ A and Y ∈ S N .Note that S N contains L [ a, b ] if b − a + 1 > N .Let us denote by A / S N the quotient category of A by S N and denote by Q N : A →A / S N the canonical functor.Note that A and A / S N are monoidal categories with the convolution as tensor products.The module R (0) ≃ k is a unit object. Note also that Q := qR (0) is an invertible centralobject of A / S N and X Q ◦ X ≃ X ◦ Q coincides with the grading shift functor.Moreover, the functors Q N is a monoidal functor. Definition 5.1.