aa r X i v : . [ m a t h . QA ] M a r INTEGRAL QUANTUM CLUSTER STRUCTURES
K. R. GOODEARL AND M. T. YAKIMOV
Abstract.
We prove a general theorem for constructing integral quantum cluster al-gebras over Z [ q ± / ], namely that under mild conditions the integral forms of quantumnilpotent algebras always possess integral quantum cluster algebra structures. These al-gebras are then shown to be isomorphic to the corresponding upper quantum cluster al-gebras, again defined over Z [ q ± / ]. Previously, this was only known for acyclic quantumcluster algebras. The theorem is applied to prove that for every symmetrizable Kac–Moody algebra g and Weyl group element w , the dual canonical form A q ( n + ( w )) Z [ q ± ] ofthe corresponding quantum unipotent cell has the property that A q ( n + ( w )) Z [ q ± ] ⊗ Z [ q ± ] Z [ q ± / ] is isomorphic to a quantum cluster algebra over Z [ q ± / ] and to the correspond-ing upper quantum cluster algebra over Z [ q ± / ]. Introduction
Problems for integral quantum custer algebras.
Cluster algebras were intro-duced by Fomin and Zelevinsky in [6] and have been applied to a number of diverseareas such as representation theory, combinatorics, Poisson and algebraic geometry, math-ematical physics and others. Their quantum counterparts, introduced by Berenstein andZelevinsky [2], are similarly the topic of intensive research from various standpoints. Inthe uniparameter quantum case it is desirable to work over the minimal ring of definition,namely over(1.1) A / := Z [ q ± / ] , where q is the quantum parameter. We will refer to such structures as integral quantumcluster algebras . Two fundamental problems that are being investigated are:(1) Given an algebra R over the rational function field F / := Q ( q / ) and an integralform R A / of R over A / (i.e., R ∼ = R A / ⊗ A / F / ), when is R A / isomorphicto an integral quantum cluster algebra?(2) When is the quantum cluster algebra A in Problem (1) equal to the correspondingupper quantum cluster algebra U defined over A / ?The best known result on Problem (1) is a theorem of Kang, Kashiwara, Kim and Oh [23]that the dual canonical forms (over A / ) of the quantum unipotent cells for all symmetricKac–Moody algebras possess integral quantum cluster algebra structures. Berenstein andZelevinsky [2] proved the equality A = U in the acyclic case. Such an equality was provedby Muller [35] for (quantum) cluster algebras that are source-sink decomposable in thecase when all frozen variables are inverted. We are not aware of any affirmative solutionsof Problem (2) for non-acyclic quantum cluster algebras when frozen variables are not Date : May 6, 2019.2000
Mathematics Subject Classification.
Primary: 13F60; Secondary: 16S38, 17B37, 81R50.
Key words and phrases.
Integral quantum cluster algebras, integral upper quantum cluster algebras,symmetrizable Kac–Moody algebras, quantum unipotent cells, dual canonical bases.The research of K.R.G was partially supported by NSF grant DMS-1601184, and that of M.T.Y. byNSF grant DMS-1601862 and Bulgarian Science Fund grant H02/15. inverted. A recent result of Geiß, Leclerc and Schr¨oer [10] establishes an equality of theform A ⊗ A / Q [ q ± / ] = U ⊗ A / Q [ q ± / ]under the assumptions that A is connected Z ≥ -graded with homogeneous cluster variablesand that such an equality holds on the classical level.1.2. Main results.
In this paper we provide affirmative answers to both Problems (1) and(2) in wide generality. As an application, affirmative answers to Problems (1) and (2) areobtained for the dual canonical forms of the quantum unipotent cells for all symmetrizableKac–Moody algebras.For an iterated skew polynomial extension R := F / [ x ][ x ; θ , δ ] · · · [ x N ; θ N , δ N ]and 1 ≤ j ≤ k ≤ N , denote by R [ j,k ] the F / -subalgebra generated by x j , . . . , x k and set R k := R [1 ,k ] . Definition.
An iterated skew polynomial extension R is called a quantum nilpotent algebra or a CGL extension if it is equipped with a rational action of an F / -torus H by F / -algebra automorphisms such that:(i) The elements x , . . . , x N are H -eigenvectors.(ii) For every k ∈ [2 , N ], δ k is a locally nilpotent θ k -derivation of the algebra R k − .(iii) For every k ∈ [1 , N ], there exists h k ∈ H such that θ k = ( h k · ) | R k − and the h k -eigenvalue of x k , to be denoted by λ k , is not a root of unity.A CGL extension is called symmetric if it has the same properties when its generatorsare adjoined in the opposite order. We will assume throughout Sections 1, 5, 6, 7 that the θ k -eigenvalues of x j belong to q Z / for j ≤ k , where we abbreviate Z / Z . Recall thata nonzero element p ∈ R is called prime if Rp = pR and the ring R/Rp is a domain.
Theorem. [14, Theorem 4.3]
For each CGL extension R and k ∈ [1 , N ] , the algebra R k has a unique ( up to rescaling ) homogeneous prime element y k which does not belong to R k − . It either equals x k or has the property that y k − y p ( k ) x k ∈ R k − for some p ( k ) ∈ [1 , k − . In the following we will work with this choice of sets of homogeneous prime elements(and not with arbitrary F / -rescalings of them). For a symmetric CGL extension thetheorem can be applied to the interval subalgebras R [ p ( k ) ,k ] to obtain that each of themhas a unique (up to rescaling) homogeneous prime element y [ p ( k ) ,k ] which does not belongto the smaller interval subalgebras. An F / -rescaling of the generators of a CGL extension R leads to another CGL extension presentation of R . The generators x k can be alwaysrescaled so that(1.2) y [ p ( k ) ,k ] = q m x p ( k ) x k − q m ′ Y i p n i i for some m, m ′ ∈ Z / n i ∈ Z ≥ where the product is over all homogeneous primeelements of R [ p ( k ) ,k ] from the theorem that are different from y [ p ( k ) ,k ] (see § R A / := A / h x , . . . , x N i ⊆ R. NTEGRAL QUANTUM CLUSTER STRUCTURES 3
Theorem A.
Let R be a symmetric CGL extension for which R A / is an A / -form of R ,that is, R A / ⊗ A / F / ∼ = R . If the sequence of homogeneous prime elements y , . . . , y N lies in R A / , then there exists a quantum cluster algebra A over A / such that R ∼ = A = U where U is the corresponding upper quantum cluster algebra over A / . For all k ∈ [1 , N ] and n ∈ Z > for which p n ( k ) is well defined ( as in the previous theorem ) , q m x k and q m ′ y [ p n ( k ) ,k ] are cluster variables of A for some m, m ′ ∈ Z / . We prove a more general result in Theorem 4.8 which deals with integral forms ofmultiparameter and arbitrary characteristic CGL extensions and quantum cluster algebras.In § Z ≥ -graded, including all quantized Weyl algebras, and with quantum cluster algebras over F p [ q ± / ].For each symmetrizable Kac–Moody algebra g and a Weyl group element w , De Concini–Kac–Procesi [5] and Lusztig [34] defined a quantum Schubert cell algebra U − [ w ] which isa subalgebra of the quantized universal enveloping algebra U q ( g ) defined over Q ( q ). Thequantum unipotent cells of Geiß–Leclerc–Schr¨oer [9] are Q ( q )-algebras A q ( n + ( w )) whichare antiisomorphic to U − [ w ]. Denote(1.3) A := Z [ q ± ] . The dual canonical forms A q ( n + ( w )) A are A -forms of A q ( n + ( w )) which are obtained bytransporting the Kashiwara–Lusztig dual canonical forms U − [ w ] ∨A of U − [ w ]. Theorem B.
Let g be an arbitrary symmetrizable Kac–Moody algebra and w a Weyl groupelement. For the dual canonical form A q ( n + ( w )) A of the corresponding quantum unipotentcell, there exists a quantum cluster algebra A over A / such that A q ( n + ( w )) A ⊗ A A / ∼ = A = U where U is the associated upper quantum cluster algebra defined over A / . Further details about the structure of the quantum cluster algebra A are given in The-orem 7.3.The following special cases of parts of the theorem were previously proved: Qin [37]proved that A q ( n + ( w )) A ⊗ A A / ∼ = A for symmetric Kac–Moody algebras g and adaptableWeyl group elements w . Kang, Kashiwara, Kim and Oh [23] proved this isomorphism forsymmetric Kac–Moody algebras g and all Weyl group elements w . Geiß, Leclerc andSchr¨oer [10] proved that A ⊗ A / Q [ q ± / ] = U ⊗ A / Q [ q ± / ]for symmetric Kac–Moody algebras g and all Weyl group elements w ; however, the factthat A = U is new even for simple cases like g = sl n . For nonsymmetric Kac–Moodyalgebras g the results in the theorem are all new, including the existence of a non-integralquantum cluster structure on A q ( n + ( w )) A ⊗ A Q ( q / ).The previous approaches to integral quantum cluster structures [4, 19, 23, 26, 36, 37]obtained monoidal categorifications of quantum cluster algebras. At the same time theyalso relied on extensive knowledge of categorifications which are available for concretefamilies of algebras. The power of Theorem A for the construction of integral quantumcluster structures lies in its flexibility to adjust to different situations and in the mildassumptions in it: one needs to verify the normalization condition (1.2), that R A / is an A / -form of R , and that the sequence of homogeneous prime elements y , . . . , y N belongsto R A / . K. R. GOODEARL AND M. T. YAKIMOV
Notation and conventions.
Throughout, K denotes an infinite field of arbitrarycharacteristic. For integers j ≤ k , set [ j, k ] := { j, . . . , k } . As above, Z / Z .An N × N matrix t = ( t kj ) over a commutative ring D is multiplicatively skew-symmetric if t jk t kj = t kk = 1 for all j, k ∈ [1 , N ]. Such a matrix gives rise to a skew-symmetricbicharacter Ω t : Z N × Z N → D ∗ for which(1.4) Ω t ( e j , e k ) = t jk , ∀ j, k ∈ [1 , N ] , where e , . . . , e N are the standard basis vectors for Z N . (We denote the group of units of D by D ∗ .) When we have need for formulas involving Z N , we view its elements as columnvectors. The transpose of an N -tuple m = ( m , . . . , m N ) is denoted m T .Given an algebra A over a commutative ring D and elements a , . . . , a k ∈ A , we write D h a , . . . , a k i to denote the unital D -subalgebra of A generated by { a , . . . , a k } . Acknowledgements.
We would like to thank Bernhard Keller for valuable suggestions.We would also like to thank the anonymous referee whose suggestions were very helpful tous in improving the paper. 2.
Quantum cluster algebras
We outline notation and conventions for quantum cluster algebras. To connect with theresults of [15], we describe a multiparameter setting which extends the uniparameter caseoriginally developed by Berenstein and Zelevinsky [2]. To allow for integral forms, we workover a commutative domain rather than over a field.Fix a commutative domain D contained in K and a positive integer N .Let F be a division algebra over D . A toric frame (of rank N ) for F (over D ) is amapping M : Z N −→ F such that(2.1) M ( f ) M ( g ) = Ω r ( f, g ) M ( f + g ) , ∀ f, g ∈ Z N , where • Ω r is a D ∗ -valued skew-symmetric bicharacter on Z N arising from a multiplicativelyskew-symmetric matrix r ∈ M N ( D ) as in (1.4), • the elements in the image of M are linearly independent over D , and • Fract D h M ( Z N ) i = F .The matrix r is uniquely reconstructed from the toric frame M , and will be denotedby r ( M ). The elements M ( e ) , . . . , M ( e N ) are called cluster variables . Fix a subset ex ⊂ [1 , N ], to be called the set of exchangeable indices ; the remaining indices, those in [1 , N ] \ ex ,will be called frozen .An integral N × ex matrix e B will be called an exchange matrix if its principal part (the ex × ex submatrix) is skew-symmetrizable. If the principal part of e B is skew-symmetric,then it is represented by a quiver whose vertices are labelled by the integers in [1 , N ].For j, k ∈ [1 , N ], there is a directed edge from the vertex j to the vertex k if and only if( e B ) jk > e B ) jk . In particular, the quiverhas no edges between any pair of vertices in [1 , N ] \ ex .A quantum seed for F (over D ) is a pair ( M, e B ) consisting of a toric frame M for F andan exchange matrix e B compatible with r ( M ) in the sense thatΩ r ( M ) ( b k , e j ) = 1 , ∀ k ∈ ex , j ∈ [1 , N ] , k = j andΩ r ( M ) ( b k , e k ) are not roots of unity , ∀ k ∈ ex , NTEGRAL QUANTUM CLUSTER STRUCTURES 5 where b k denotes the k -th column of e B .The mutation in direction k ∈ ex of a quantum seed ( M, e B ) is the quantum seed( µ k ( M ) , µ k ( e B )) where µ k ( M ) is described below and µ k ( e B ) is the N × ex matrix ( b ′ ij )with entries b ′ ij := ( − b ij , if i = k or j = kb ij + | b ik | b kj + b ik | b kj | , otherwise , [6]. If the principal part of e B is skew-symmetric, then µ k ( e B ) has the same property andthe pair of quivers corresponding to e B and µ k ( e B ) are obtained from each other by quivermutation at the vertex k , see [7, §§ b k of e B are D -algebra automorphisms ρ b k , ± of F such that ρ b k ,ǫ ( M E ǫ ( e j )) = ( M E ǫ ( e k ) + M E ǫ ( e k + ǫb k ) , if j = kM E ǫ ( e j ) , if j = k, [2, Proposition 4.2] and [15, Lemma 2.8], where E ǫ = E e Bǫ is the N × N matrix with entries( E ǫ ) ij = δ ij , if j = k − , if i = j = k max(0 , − ǫb ik ) , if i = j = k. The toric frame µ k ( M ) is defined as µ k ( M ) := ρ b k ,ǫ M E ǫ : Z N −→ F . It is independent of the choice of ǫ , and, paired with µ k ( e B ), forms a quantum seed over K [15, Proposition 2.9]. (See also [15, Corollary 2.11], and compare with [2, Proposition4.9] for the uniparameter case.) By [15, Proposition 2.9 and Eq. (2.22)], the entries of r ( µ k ( M )) = µ k ( r ( M )) are products of powers of the entries of r ( M ), so r ( µ k ( M )) ∈ M N ( D ). It follows that µ k ( M ) is a toric frame for F over D , so that ( µ k ( M ) , µ k ( e B )) is aquantum seed over D .Fix a subset inv of the set [1 , N ] \ ex of frozen indices – the corresponding clustervariables will be inverted. The quantum cluster algebra A ( M, e B, inv ) D is the unital D -subalgebra of F generated by the cluster variables of all seeds obtained from ( M, e B ) byiterated mutations and by { M ( e k ) − | k ∈ inv } . To each quantum seed ( M, e B ) and choiceof inv , one associates the mixed quantum torus/quantum affine space algebra(2.2) D T ( M, e B, inv ) := D h M ( e k ) ± , M ( e j ) | k ∈ ex ∪ inv , j ∈ [1 , N ] \ ( ex ∪ inv ) i ⊂ F . The intersection of all such subalgebras of F associated to all seeds that are obtainedby iterated mutation from the seed ( M, e B ) is called the upper quantum cluster algebra of( M, e B ) and is denoted by U ( M, e B, inv ) D . The corresponding Laurent Phenomenon [15,Theorem 2.15] says that(2.3) A ( M, e B, inv ) D ⊆ U ( M, e B, inv ) D . If K is the quotient field of D , then F is also a division algebra over K , and the aboveconstructions may be performed over K . The corresponding quantum cluster algebras over K are just the K -subalgebras of F generated by the quantum cluster algebras over D : A ( M, e B, inv ) K = K · A ( M, e B, inv ) D . The uniparameter quantum cluster algebras of Berenstein and Zelevinsky [2] come fromthe above axiomatics when the following two conditions are imposed:
K. R. GOODEARL AND M. T. YAKIMOV (1) The base ring is taken to be D = A / = Z [ q ± / ] . So, D ∗ = ( A / ) ∗ = ± q Z / .(2) The toric frame of one seed (and thus of any seed) has a multiplicatively skew-symmetric matrix r ∈ M N ( D ) of the form r = ( q m ij / ) Ni,j =1 for some m ij ∈ Z . Quantum nilpotent algebras
Quantum nilpotent algebras are iterated skew polynomial algebras over a base field,which we take to be K in this section. We use the standard notation S [ x ; θ, δ ] for a skewpolynomial ring , or Ore extension ; it denotes a ring generated by a subring S and anelement x satisfying xs = θ ( s ) x + δ ( s ) for all s ∈ S , where θ is a ring endomorphismof S and δ is a (left) θ -derivation of S . The ring S [ x ; θ, δ ] is a free left S -module, withthe nonnegative powers of x forming a basis. For all skew polynomial rings S [ x ; θ, δ ]considered in this paper, we assume that θ is an automorphism of S . Moreover, we workin the context of algebras over a commutative ring D , so our coefficient rings S will be D -algebras, the maps θ will be D -algebra automorphisms, and the maps δ will be D -linear θ -derivations. Under these assumptions, S [ x ; θ, δ ] is naturally a D -algebra. Throughoutthe present section, D = K .3.1. CGL extensions.
We focus on iterated skew polynomial extensions(3.1) R := K [ x ][ x ; θ , δ ] · · · [ x N ; θ N , δ N ] , where K [ x ] = K [ x ; id K , R k := K h x , . . . , x k i = K [ x ][ x ; θ , δ ] · · · [ x k ; θ k , δ k ] for k ∈ [0 , N ];in particular, R = K . Definition 3.1.
An iterated skew polynomial extension (3.1) is called a quantum nilpotentalgebra or a
CGL extension [31, Definition 3.1] if it is equipped with a rational action ofa K -torus H by K -algebra automorphisms such that:(i) The elements x , . . . , x N are H -eigenvectors.(ii) For every k ∈ [2 , N ], δ k is a locally nilpotent θ k -derivation of the algebra R k − .(iii) For every k ∈ [1 , N ], there exists h k ∈ H such that θ k = ( h k · ) | R k − and the h k -eigenvalue of x k , to be denoted by λ k , is not a root of unity.Conditions (i) and (iii) imply that θ k ( x j ) = λ kj x j for some λ kj ∈ K ∗ , ∀ ≤ j < k ≤ N. We then set λ kk := 1 and λ jk := λ − kj for j < k . This gives rise to a multiplicativelyskew-symmetric matrix λ := ( λ kj ) ∈ M N ( K ∗ ) and the corresponding skew-symmetricbicharacter Ω λ from (1.4). The elements h k ∈ H interact with the skew derivations δ k asfollows:(3.2) ( h k · ) ◦ δ k = λ k δ k ◦ ( h k · ) , ∀ k ∈ [1 , N ] , see [15, Eq. (3.15)].The length of R is N and its rank is given by by(3.3) rk( R ) := { k ∈ [1 , N ] | δ k = 0 } ∈ Z > (cf. [14, Eq. (4.3)]). Denote the character group of the torus H by X ( H ). The actionof H on R gives rise to an X ( H )-grading of R , and the H -eigenvectors are precisely the NTEGRAL QUANTUM CLUSTER STRUCTURES 7 nonzero homogeneous elements with respect to this grading. The H -eigenvalue of a nonzerohomogeneous element u ∈ R will be denoted by χ u ; this equals its X ( H )-degree relativeto the X ( H )-grading.By [31, Proposition 3.2, Theorem 3.7], every CGL extension R is an H -UFD, meaningthat each nonzero H -prime ideal of R contains a homogeneous prime element. (A primeelement of a domain R is a nonzero element p ∈ R such that Rp = pR – i.e., p is a normalelement of R – and the ring R/Rp is a domain.) A recursive description of the sets ofhomogeneous prime elements of the intermediate algebras R k of a CGL extension R wasobtained in [14]. To state the result, we require the standard predecessor and successorfunctions, p = p η and s = s η , of a function η : [1 , N ] → Z , defined as follows:(3.4) p ( k ) := max { j < k | η ( j ) = η ( k ) } ,s ( k ) := min { j > k | η ( j ) = η ( k ) } , where max ∅ = −∞ and min ∅ = + ∞ . Corresponding order functions O ± : [1 , N ] → Z ≥ are defined by(3.5) O − ( k ) := max { m ∈ Z ≥ | p m ( k ) = −∞} ,O + ( k ) := max { m ∈ Z ≥ | s m ( k ) = + ∞} . Theorem 3.2. [14, Theorem 4.3]
Let R be a CGL extension of length N and rank rk( R ) as in (3.1) . There exist a function η : [1 , N ] → Z whose range has cardinality rk( R ) andelements c k ∈ R k − for all k ∈ [2 , N ] with p ( k ) = −∞ such that the elements y , . . . , y N ∈ R , recursively defined by (3.6) y k := ( y p ( k ) x k − c k , if p ( k ) = −∞ x k , if p ( k ) = −∞ , are homogeneous and have the property that for every k ∈ [1 , N ] , (3.7) { y j | j ∈ [1 , k ] , s ( j ) > k } is a list of the homogeneous prime elements of R k up to scalar multiples.The elements y , . . . , y N ∈ R with these properties are unique. The function η satisfyingthe above conditions is not unique, but the partition of [1 , N ] into a disjoint union of thelevel sets of η is uniquely determined by the presentation (3.1) of R , as are the predecessorand successor functions p and s . The function p has the property that p ( k ) = −∞ if andonly if δ k = 0 . The statement of Theorem 3.2 is upgraded as in [15, Theorem 3.6 and following com-ments]. In the setting of the theorem, the rank of R is also given by(3.8) rk( R ) = |{ j ∈ [1 , N ] | s ( j ) > N }| [14, Eq. (4.3)]. Definition 3.3.
Denote by ≺ the reverse lexicographic order on Z N ≥ :(3.9) ( m ′ , . . . , m ′ N ) ≺ ( m , . . . , m N ) iff there exists n ∈ [1 , N ] with m ′ n < m n , m ′ n +1 = m n +1 , . . . , m ′ N = m N .A CGL extension R as in (3.1) has the K -basis { x f := x m · · · x m N N | f = ( m , . . . , m N ) T ∈ Z N ≥ } . K. R. GOODEARL AND M. T. YAKIMOV
We say that a nonzero element b ∈ R has leading term tx f and leading coefficient t where t ∈ K ∗ and f ∈ Z N ≥ if b = tx f + X g ∈ Z N ≥ , g ≺ f t g x g for some t g ∈ K , and we set lc( b ) := t and lt( b ) := tx f .The leading terms of the prime elements y k in Theorem 3.2 are given by(3.10) lt( y k ) = x p O − ( k ) ( k ) · · · x p ( k ) x k , ∀ k ∈ [1 , N ] . The leading terms of reverse-order monomials x m N N · · · x m involve symmetrization scalarsin K ∗ defined by(3.11) S λ ( f ) := Y ≤ j Given an iterated skew polynomial extension R asin (3.1), denote its interval subalgebras R [ j,k ] := K h x i | j ≤ i ≤ k i , ∀ j, k ∈ [1 , N ];in particular, R [ j,k ] = K if j (cid:2) k . Definition 3.4. A CGL extension R as in Definition 3.1 is called symmetric provided(i) For all 1 ≤ j < k ≤ N , δ k ( x j ) ∈ R [ j +1 ,k − . (ii) For all j ∈ [1 , N ], there exists h ∗ j ∈ H such that h ∗ j · x k = λ − kj x k = λ jk x k , ∀ k ∈ [ j + 1 , N ]and h ∗ j · x j = λ ∗ j x j for some λ ∗ j ∈ K ∗ which is not a root of unity.Under these conditions, R has a CGL extension presentation with the variables x k indescending order:(3.13) R = K [ x N ][ x N − ; θ ∗ N − , δ ∗ N − ] · · · [ x ; θ ∗ , δ ∗ ] , see [14, Corollary 6.4]. Proposition 3.5. [15, Proposition 5.8] Let R be a symmetric CGL extension of length N .If l ∈ [1 , N ] with O + ( l ) = m > , then (3.14) λ ∗ l = λ ∗ s ( l ) = · · · = λ ∗ s m − ( l ) = λ − s ( l ) = λ − s ( l ) = · · · = λ − s m ( l ) . Definition 3.6. Define the following subset of the symmetric group S N :(3.15) Ξ N := { σ ∈ S N | σ ( k ) = max σ ([1 , k − σ ( k ) = min σ ([1 , k − − , ∀ k ∈ [2 , N ] } . In other words, Ξ N consists of those σ ∈ S N such that σ ([1 , k ]) is an interval for all k ∈ [2 , N ]. The following subset of Ξ N will also be needed:(3.16) Γ N := { σ i,j | ≤ i ≤ j ≤ N } , where σ i,j := [ i + 1 , . . . , j, i, j + 1 , . . . , N, i − , i − , . . . , . NTEGRAL QUANTUM CLUSTER STRUCTURES 9 If R is a symmetric CGL extension of length N , then for each σ ∈ Ξ N there is a CGLextension presentation(3.17) R = K [ x σ (1) ][ x σ (2) ; θ ′′ σ (2) , δ ′′ σ (2) ] · · · [ x σ ( N ) ; θ ′′ σ ( N ) , δ ′′ σ ( N ) ] , see [14, Remark 6.5], [15, Proposition 3.9]. Moreover, if 1 ≤ i ≤ k ≤ N , then thesubalgebra R [ i,k ] of R is a symmetric CGL extension, to which Theorem 3.2 applies. Inthe case k = s m ( i ) we have Proposition 3.7. [15, Theorem 5.1] Assume that R is a symmetric CGL extension oflength N , and i ∈ [1 , N ] and m ∈ Z ≥ are such that s m ( i ) ∈ [1 , N ] . Then there is a uniquehomogeneous prime element y [ i,s m ( i )] ∈ R [ i,s m ( i )] such that (i) y [ i,s m ( i )] / ∈ R [ i,s m ( i ) − and y [ i,s m ( i )] / ∈ R [ i +1 ,s m ( i )] . (ii) lt( y [ i,s m ( i )] ) = x i x s ( i ) · · · x s m ( i ) . The elements y [ i,s m ( i )] ∈ R will be called interval prime elements . Certain combinationsof the homogeneous prime elements from Proposition 3.7 play an important role in themutation formulas for quantum cluster variables of symmetric CGL extensions. They aregiven in the following theorem, where we denote(3.18) e [ j,s l ( j )] := e j + e s ( j ) + · · · + e s l ( j ) ∈ Z N , ∀ j ∈ [1 , N ] , l ∈ Z ≥ such that s l ( j ) ∈ [1 , N ] , and where we set y [ s ( i ) ,i ] := 1. Theorem 3.8. [15, Corollary 5.11] Assume that R is a symmetric CGL extension of length N , and i ∈ [1 , N ] and m ∈ Z > are such that s m ( i ) ∈ [1 , N ] . Then (3.19) u [ i,s m ( i )] := y [ i,s m − ( i )] y [ s ( i ) ,s m ( i )] − Ω λ ( e i , e [ s ( i ) ,s m − ( i )] ) y [ s ( i ) ,s m − ( i )] y [ i,s m ( i )] is a nonzero homogeneous normal element of R [ i +1 ,s m ( i ) − which is not a multiple of y [ s ( i ) ,s m − ( i )] if m ≥ . The form and properties of the elements u [ i,s m ( i )] mainly enter into the proofs of themutation formulas for symmetric CGL extensions. However, an explicit normalization con-dition for the leading coefficients of these elements is required; see (3.28) and Proposition3.10.3.3. Rescaling of generators. Assume R is a CGL extension of length N as in (3.1).Given scalars t , . . . , t N ∈ K ∗ , one can rescale the generators x j of R in the fashion(3.20) x j t j x j , ∀ j ∈ [1 , N ] , meaning that R is also an iterated Ore extension with generators t j x j ; in fact,(3.21) R = K [ t x ][ t x ; θ , t δ ] · · · [ t N x N ; θ N , t N δ N ] . This is also a CGL extension presentation of R , and if (3.1) is a symmetric CGL extension,then so is (3.21).Rescaling as in (3.20), (3.21) does not affect the H -action or the matrix λ , but variouselements computed in terms of the new generators are correspondingly rescaled, such asthe homogeneous prime elements from Theorem 3.2 and Proposition 3.7. These transformas follows:(3.22) y k (cid:18) O − ( k ) Y l =0 t p l ( k ) (cid:19) y k and y [ i,s m ( i )] (cid:18) m Y l =0 t s l ( i ) (cid:19) y [ i,s m ( i )] . Consequently, the homogeneous normal elements (3.19) transform via(3.23) u [ i,s m ( i )] (cid:0) t i t s ( i ) · · · t s m − ( i ) t s m ( i ) (cid:1) u [ i,s m ( i )] . Normalization conditions. In order for the homogeneous prime elements y k fromTheorem 3.2 to function as quantum cluster variables, some normalizations are required.Throughout this subsection, assume that R is a symmetric CGL extension of length N asin Definitions 3.1 and 3.4. Assume also that the following mild conditions on scalars aresatisfied: Condition (A). The base field K contains square roots ν kl = √ λ kl of the scalars λ kl for1 ≤ l < k ≤ N such that the subgroup of K ∗ generated by the ν kl contains no elements oforder 2. Then set ν kk := 1 and ν kl := ν − lk for k < l , so that ν := ( ν kl ) is a multiplicativelyskew-symmetric matrix. Condition (B). There exist positive integers d n , for n ∈ η ([1 , N ]), such that λ d η ( l ) k = λ d η ( k ) l , ∀ k, l ∈ [1 , N ] with p ( k ) , p ( l ) = −∞ . In view of Proposition 3.5, this is equivalent to the condition( λ ∗ k ) d η ( l ) = ( λ ∗ l ) d η ( k ) , ∀ k, l ∈ [1 , N ] with s ( k ) , s ( l ) = + ∞ . Remark 3.9. Note that Condition (B) is always satisfied if all λ k = q m k for some m k ∈ Z and q ∈ K (which has to be a non-root of unity due to assumption (iii) in Definition 3.1).This is the setting of Theorem A in the introduction.In parallel with (3.11), define(3.24) S ν ( f ) := Y ≤ j Let R be a symmetric CGL extension oflength N , satisfying condition (A) . (i) There exist N -tuples ( t , . . . , t N ) ∈ ( K ∗ ) N such that after the rescaling (3.20) , con-dition (3.28) holds. NTEGRAL QUANTUM CLUSTER STRUCTURES 11 (ii) If (3.28) holds, then π [ i,s m ( i )] = S ν ( e [ s ( i ) ,s m ( i )] ) − S ν ( − e i + f [ i,s m ( i )] ) for all i ∈ [1 , N ] , m ∈ Z ≥ with s m ( i ) ∈ [1 , N ] . Quantum cluster algebra structures for symmetric CGL extensions. Wepresent in this subsection the main theorem from [15].Recall the notation on quantum cluster algebras from Section 2. There is a right actionof S N on the set of toric frames for a division algebra F , given by re-enumeration,(3.29) ( M · τ )( e k ) := M ( e τ ( k ) ) , r ( M · τ ) jk = r ( M ) τ ( j ) ,τ ( k ) , τ ∈ S N , j, k ∈ [1 , N ] . Fix a symmetric CGL extension R of length N such that Conditions (A) and (B) hold.Define the multiplicatively skew-symmetric matrix ν as in Condition (A), with associatedbicharacter Ω ν as in (1.4), and define a second multiplicatively skew-symmetric matrix r = ( r kj ) by(3.30) r kj := Ω ν ( e k , e j ) , ∀ k, j ∈ [1 , N ] . Let y , . . . , y N be the sequence of normalized homogeneous prime elements given in (3.25).(We recall that each of these is a prime element in some of the subalgebras R l , but notnecessarily in the full algebra R = R N .) There is a unique toric frame M : Z N → Fract( R )whose matrix is r ( M ) := r and such that M ( e k ) := y k for all k ∈ [1 , N ] [15, Proposition4.6].Next, consider an arbitrary element σ ∈ Ξ N ⊂ S N , recall (3.15). For any k ∈ [1 , N ], wesee that(3.31) η − ησ ( k ) ∩ σ ([1 , k ]) = ( { p n ( σ ( k )) , . . . , p ( σ ( k )) , σ ( k ) } , if σ (1) ≤ σ ( k ) { σ ( k ) , s ( σ ( k )) , . . . , s n ( σ ( k )) } , if σ (1) ≥ σ ( k )for some n ∈ Z ≥ . Corresponding to σ , we have the CGL extension presentation (3.17),whose λ -matrix is the matrix λ σ with entries ( λ σ ) ij := λ σ ( i ) σ ( j ) . Analogously we define thematrix ν σ , and denote by r σ the corresponding multiplicatively skew-symmetric matrixderived from ν σ by applying (3.30) to the presentation (3.17). Explicitly,(3.32) ( r σ ) kj = Y { ν il | i ∈ σ ([1 , k ]) , η ( i ) = ησ ( k ) , l ∈ σ ([1 , j ]) , η ( l ) = ησ ( j ) } , cf. (3.31). Let y σ, , . . . , y σ,N be the sequence of normalized prime elements given by ap-plying (3.25) to the presentation (3.17). By [15, Proposition 4.6], there is a unique toricframe M σ : Z N → Fract( R ) whose matrix is r ( M σ ) := r σ and such that for all k ∈ [1 , N ],(3.33) M σ ( e k ) := y σ,k = ( y [ p n ( σ ( k )) ,σ ( k )] , if σ (1) ≤ σ ( k ) y [ σ ( k ) ,s n ( σ ( k ))] , if σ (1) ≥ σ ( k )in the two cases of (3.31), respectively. The last equality is proved in [15, Theorem 5.2].Recall that the set P ( N ) := { k ∈ [1 , N ] | s ( k ) = + ∞} parametrizes the set of homoge-neous prime elements of R , i.e., { y k | k ∈ P ( N ) } is a list of the homogeneous prime elements of R up to scalar multiples (Theorem 3.2). Define ex := [1 , N ] \ P ( N ) = { l ∈ [1 , N ] | s ( l ) = + ∞} . Since | P ( N ) | = rk( R ), the cardinality of the set ex is N − rk( R ). For σ ∈ Ξ N , define theset ex σ = { l ∈ [1 , N ] | ∃ k > l with ησ ( k ) = ησ ( l ) } of the same cardinality. Finally, recall the notation χ u from Definition 3.1.In [15, Theorem 8.2] we re-indexed all toric frames M σ in such a way that the rightaction in Theorem 3.11 (c) was trivialized and the exchangeable variables in all such seedswere parametrized just by ex , rather than by ex σ . We omit the re-indexing here, tosimply the exposition. This affects the upper cluster algebra in the following way: Whenconsidering the quantum seed ( M σ , e B σ ), the set ex must be replaced by ex σ in relationssuch as (2.2). Theorem 3.11. [15, Theorem 8.2] Let R be a symmetric CGL extension of length N andrank rk( R ) as in Definitions and . Assume that Conditions (A), (B) hold, and thatthe sequence of generators x , . . . , x N of R is normalized ( rescaled ) so that condition (3.28) is satisfied. Then the following hold: (a) For all σ ∈ Ξ N ( see (3.15)) and l ∈ ex σ , there exists a unique vector b lσ ∈ Z N suchthat χ M σ ( b lσ ) = 1 and (3.34) Ω r σ ( b lσ , e j ) = 1 , ∀ j ∈ [1 , N ] , j = l and Ω r σ ( b lσ , e l ) = λ ∗ min η − η ( σ ( l )) . Denote by e B σ ∈ M N ×| ex | ( Z ) the matrix with columns b lσ , l ∈ ex σ . Let e B := e B id . (b) For all σ ∈ Ξ N , the pair ( M σ , e B σ ) is a quantum seed for Fract( R ) . The principalpart of e B σ is skew-symmetrizable via the integers d η ( k ) , k ∈ ex σ from Condition (B) . (c) All such quantum seeds are mutation-equivalent to each other up to the S N action.They are linked by the following one-step mutations. Let σ, σ ′ ∈ Ξ N be such that σ ′ = ( σ ( k ) , σ ( k + 1)) ◦ σ = σ ◦ ( k, k + 1) for some k ∈ [1 , N − . If η ( σ ( k )) = η ( σ ( k + 1)) , then M σ ′ = M σ · ( k, k + 1) in terms ofthe action (3.29) . If η ( σ ( k )) = η ( σ ( k + 1)) , then M σ ′ = µ k ( M σ ) · ( k, k + 1) . (d) The CGL extension R equals the quantum cluster and upper cluster algebras associ-ated to M , e B , ∅ : R = A ( M, e B, ∅ ) K = U ( M, e B, ∅ ) K . In particular, A ( M, e B, ∅ ) K and U ( M, e B, ∅ ) K are affine and noetherian, and more precisely A ( M, e B, ∅ ) K is generated by the cluster variables in the seeds parametrized by the finitesubset Γ N of Ξ N , recall (3.16) . (e) Let inv be any subset of the set P ( N ) of frozen variables. Then R [ y − k | k ∈ inv ] = A ( M, e B, inv ) K = U ( M, e B, inv ) K . Integral quantum cluster structures on quantum nilpotent algebras We introduce integral forms of CGL extensions and show that the quantum clusteralgebra structure on a symmetric CGL extension R satisfying the hypotheses of Theorem3.11 passes to appropriate integral forms of R .Throughout the section, let R be a CGL extension of length N as in Definition 3.1, withassociated torus H , scalars λ kj and λ k , and other notation as in Section 3. Let D ⊆ K bea unital subring of K , and write D ∗ for the group of units of D .4.1. Integral forms of CGL extensions.Definition 4.1. We say that the D -subalgebra D h x , . . . , x N i of R is a D -form of the CGLpresentation (3.1) – and therefore that (3.1) has a D -form – provided this subalgebra isan iterated skew polynomial extension of the form(4.1) D h x , . . . , x N i = D [ x ][ x ; θ , δ ] · · · [ x N ; θ N , δ N ] , NTEGRAL QUANTUM CLUSTER STRUCTURES 13 where we let θ k (resp., δ k ) also denote the restriction of the original θ k (resp., δ k ) to a D -algebra automorphism (resp., θ k -derivation) of D h x , . . . , x k − i . Remark 4.2. (a) The CGL presentation (3.1) has a D -form if and only if • λ kj ∈ D ∗ for 1 ≤ j < k ≤ N ; • δ k maps D h x , . . . , x k − i into itself for each k ∈ [2 , N ].(b) Whether (3.1) has a D -form depends on the choice of D as well as the choice of CGLpresentation (3.1). For instance, if N = 2 and δ ( x ) ∈ K \ D , then (3.1) does not have a D -form. However, if γ = δ ( x ), then R has the CGL presentation K [ x ][ γ − x ; σ , γ − δ ],which does have a D -form.(c) Even if (3.1) has a D -form, the homogeneous prime elements y , . . . , y N from The-orem 3.2 need not belong to D h x , . . . , x N i . For instance, if R is the quantized Weylalgebra A q ( K ) = K h x , x | x x = qx x + 1 i with q ∈ K ∗ transcendental over the prime field of K and D = ( Z · K )[ q ± ], then theabove CGL presentation has a D -form, but D h x , x i does not contain the element y = x x + ( q − − .The problems indicated in Remark 4.2 can typically be corrected by rescaling the gen-erators x k as in § D -subalgebra R ′ = D h x , . . . , x N i of R , we adapt previous notationand write R ′ k := D h x , . . . , x k i and R ′ [ j,k ] := D h x j , . . . , x k i , ∀ j ≤ k ∈ [1 , N ] . Proposition 4.3. Assume that K = Fract D and that λ kj ∈ D ∗ for ≤ j < k ≤ N . Thenthere exist t , . . . , t N ∈ D \ { } such that (a) R D := D h t x , . . . , t N x N i is a D -form of the CGL presentation (3.21) . (b) The elements y , . . . , y N from Theorem for the presentation (3.21) all lie in R D .Proof. Set R ′ := R D for the proof. We induct on N . The case N = 1 holds trivially bytaking t = 1.Now assume that N > t , . . . , t N − ∈ D \ { } such that thealgebra R ′ N − := D h t x , . . . , t N − x N − i satisfies condtions (a), (b). In particular, R ′ N − is a D -form of the CGL presentation(4.2) R N − = K [ t x ][ t x ; θ , t δ ] · · · [ t N − x N − ; θ N − , t N − δ N − ] . Since λ ± Nj ∈ D for all j ∈ [1 , N − θ N restricts to an automorphismof R ′ N − .Write δ N ( x ) , . . . , δ N ( x N − ) as K -linear combinations of monomials( t x ) m · · · ( t N − x N − ) m N − in the standard PBW basis for the presentation (4.2), and let κ i for i ∈ I be a list of thenonzero coefficients that appear. Choose a nonzero element b ∈ D such that bκ i ∈ D forall i . Set t N := ( b (if p ( N ) = −∞ )( λ N − b (if p ( N ) = −∞ ) . Since bκ i ∈ D for all i , we have bδ N ( x j ) ∈ R ′ N − for all j ∈ [1 , N − bδ N maps R ′ N − into itself. Then also t N δ N maps R ′ N − into itself. Therefore R ′ = R ′ N − h t N x N i isan Ore extension R ′ N − [ t N x N ; θ N , t N δ N ] and (a) holds. It remains to show that the element y N for the CGL presentation (3.21) lies in R ′ . If p ( N ) = −∞ , then y N = t N x N and we are done. Now assume that p ( N ) = −∞ . Then y N = y p ( N ) x N − c N where c N ∈ R N and y p ( N ) is the p ( N )-th y -element for (3.21). By ourinduction hypotheses, y p ( N ) ∈ R ′ N − . From [14, Proposition 4.7(b)], we have( λ N − bδ N ( y p ( N ) ) = t N δ N ( y p ( N ) ) = O − ( N ) Y m =1 λ N,p m ( N ) ( λ N − c N . Since λ N,p m ( N ) ∈ D ∗ for all m ∈ [1 , O − ( N )] and bδ N ( y p ( N ) ) ∈ R ′ N − , we conclude that c N ∈ R ′ N − . Therefore y N ∈ R ′ , as required. (cid:3) Lemma 4.4. If the CGL presentation (3.1) has a D -form, then λ k ∈ D ∗ for all k ∈ [2 , N ] such that p ( k ) = −∞ .Proof. If k ∈ [2 , N ] and p ( k ) = −∞ , then δ k = 0 (recall Theorem 3.2). Choose i ∈ [1 , k − 1] such that δ k ( x i ) = 0, and choose a monomial x f , for some f = ( m , . . . , m k − ) T ∈ Z k − ≥ , which appears with a nonzero coefficient in δ k ( x i ). In view of (3.2), h k .δ k ( x i ) = λ k λ ki δ k ( x i ). Since all monomials in x , . . . , x N are h k -eigenvectors, it follows that h k .x f = λ k λ ki x f . On the other hand, h k .x f = θ k ( x f ) = Q k − j =1 λ m j kj x f , and consequently λ k = λ − ki k − Y j =1 λ m j kj ∈ D ∗ , since all λ kj ∈ D ∗ (Remark 4.2(a)). (cid:3) In case R is symmetric and (3.1) has a D -form, the alternative CGL extension presen-tations of R given in (3.17) also have D -forms, as we now show. Lemma 4.5. Assume that R D = D h x , . . . , x N i is a D -form for (3.1) , and that R is asymmetric CGL extension. (a) For ≤ j < k ≤ N , the algebra ( R D ) [ j,k ] is a D -form for the CGL presentation (4.3) R [ j,k ] = K [ x j ][ x j +1 ; θ j +1 , δ j +1 ] · · · [ x k ; θ k , δ k ] . (b) For each σ ∈ Ξ N , the algebra R D is a D -form for the CGL presentation (3.17) of R .Proof. Set R ′ := R D .(a) The symmetry assumption on R implies that the K -subalgebra R [ j,k ] of R is itselfa CGL extension of the form (4.3), as noted following Definition 3.6. For l ∈ [ j + 1 , k ],closure of both R [ j,l − and R ′ l − under θ ± l and δ l implies that R ′ [ j,l − is closed under θ ± l and δ l . It follows that R ′ [ j,k ] is an iterated Ore extension of the form D [ x j ] · · · [ x k ; θ k , δ k ],as required.(b) We first consider the reverse CGL extension presentation (3.13). As shown in [14, § θ ∗ j , δ ∗ j are denoted σ ′ j , δ ′ j ), we have θ ∗ j ( x k ) = λ jk x k and δ ∗ j ( x k ) = − λ jk δ k ( x j ) , ∀ ≤ j < k ≤ N. Consequently, D h x j +1 , . . . , x N i is stable under ( θ ∗ j ) ± and δ ∗ j for each j ∈ [1 , N − R ′ as an iterated Ore extension in the form(4.4) R ′ = D [ x N ][ x N − ; θ ∗ N − , δ ∗ N − ] · · · [ x ; θ ∗ , δ ∗ ] , which shows that R ′ is a D -form for (3.13).Now let σ be an arbitrary element of Ξ N and consider the corresponding CGL extensionpresentation (3.17) of R . As indicated in [14, Remark 6.5], the automorphisms θ ′′ j and skew NTEGRAL QUANTUM CLUSTER STRUCTURES 15 derivations δ ′′ j appearing in (3.17) are restrictions of either θ j , δ j or θ ∗ j , δ ∗ j . Combined withthe results of the previous paragraph, we conclude that R ′ is an iterated Ore extension ofthe form D [ x σ (1) ][ x σ (2) ; θ ′′ σ (2) , δ ′′ σ (2) ] · · · [ x σ ( N ) ; θ ′′ σ ( N ) , δ ′′ σ ( N ) ] . Therefore R ′ is a D -form for (3.17). (cid:3) Lemma 4.6. Assume that the CGL presentation (3.1) has a D -form R D = D h x , . . . , x N i which contains the elements y , . . . , y N from Theorem . (a) For each k ∈ [1 , N ] , the element y k is normal in ( R D ) k . (b) For any subset I ⊆ [1 , N ] , the multiplicative set generated by D ∗ ∪ { y i | i ∈ I } is adenominator set in R D .Proof. Set R ′ := R D .(a) By [14, Corollary 4.8], y k quasi-commutes with those x j such that j < s ( k ) accordingto the rule y k x j = (cid:18) O − ( k ) Y m =0 λ j,p m ( k ) (cid:19) − x j y k . Since the λ j,p m ( k ) all lie in D ∗ , it follows that y k R ′ k = R ′ k y k .(b) It suffices to show that D ∗ y N k , the multiplicative set generated by D ∗ ∪ { y k } , is adenominator set in R ′ for each k ∈ [1 , N ]. By part (a), D ∗ y N k is a denominator set in R ′ k .Since y k is homogeneous (with respect to the X ( H )-grading on R ), it is an eigenvectorfor each h ∈ H and thus for θ k +1 , . . . , θ N . The leading term of y k is x p O − ( k ) ( k ) · · · x p ( k ) x k ,and so θ l ( y k ) = (cid:18) O − ( k ) Y m =0 λ l,p m ( k ) (cid:19) y k , for 1 ≤ k < l ≤ N. Consequently, θ l ( D ∗ y N k ) = D ∗ y N k for all l > k . It therefore follows from [11, Lemma 1.4],by induction on l , that D ∗ y N k is a denominator set in R ′ l for l = k + 1 , . . . , N . (cid:3) Proposition 4.7. Assume that R is a symmetric CGL extension and that the CGL pre-sentation (3.1) has a D -form R D = D h x , . . . , x N i which contains the elements y , . . . , y N . (a) The elements y [ i,s m ( i )] of Proposition all belong to R D . (b) The elements u [ i,s m ( i )] of Theorem all belong to R D , and their leading coefficients π [ i,s m ( i )] belong to D . (c) The elements y σ,k , for σ ∈ Ξ N and k ∈ [1 , N ] , all belong to R D .Proof. Set R ′ := R D .(a) We first recall that by the case τ = id of [15, Theorem 5.3], y k is a scalar multipleof y [ p O − ( k ) ( k ) ,k ] for all k ∈ [1 , N ]. However, these elements both have leading coefficient 1,so they are equal. Taking k = s m ( i ), we obtain(4.5) y [ i,s m ( i )] = y s m ( i ) , ∀ i ∈ [1 , N ] with p ( i ) = −∞ . This verifies that y [ i,s m ( i )] ∈ R ′ whenever p ( i ) = −∞ .We next show, by induction on i , that all y [ i,s m ( i )] ∈ R ′ . The case i = 1 follows fromthe previous result, since p (1) = −∞ . Now assume that i > y [ j,s m ( j )] ∈ R ′ forall j ∈ [1 , i − 1] and m ∈ [0 , O + ( j )]. If p ( i ) = −∞ , we are done by the previous result,so we may assume that p ( i ) = j ∈ [1 , i − k = s m ( i ) = s m +1 ( j ). By the inductionhypothesis, y [ j,k ] ∈ R ′ . According to [15, Theorem 5.1(d)], y [ j,k ] = x j y [ i,k ] − c ′ for some c ′ ∈ R [ j +1 ,k ] . Since R [ j,k ] (resp., R ′ [ j,k ] ) is a free right module over R [ j +1 ,k ] (resp., R ′ [ j +1 ,k ] ) with basis { , x j , x j , . . . } , the assumption y [ j,k ] ∈ R ′ implies y [ i,k ] ∈ R ′ . Thisconcludes the induction step.(b) Since all values of the bicharacter Ω λ lie in D ∗ , the formula (3.19) together withpart (a) yields u [ i,s m ( i )] ∈ R ′ . Consequently, its leading coefficient, π [ i,s m ( i )] , must lie in D .(c) Fix σ ∈ Ξ N . We proceed by induction on k ∈ [1 , N ] to show that y σ,k ∈ R ′ . Thecase k = 1 holds trivially, since y σ, = x σ (1) . Now let k > y σ,j ∈ R ′ forall j ∈ [1 , k − p ( σ ( k )) / ∈ σ ([1 , k − y σ,k = x σ ( k ) and we are done. Assume now that p ( σ ( k )) = σ ( l ) for some l ∈ [1 , k − y σ,l ∈ R ′ by induction, and(4.6) y σ,k = y σ,l x σ ( k ) − c, = c ∈ R σ ([1 ,k − . By [15, Theorem 5.3], one of the following cases holds:(i) σ ( k ) > σ (1), y σ,k = λy [ p m ( σ ( k )) ,σ ( k )] , m = max { n ∈ Z ≥ | p n ( σ ( k )) ∈ σ ([1 , k ]) } ,(ii) σ ( k ) < σ (1), y σ,k = λy [ σ ( k ) ,s m ( σ ( k ))] , m = max { n ∈ Z ≥ | s n ( σ ( k )) ∈ σ ([1 , k ]) } ,for some λ ∈ K ∗ .Case (i). By the definition (3.15) of Ξ N , σ ( k ) = max σ ([1 , k ]) and σ ([1 , k − ⊆ [1 , σ ( k ) − . As p ( σ ( k )) = σ ( l ) ∈ σ ([1 , k ]), we also have m ≥ 1, and so y [ p m ( σ ( k )) ,σ ( k )] = y [ p m ( σ ( k )) ,p ( σ ( k ))] x σ ( k ) − c ′ , = c ′ ∈ R [ p m ( σ ( k )) ,σ ( k ) − . Comparing terms in R [1 ,σ ( k )] = R [1 ,σ ( k ) − x σ ( k ) + R [1 ,σ ( k ) − , we find that y σ,l = λy [ p m ( σ ( k )) ,p ( σ ( k ))] . Since lc( y [ p m ( σ ( k )) ,p ( σ ( k ))] ) = 1 and y σ,l ∈ R ′ , we find that λ ∈ D . In view of part (a), weconclude that y σ,k = λy [ p m ( σ ( k )) ,σ ( k )] ∈ R ′ . Case (ii). Now σ ( k ) = min σ ([1 , k ]) and σ ([1 , k − ⊆ [ σ ( k ) + 1 , N ]. If m = 0, we wouldhave y σ,k = λy σ ( k ) ,σ ( k )] = λx σ ( k ) , contradicting (4.6). Thus, m > 0. By [15, Theorem5.1(d)], y [ σ ( k ) ,s m ( σ ( k ))] = x σ ( k ) y [ s ( σ ( k )) ,s m ( σ ( k ))] − c ′ , = c ′ ∈ R [ σ ( k )+1 ,s m ( σ ( k ))] . We may rewrite y σ,k in the form y σ,k = µ − x σ ( k ) y σ,l − e c, where µ ∈ K ∗ arises from θ ′′ σ ( k ) ( y σ,l ) = µy σ,l and e c = µ − δ ′′ σ ( k ) ( y σ,l ) + c ∈ R σ ([1 ,k − ⊆ R [ σ ( k )+1 ,N ] . Now y σ,l is a homogeneous element of R ′ and R ′ is a D -form for (3.17). Moreover, y σ,l has leading coefficient 1 with respect to the presentation (3.17), so θ ′′ σ ( k ) ( y σ,l ) must be a D ∗ -multiple of y σ,l . Hence, µ ∈ D ∗ .Comparing terms in R [ σ ( k ) ,N ] = x σ ( k ) R [ σ ( k )+1 ,N ] + R [ σ ( k )+1 ,N ] , we find that µ − y σ,l = λy [ s ( σ ( k )) ,s m ( σ ( k ))] . Since lc( y [ s ( σ ( k )) ,s m ( σ ( k ))] ) = 1 while y σ,l ∈ R ′ and µ ∈ D ∗ , we obtain λ ∈ D , and therefore y σ,k = λy [ σ ( k ) ,s m ( σ ( k ))] ∈ R ′ in view of part (a). This concludes the second case of theinductive step. (cid:3) NTEGRAL QUANTUM CLUSTER STRUCTURES 17 Quantum cluster algebra structures on integral forms. For integral forms ofappropriately normalized symmetric CGL extensions, we have the following exact analogof Theorem 3.11. Fix a symmetric CGL extension R of length N such that Conditions(A) and (B) hold. Set F := Fract( R ), and let D be a commutative domain whose fieldof fractions is K . Define toric frames M σ : Z N → F , multiplicatively skew-symmetricmatrices r σ ∈ M N ( K ), and sets ex σ ⊆ [1 , N ] as in Subsection 3.5. (Recall the notation M = M id , r = r id , ex = ex id .) Provided the matrices r σ have entries from D ∗ , the frames M σ also qualify as toric frames over D , and we shall view them as such. Theorem 4.8. Let R be a symmetric CGL extension of length N as in Definitions , and assume that Conditions (A), (B) , and (3.28) hold. Let D be a ( commutative ) domain with quotient field Fract( D ) = K , such that the scalars ν kl in Condition (A) all liein D ∗ . Assume that the CGL presentation (3.1) has a D -form R D = D h x , . . . , x N i whichcontains the homogeneous prime elements y , . . . , y N from Theorem . (a) For each σ ∈ Ξ N , let e B σ be the N × | ex | integer matrix determined as in Theorem . Then the pair ( M σ , e B σ ) is a quantum seed for F := Fract( R ) = Fract( R D ) over D , and the principal part of e B σ is skew-symmetrizable via the integers d η ( k ) , k ∈ ex σ fromCondition (B) . (b) All the quantum seeds ( M σ , e B σ ) from part (a) are mutation-equivalent to each otherup to the S N action. They are linked by sequences of one-step mutations of the followingkind. Suppose σ, σ ′ ∈ Ξ N are such that σ ′ = ( σ ( k ) , σ ( k + 1)) ◦ σ = σ ◦ ( k, k + 1) for some k ∈ [1 , N − . If η ( σ ( k )) = η ( σ ( k + 1)) , then M σ ′ = M σ · ( k, k + 1) in terms ofthe action (3.29) . If η ( σ ( k )) = η ( σ ( k + 1)) , then M σ ′ = µ k ( M σ ) · ( k, k + 1) . (c) The algebra R D equals the quantum cluster and upper cluster algebras over D asso-ciated to M , e B , ∅ : R D = A ( M, e B, ∅ ) D = U ( M, e B, ∅ ) D . In particular, A ( M, e B, ∅ ) D is a finitely generated D -algebra, and it is noetherian if D is noetherian. In fact, A ( M, e B, ∅ ) D is generated by the cluster variables in the seedsparametrized by the finite subset Γ N of Ξ N , recall (3.16) . (d) For any subset inv of the set P ( N ) of frozen variables, there are equalities R D [ y − k | k ∈ inv ] = A ( M, e B, inv ) D = U ( M, e B, inv ) D . Proof. (a) We already have from Theorem 3.11(a) that ( M σ , e B σ ) is a quantum seed for F over K and that the principal part of e B σ is skew-symmetrizable via the d η ( k ) , k ∈ ex σ . Theentries of r ( M σ ) = r σ , given in (3.32), lie in D ∗ due to the assumption that all ν kl ∈ D ∗ .Since K = Fract( D ), we have Fract D h M σ ( Z N ) i = Fract K h M σ ( Z N ) i = F , and so ( M σ , e B σ )is also a quantum seed for F over D .(b) This is immediate from Theorem 3.11(c).(c) and (d) are proved below. (cid:3) Examples.Example 4.9. Consider a uniparameter quantized Weyl algebra R = A q, α n ( K ), for a non-root of unity q ∈ K ∗ and a skewsymmetric matrix α = ( a ij ) ∈ M n ( Z ). This algebra is presented by generators v , w , . . . , v n , w n and relations(4.7) w i w j = q a ij w j w i , (all i, j ) ,v i v j = q a ij v j v i , ( i < j ) ,v i w j = q − a ij w j v i , ( i < j ) ,v i w j = q − a ij w j v i , ( i > j ) ,v j w j = 1 + qw j v j + ( q − X l Let R = A q, α n ( K ) and H = ( K ∗ ) n as in Example 4.9, and take thesymmetric CGL presentation (4.8). Set D = ( Z · K )[ q ± ]. Then D h ( q − w n , . . . , ( q − w , v , . . . , v n i is a D -form for the presentation (4.8) which contains the homogeneous prime elements y , . . . , y n from Theorem 3.2.The CGL presentation (4.8) satisfies Condition (B) with all d i = 1, and to obtainCondition (A) we just need to assume that K contains a square root of q . Choose one,and label it q / . The condition (3.28), however, only holds after a further rescaling of thegenerators. Namely, write R as an iterated Ore extension with variables x , . . . , x n where x i := ( ( q − w n +1 − i (if i ∈ [1 , n ]) , ( − i − n q ( i − n − / v i − n (if i ∈ [ n + 1 , n ]) . In order to express the relations among these x i in a convenient form, we use the followingnotation: l ′ := 2 n + 1 − l (for l ∈ [1 , n ]) and c ij := a n +1 − i,n +1 − j (for i, j ∈ [1 , n ]) . NTEGRAL QUANTUM CLUSTER STRUCTURES 19 Then R has the presentation with generators x , . . . , x n and defining relations(4.9) x i x j = q c ij x j x i , ( i, j ∈ [1 , n ]) x i x j = q c i ′ j ′ x j x i , ( n < i < j ≤ n ) x i x j = q − c i ′ j x j x i , ( j ≤ n < i < j ′ ≤ n ) x i x j = q − c i ′ j x j x i , ( j ≤ n < j ′ < i ≤ n ) x j ′ x j = ( − n +1 − j q ( n − j ) / ( q − 1) + qx j x j ′ + ( q − X ≤ j Recall the notation F / := Q ( q / ). Let R be the F / -algebra withgenerators x , . . . , x and relations x x = qx x , x x = qx x + (1 − q ) x , x x = qx x ,x x = qx x + (1 − q ) x x , x x = qx x + ( q − x , x x = qx x ,x x = q − x x , x x = q − x x , x x = q − x x ,x x = q − x x + ( q − ,x x = q − x x + ( q − − q ) x yx + (1 − q ) x x ,x x = q − x x + ( q − q − ) x yx ,x x = q − x x + ( q − y , x x = q − x x , x x = qx x , where y := x x − q (1 − q ) . The algebra R is a symmetric CGL extension for the torus H := (( F / ) × ) acting so thatfor the corresponding grading by X ( H ) ∼ = Z , the variables x , . . . , x have degrees(4 , , (3 , , (2 , , (1 , , ( − , , ( − , − . The h -elements for this CGL extension are h = h = ( q, q − ) , h = h = ( q − , q ) ∈ H . Consequently, λ k = q for k ∈ [3 , h , h ∈ H can be alsochosen so that λ k = q for k = 1 , 2. Obviously Conditions (A) and (B) hold.Denote by R A / the A / -subalgebra of R generated by x , . . . , x . The homogeneousprime elements y , . . . , y belong to R A / and are given by y = x , y = x , y = x x + q − x ,y = x x − q − x , y = x x x − q − x x − q (1 − q ) x ,y = x x x + q − x x − qx y − (1 + q − ) x x yx + q − x x . (The element y is precisely the interval prime element y [3 , .) Consequently, the η -functionfrom Theorem 3.2 is given by η (1) = η (3) = η (6) = 1 and η (2) = η (4) = η (5) = 2. Hence,the predecessor function p maps 6 2. So, ex = [1 , R A / from Theorem 4.8 is given by r = s − s − s − s − s s − ss s − s s s s s s s s − s s − s − s − s − , where s := q / . The quiver of the initial quantum seed of R A / is1 / / / / (cid:15) (cid:15) (cid:15) (cid:15) o o / / e e ❑❑❑❑❑❑❑❑❑❑❑❑ / / / / O O NTEGRAL QUANTUM CLUSTER STRUCTURES 21 where the vertices 5 , R A / is isomorphic to the corresponding cluster and upper cluster algebras over A / where thetwo frozen variables are not inverted.All statements in the example hold if A / and F / are replaced by F p [ q ± / ] and F p ( q / ), respectively. (cid:3) Remark 4.12. The algebras in Examples 4.9–4.11 do not come from quantum unipotentcells in any symmetrizable Kac–Moody algebra, because the algebras in those examplesare Z -graded but they are not Z ≥ -graded connected algebras while all quantum unipotentcells are Z ≥ -graded connected algebras. In particular, these examples concern applicationsof Theorem 4.8 that are not covered by [23] or the results in Sect. 7 of this paper. Remark 4.13. There are also simple examples of symmetric CGL extensions R whichcannot be “untwisted” into a uniparameter form. More precisely, there are such R forwhich no twist of R relative to a K ∗ -valued cocycle on a natural grading group turns R into a uniparameter CGL extension. For instance, this is true of the multiparameterquantized Weyl algebra A Q,Pn ( K ) when the parameters in the vector Q = ( q , . . . , q n )generate a non-cyclic subgroup of K ∗ (see [16, Example 5.10]). One can show that thequantized Weyl algebras A Q,Pn ( K ) have integral forms over subrings Z [ q ± / , . . . , q ± / n ] of K . Theorem 4.8 can be applied to prove that the integral forms are isomorphic to quantumcluster algebras over Z [ q ± / , . . . , q ± / n ].4.4. Proof of parts (c), (d) of Theorem 4.8. For the first part of this subsection, weassume only that K = Fract( D ). The normalization assumptions in Theorem 4.8 will beinvoked only in the proof of parts (c), (d) of the theorem.In the following lemma and proposition, divisibility refers to divisibility within the ring R D . Lemma 4.14. Assume that (3.1) has a D -form R D = D h x , . . . , x N i . Let d ∈ D \{ } and u, v ∈ R D \{ } such that d | uv . If lc( v ) ∈ D ∗ , then d | u .Proof. Let lt( u ) = bx f and lt( v ) = cx g where b, c ∈ D \{ } and f, g ∈ Z N ≥ . By assumption, c ∈ D ∗ and uv = dw for some w ∈ R ′ \{ } . We proceed by induction on f with respectto ≺ . If f = 0, we have u = b and bcx g = lt( uv ) = d lt( w ). In this case, d | bc , whence d divides b = u because c is a unit in D .Now assume that f ≻ 0. In view of [15, Eq. (3.20)], we have λbc = lc( uv ) = d lc( w )for some λ ∈ D which is a product of λ k,j s. By assumption, λ is a unit in D , whence b = de for some e ∈ D . Now u = dex f + u ′ where either u ′ = 0 or lt( u ′ ) = b ′ x f ′ with b ′ ∈ D and f ′ ≺ f . In the second case, u ′ v = uv − dex f v = d ( w − ex f v ) . By induction, d | u ′ , and thus d | u . This verifies the induction step. (cid:3) Proposition 4.15. Assume that (3.1) has a D -form R D = D h x , . . . , x N i which contains y , . . . , y N . If Y is the multiplicative set generated by D ∗ ∪ { y , . . . , y N } , then (4.11) R D [ Y − ] ∩ R = R D . Recall from Lemma 4.6(b) that Y is a denominator set in R D . Proof. If r ∈ R D [ Y − ] ∩ R , then r = ay − for some a ∈ R D and y ∈ Y . Since r ∈ K R D , wealso have r = d − b for some d ∈ D \{ } and b ∈ R D . Now da = by . Since lc( y j ) = 1 for all j ∈ [1 , N ], we see via [15, Eq. (3.20)] that lc( y ) ∈ D ∗ . By Lemma 4.14, b = db ′ for some b ′ ∈ R D . Thus a = b ′ y and therefore r = ay − = b ′ ∈ R D . (cid:3) From now on, assume that R is a symmetric CGL extension and that (3.1) has a D -form R D = D h x , . . . , x N i which contains y , . . . , y N .. For each σ ∈ Ξ N , we have theCGL presentation (3.17) for R , and R D is a D -form of this presentation by Lemma 4.5(b).Let y σ, , . . . , y σ,N be the (unnormalized) sequence of homogeneous prime elements fromTheorem 3.11 for the presentation (3.17), and let E σ denote the multiplicative set generatedby D ∗ ∪ { y σ,l | l ∈ [1 , N ] , s σ ( l ) = + ∞} = D ∗ ∪ { y σ,l | l ∈ ex σ } , where s σ is the successor function for the level sets of ησ . (By [15, Corollary 5.6(b)], ησ can be chosen as the η -function for the presentation (3.17).) By Proposition 4.7(c) andLemma 4.6(b), E σ is a denominator set in R D . Proposition 4.16. The ring R D equals the following intersection of localizations: (4.12) R D = \ σ ∈ Γ N R D [ E − σ ] . Proof. Let T denote the right hand side of (4.12). Since T σ ∈ Γ N R [ E − σ ] = R by [15,Theorem 8.19(d)], we have T ⊆ R . On the other hand, E id is contained in the denominatorset Y of Proposition 4.15, and so T ⊆ R D [ Y − ]. Proposition 4.15 thus implies T ⊆ R D ,yielding (4.12). (cid:3) Corollary 4.17. If inv is any subset of [1 , N ] \ ex , then (4.13) R D [ y − k | k ∈ inv ] = \ σ ∈ Γ N R D [ E − σ ][ y − k | k ∈ inv ] . Proof. This follows from Proposition 4.16 in the same way that [15, Theorem 8.19(e)]follows from [15, Theorem 8.19(d)]. (cid:3) Proof of Theorem . Note that the scalars S ν ( f ) from (3.24), for f ∈ Z N , lie in D ∗ because of our assumption that all ν kl ∈ D ∗ . Hence, invoking Proposition 4.7(a), thenormalized elements y j and y [ i,s m ( i )] from (3.25) and (3.26) belong to R D . By (3.33), wethus have y σ,k ∈ R D for all σ ∈ Ξ N and k ∈ [1 , N ].We next show that(4.14) R D = D h y σ,k | σ ∈ Γ N , k ∈ [1 , N ] i . The proof is parallel to that for the corresponding statement in [15, Theorem 8.2(b)]. Foreach j ∈ [1 , N ], there is an element σ ∈ Γ N with σ (1) = j . By (3.33), y σ, is a D ∗ -multipleof y [ j,j ] = x j , and so x j ∈ D ∗ y σ, . Therefore all x j lie in the right hand side of (4.14), andthe equation is established. Since all the y σ,k = M σ ( e k ) are cluster variables, it followsthat R D ⊆ A ( M, e B, ∅ ) D .We have A ( M, e B, ∅ ) D ⊆ U ( M, e B, ∅ ) D by the Laurent Phenomenon (2.3), and U ( M, e B, ∅ ) D ⊆ \ σ ∈ Ξ N D T ( M σ , e B σ , ∅ ) = \ σ ∈ Ξ N D h y ± σ,k , y σ,j | k ∈ ex σ , j ∈ [1 , N ] \ ex σ i , NTEGRAL QUANTUM CLUSTER STRUCTURES 23 where ex σ appears instead of ex for the indexing reasons explained ahead of Theorem3.11. Since D h y ± σ,k , y σ,j | k ∈ ex σ , j ∈ [1 , N ] \ ex σ i ⊆ R D [ E − σ ] for each σ ∈ Ξ N , we obtain U ( M, e B, ∅ ) D ⊆ \ σ ∈ Ξ N R D [ E − σ ] . In view of Proposition 4.16, we have the following sequence of inclusions:(4.15) R D ⊆ A ( M, e B, ∅ ) D ⊆ U ( M, e B, ∅ ) D ⊆ \ σ ∈ Γ N R D [ E − σ ] = R D . All the inclusions in (4.15) must be equalities, which establishes the first part of Theorem4.8(c). The finite generation statements concerning A ( M, e B, ∅ ) D now follow from (4.14). If D is noetherian, the iterated Ore extension R D is noetherian by standard skew polynomialring results. This concludes the proof of part (c).Part (d) is proved analogously, using Corollary 4.17 in place of Proposition 4.16. (cid:3) Quantum Schubert cell algebras, canonical bases and quantum functionalgebras Quantized universal enveloping algebras. Fix a (finite) index set I = [1 , r ] andconsider a Cartan datum ( A, P, Π , P ∨ , Π ∨ ) consisting of the following:(i) A generalized Cartan matrix A = ( a ij ) i,j ∈ I such that a ii = 2 for i ∈ I , − a ij ∈ Z ≥ for i = j ∈ I , and there exists a diagonal matrix D = diag( d i ) i ∈ I with relativelyprime entries d i ∈ Z > for which DA is symmetric.(ii) A free abelian group P ( weight lattice ).(iii) A subset Π = { α i | i ∈ I } ⊂ P ( set of simple roots ).(iv) The dual group P ∨ := Hom Z ( P, Z ) ( coweight lattice ).(v) Two linearly independent subsets Π ∨ = { h i | i ∈ I } ⊂ P ∨ ( set of simple coroots )such that h h i , α j i = a ij for i, j ∈ I , and { ̟ i ∈ P | i ∈ I } ( set of fundamentalweights ) such that h h i , ̟ j i = δ ij .Let g be the symmetrizable Kac–Moody algebra over Q corresponding to this Cartandatum. Denote Q := ⊕ i ∈ I Z α i ⊂ P, Q + := ⊕ i ∈ I Z ≥ α i and P + := { γ ∈ P | h h i , γ i ∈ Z ≥ , ∀ i ∈ I } , P ++ := { γ ∈ P | h h i , γ i ∈ Z > , ∀ i ∈ I } . Set h := Q ⊗ Z P ∨ . There exists a Q -valued nondegenerate symmetric bilinear form ( ., . )on h ∗ = Q ⊗ Z P such that(5.1) h h i , µ i = 2( α i , µ )( α i , α i ) and ( α i , α i ) = 2 d i for i ∈ I, µ ∈ h ∗ . Set k γ k := ( γ, γ ) for γ ∈ h ∗ . Denote by W the Weyl group of g acting by isometries on( h ∗ , ( ., . )). Denote by s i its generators, by ℓ : W → Z ≥ the length function on W , and by ≥ the Bruhat order on W . We will also denote by ( ., . ) the transfer of this bilinear formto h , satisfying ( h i , h j ) = ( α i , α j ) /d i d j for all i, j ∈ I .Let U q ( g ) be the quantized universal enveloping algebra of g over the rational functionfield Q ( q ). It has generators q h , e i , f i for i ∈ I , h ∈ P ∨ and the following relations for h, h ′ ∈ P ∨ , i, j ∈ I : q = 1 , q h q h ′ = q h + h ′ ,q h e i q − h = q h h,α i i e i , q h f i q − h = q −h h,α i i f i ,e i f j − f j e i = δ ij q d i h i − q − d i h i q i − q − i , − a ij X k =0 ( − k (cid:20) − a ij k (cid:21) i e − a ij − ki e j e ki = 0 , i = j, − a ij X k =0 ( − k (cid:20) − a ij k (cid:21) i f − a ij − ki f j f ki = 0 , i = j, where q i := q d i , [ n ] i := q ni − q − ni q i − q − i , [ n ] i ! := [1] i · · · [ n ] i and (cid:20) nk (cid:21) i := [ n ] i [ k ] i [ n − k ] i for k ≤ n in Z ≥ and i ∈ I . The algebra U q ( g ) is a Hopf algebra with coproduct, antipodeand counit such that∆( q h ) = q h ⊗ q h , ∆( e i ) = e i ⊗ q d i h i ⊗ e i , ∆( f i ) = f i ⊗ q − d i h i + 1 ⊗ f i ,S ( q h ) = q − h , S ( e i ) = − q − d i h i e i , S ( f i ) = − f i q d i h i ,ǫ ( q h ) = 1 , ǫ ( e i ) = ǫ ( f i ) = 0for h ∈ P ∨ , i ∈ I . The Hopf algebra U q ( g ) is Q -graded with(5.2) deg e i = α i , deg f i = − α i , deg q h = 0 . For a Q -graded subalgebra R of U q ( g ), its graded components will be denoted by R γ ,where γ ∈ Q . For a homogeneous x ∈ U q ( g ) γ , set wt x := γ . Define the torus H := ( Q ( q ) × ) I . For γ = P n i α i ∈ Q , let t t γ denote the character of H given by ( r i ) i ∈ I Q i r n i i . Thisidentifies the rational character lattice of H with Q . The torus H acts on U q ( g ) by(5.3) t · x = t γ x for x ∈ U q ( g ) γ , γ ∈ Q. Let ∆ + ⊂ Q + be the set of positive roots of g . For w ∈ W , denote the following Liesubalgebras of the Kac–Moody algebra g ,(5.4) n ± := ⊕ α ∈ ∆ + g ± α , n ± ( w ) := ⊕ α ∈ ∆ + ∩ w − ( − ∆ + ) g ± α , where for α ∈ ∆ + , g ± α are the corresponding root spaces in g . Let b ± be the correspondingBorel subalgebras of g . Denote by U q ( n ± ) and U q ( h ) the unital subalgebras of U q ( g )respectively generated by { e i | i ∈ I } , { f i | i ∈ I } and { q h | h ∈ P ∨ } . Denote the Hopfsubalgebras U q ( b ± ) := U q ( n ± ) U q ( h ) of U q ( g ).Consider the Q ( q )-linear anti-automorphisms ∗ and ϕ of U q ( g ) defined by e ∗ i := e i , f ∗ i := f i , ( q h ) ∗ := q − h , and ϕ ( e i ) := f i , ϕ ( f i ) := e i , ϕ ( q h ) := q h for i ∈ I , h ∈ P ∨ . Their composition ϕ ∗ := ϕ ◦ ∗ = ∗ ◦ ϕ is the Q ( q )-linear automorphismof U q ( g ) satisfying ϕ ∗ ( e i ) = f i , ϕ ∗ ( f i ) = e i and ϕ ∗ ( q h ) = q − h . NTEGRAL QUANTUM CLUSTER STRUCTURES 25 Denote by c c the automorphism of the field Q ( q ) given by q = q − . The bar involution x x of U q ( g ) is its Q ( q )-skewlinear automorphism such that cx = c x for c ∈ Q ( q ), x ∈ U q ( g ) and f i = f i , e i = e i , q h = q − h for i ∈ I , h ∈ P ∨ . Denote the Q ( q )-skewlinearantiautomorphism ϕ of U q ( g ), ϕ ( x ) := ϕ ( x ) = ϕ ( x ) , ∀ x ∈ U q ( g ) . A U q ( g )-module V is called integrable if e i and f i act locally nilpotently on V and V = ⊕ µ ∈ P V µ with dim V µ < ∞ , where V µ = { v ∈ M | q h · v = q h h,µ i v, ∀ h ∈ P ∨ } . The category O int ( g ) consists of the integrable U q ( g ) modules whose nontrivial gradedsubspaces have weights in ∪ j ( µ j + Q ) for finitely many µ , . . . , µ n ∈ P (depending onthe module). It is a semisimple monoidal category with respect to the tensor product of U q ( g )-modules and with simple objects given by the irreducible highest weight modules V ( µ ) with highest weights µ ∈ P + .For V ∈ O int ( g ) its restricted dual module with respect to the antiautomorphism ϕ is amodule in O int ( g ) defined by D ϕ V := ⊕ µ ∈ P V ∗ µ , where V ∗ µ is the dual Q ( q )-vector space of V µ . The U q ( g )-action on D ϕ V is given by h x · ξ, v i = h ξ, ϕ ( x ) · v i for v ∈ V , ξ ∈ D ϕ V .Denote by { T i | i ∈ I } the generators of the braid group of W . For w ∈ W , let T w := T i · · · T i N for a reduced expression s i · · · s i N of w . We will denote by the samenotation Lustig’s braid group action [34] on U q ( g ) and on the modules in O int ( g ). We willfollow the conventions of [20].5.2. Two bilinear forms. Consider the Q ( q )-linear skew-derivations e ′′ i of U q ( n − ), e ′′ i ( f j ) = δ ij and e ′′ i ( xy ) = e ′′ i ( x ) y + q −h h i ,γ i i xe ′′ i ( y )for all i, j ∈ I , x ∈ U q ( n − ) γ , y ∈ U q ( n − ). The Kashiwara–Lusztig nondegenerate, sym-metric bilinear form ( − , − ) KL : U q ( n − ) × U q ( n − ) → Q ( q ) is the unique bilinear form suchthat(1 , KL = 1 and ( f i x, y ) KL = ( q − i − q i ) − ( x, e ′′ i ( y )) KL , ∀ i ∈ I, x, y ∈ U q ( n − ) . Remark 5.1. The Lusztig form uses the scalars (1 − q − i ) − instead of ( q − i − q i ) − , see[33, Eq. (1.2.13)(a)]. For the Kashiwara form ( q − i − q i ) − is replaced by 1, and e ′′ i arereplaced by the skew-derivations e ′ i of U q ( n − ) satisfying e ′ i ( xy ) = e ′ i ( x ) y + q ( α i ,γ ) xe ′ i ( y ), see[24, Eq. (3.4.4) and Proposition 3.4.4].The use of the above form leads to minimal rescaling of dual PBW generators, quantumminors, and cluster variables.Let d ∈ Z > be such that ( P ∨ , P ∨ ) ⊆ Z /d . The Rosso–Tanisaki form [20, § − , − ) RT : U q ( b − ) × U q ( b + ) → Q ( q /d )is the Hopf algebra pairing satisfying(5.5) ( x, yy ′ ) RT = (∆( x ) , y ′ ⊗ y ) RT , ( xx ′ , y ) RT = ( x ⊗ x ′ , ∆( y )) RT for x, x ′ ∈ U q ( b − ), y, y ′ ∈ U q ( b + ), and normalized by( f i , e j ) RT = δ ij ( q − i − q i ) − , ( q h , q h ′ ) RT = q − ( h,h ′ ) , ( f i , q h ) RT = ( q h , e i ) RT = 0for all i, j ∈ I , h ∈ P ∨ . Its restrictions to U q ( n − ) × U q ( b + ) and U q ( b − ) × U q ( n + ) takevalues in Q ( q ). The above two forms are related by(5.6) ( x, x ′ ) KL = ( x, ϕ ∗ ( x ′ )) RT , ∀ x, x ′ ∈ U q ( n − ) , see e.g. [29, Lemma 3.8] or [39, Proposition 8.3].5.3. Integral forms and canonical bases. Recall the notation (1.3). The (dividedpower) integral forms U q ( n ± ) A of U q ( n ± ) are the A -subalgebras generated by e ( k ) i := e ki / [ k ] i ! (resp. f ( k ) i := f ki / [ k ] i !) for i ∈ I , k ∈ Z > . We have ϕ ∗ ( U q ( n − ) A ) = U q ( n + ) A . The dual integral form U q ( n − ) ∨A of U q ( n − ) is the A -subalgebra U q ( n − ) ∨A = { x ∈ U q ( n − ) | ( x, U q ( n − ) A ) KL ⊂ A} (5.7) = { x ∈ U q ( n − ) | ( x, U q ( n + ) A ) RT ⊂ A} . Kashiwara [24] defined a lower global basis B low of U q ( n − ) A and an upper global basis B up of U q ( n − ) ∨A . The basis B up is defined from B low as the dual basis with respect to theform ( − , − ) KL . Lusztig [33] defined related canonical and dual canonical bases of U q ( n + ) A and a dual integral form of U q ( n + ).5.4. Quantum Schubert cell algebras, dual integral forms and CGL extensions. To each w ∈ W , De Concini–Kac–Procesi [5] and Lusztig [34, § quantumSchubert cell subalgebras of U q ( n ± ). Given a reduced expression(5.8) w = s i . . . s i N , define w ≤ k := s i . . . s i k , w [ j,k ] := s i j . . . s i k , w − ≤ k := ( w ≤ k ) − , w − j,k ] := ( w [ j,k ] ) − ∈ W for 0 ≤ j ≤ k ≤ N . Denote the roots and root vectors(5.9) β k := w ≤ k − ( α i k ) , f β k := T − w − ≤ k − ( f i k ) ∈ U q ( n − ) A , e β k := T − w − ≤ k − ( e i k ) ∈ U q ( n + ) A for k ∈ [1 , N ]. The algebras U q ( n ± ( w )) are the unital Q ( q )-subalgebras of U q ( n ± ) gener-ated by e β , . . . , e β N and f β , . . . , f β N , respectively. These definitions are independent ofthe choice of reduced expression of w . Furthermore,(5.10) U q ( n ± ( w )) = U q ( n ± ) ∩ T − w − ( U q ( n ∓ )) ,U q ( n ± ) = (cid:0) U q ( n ± ) ∩ T − w − ( U q ( n ± )) (cid:1) U q ( n ± ( w )) . This was conjectured in [1, Conjecture 5.3] and proved in [28, 38].Note that the algebras considered in [5] (see also [20]) are U ± q [ w ] = ∗ (cid:0) U q ( n ± ( w )) (cid:1) . We use U q ( n ± ( w )) instead, to avoid making all algebras here antiisomorphic to the onesin [9]. The A -algebra U q ( n − ( w )) ∨A := U q ( n − ( w )) ∩ U q ( n − ) ∨A is called the dual integral form of U q ( n − ( w )). Define the dual PBW generators of U q ( n − ( w ))(5.11) f ∗ β k := 1( f β k , e β k ) RT f β k = 1( ϕ ∗ ( e β k ) , ϕ ∗ ( e β k )) KL ϕ ∗ ( e β k ) = ( q − i k − q i k ) f β k for k ∈ [1 , N ]. Note that ϕ ∗ ( e β k ) differs from f β k by a unit of A , namely ϕ ∗ ( e β k ) =( − q i k ) Q i ( − q i ) n i f β k where n i ∈ Z ≥ are such that β k = P n i α i (see e.g [20, Eq. 8.14(9)]).The inner products between the dual PBW monomials and the divided-power PBW mono-mials are given by(5.12) (cid:16) ( f ∗ β ) m · · · ( f ∗ β N ) m N , e ( l ) β · · · e ( l N ) β N (cid:17) RT = N Y k =1 δ m k l k q m k ( m k − / i k , ∀ m k , l k ∈ Z ≥ (see e.g. [20, § e ( l k ) β k := e l k β k / [ l k ] i k . Theorem 5.2. (Kimura) [27, Prop. 4.26, Thms. 4.25 and 4.27] The algebras U q ( n − ( w )) ∨A have the following decompositions as free A -modules: U q ( n − ( w )) ∨A = M m ,...,m N ∈ Z ≥ A · ( f ∗ β ) m · · · ( f ∗ β N ) m N (5.13) = M d ∈ B up ∩ U q ( n − ( w )) A · d. The Levendorskii–Soibelman straightening law takes on the form (5.14) f ∗ β k f ∗ β j − q ( β k ,β j ) f ∗ β j f ∗ β k = X m =( m j +1 ,...,m k − ) ∈ Z k − j − ≥ b m ( f ∗ β j +1 ) m j +1 . . . ( f ∗ β k − ) m k − , b m ∈ A for all ≤ j < k ≤ N . Remark 5.3. Recall (5.3) and denote(5.15) t := ( q − i − q i ) i ∈ I ∈ H . The objects associated to U q ( n + ) used by Geiß, Leclerc and Schr¨oer in [9] are precisely theimages under the isomorphism( t · ) ◦ ϕ ∗ : U q ( n − ) ∼ = −→ U q ( n + )of the objects associated to U q ( n − ) which we consider. Firstly, [9] uses the canonical basis ϕ ( B low ) = ϕ ∗ ( B low ) of U q ( n + ) and the PBW generators e β k = ϕ ( f β k ). They use thebilinear form ( − , − ) on U q ( n + ) defined by(5.16) ( y, y ′ ) := ( ϕ ∗ ( y ) , t − ϕ ∗ ( y ′ )) KL , ∀ y, y ′ ∈ U q ( n + )leading to the following:(1) The dual canonical basis of U q ( n + ) constructed from the canonical basis ϕ ( B low ) = ϕ ∗ ( B low ) and the bilinear form (5.16), thus giving the basis ( t · ) ◦ ϕ ∗ ( B up );(2) The dual PBW generators e ∗ β k := e β k / ( e β k , e β k ) = ( t · ) ◦ ϕ ∗ ( f ∗ β k ) of U q ( n + ( w ));(3) The dual integral forms ( t · ) ◦ ϕ ∗ ( U q ( n − ) ∨A ) = { y ∈ U q ( n + ) | ( x, U q ( n + ) A ) ⊂ A} of U q ( n + ) and ( t · ) ◦ ϕ ∗ ( U q ( n − ( w )) ∨A ) of U q ( n + ( w )). (cid:3) For a reduced expression (5.8) of w ∈ W and k ∈ [1 , N ], fix elements t k , t ∗ k ∈ H suchthat(5.17) t β j k = q ( β k ,β j ) for j ∈ [1 , k ] and ( t ∗ k ) β l = q − ( β k ,β l ) for l ∈ [ k, N ] , cf. (5.3); such t k , t ∗ k exist but are not unique since the restriction of the form ( ., . ) to Q is degenerate when g is not finite dimensional. Note that the algebras U q ( n − ( w )) ∨A arepreserved by the automorphisms ( t k · ) , ( t ∗ k · ). Lemma 5.4. Let w ∈ W , (5.8) a reduced expression of w , and t k ∈ H satisfying (5.17) . (a) For k ∈ [1 , N ] , the algebra U q ( n − ( w ≤ k )) ∨A is an Ore extension U q ( n − ( w ≤ k )) ∨A ∼ = U q ( n − ( w ≤ k − )) ∨A [ f ∗ β k ; ( t k · ) , δ k ] , where δ k is the locally nilpotent ( t k · ) -derivation of U q ( n − ( w ≤ k − )) ∨A given by δ k ( x ) := f ∗ β k x − q ( β k , wt x ) xf ∗ β k for homogeneous x ∈ U q ( n + ( w ≤ k − )) ∨A . The t k -eigenvalue of f ∗ β k equals q i k , which is not a root of unity. (b) The algebra (5.18) U q ( n − ( w )) ∼ = Q ( q )[ f ∗ β ][ f ∗ β ; ( t · ) , δ ] · · · [ f ∗ β N ; ( t N · ) , δ N ] is a symmetric CGL extension. The algebra (5.19) U q ( n − ( w )) ∨A ∼ = A [ f ∗ β ][ f ∗ β ; ( t · ) , δ ] · · · [ f ∗ β N ; ( t N · ) , δ N ] with the generators f ∗ β , . . . , f ∗ β N is an A -form of the CGL extension (5.18) . (c) The interval subalgebras of U q ( n − ( w )) ∨A are (5.20) ( U q ( n − ( w )) ∨A ) [ j,k ] = T − w − ≤ j − (cid:0) U q ( n − ( w [ j,k ] )) ∨A (cid:1) for ≤ j ≤ k ≤ N. Proof. Part (a) follows from (5.13) and (5.14).(b) The facts that U q ( n − ( w )) is a CGL extension and that U q ( n − ( w )) ∨A with the gen-erators f ∗ β , . . . , f ∗ β N is an A -form of it follow by iterating (a). Its symmetricity is provedanalogously to (a).(c) Applying twice (5.13) and using (5.11), we obtain( U q ( n − ( w )) ∨A ) [ j,k ] = ⊕ m j ,...,m k ∈ Z ≥ A · ( f ∗ β j ) m j · · · ( f ∗ β k ) m k = T − w − ≤ j − (cid:0) ⊕ m j ,...,m k ∈ Z ≥ A · (( q − i j − q i j ) f i j ) m j · · · (( q − i k − q i k ) T − w − j,k − f i k ) m k (cid:1) = T − w − ≤ j − (cid:0) U q ( n − ( w [ j,k ] )) ∨A (cid:1) , which proves (5.20). (cid:3) An important feature of the normalization of ( − , − ) KL is that there are no additionalscalars in Lemma 5.4(c) due to the braid group action.5.5. The quantum function algebra of g . Consider the full dual Q ( q )-vector space U q ( g ) ∗ which is canonically a unital algebra using the coproduct and counit of U q ( g ). It isa U q ( g )-bimodule by(5.21) h x · c · y, z i := h c, yzx i for c ∈ U q ( g ) ∗ , x, y, z ∈ U q ( g ) . For a right U q ( g )-module V , let V ϕ be the left U q ( g )-module structure on the vector space V such that x · v = v · ϕ ( x ) for v ∈ V, x ∈ U q ( g ) . For each µ ∈ P + , there exists a unique irreducible right U q ( g ) module V r ( µ ) such that V r ( µ ) ϕ ∼ = V ( µ ). Analogously to O int ( g ), one defines an O -type category of integrable right U q ( g )-modules; it is denoted by O int ( g op ).Kashiwara defined [25, Sect. 7] the quantized coordinate ring A q ( g ) of the Kac–Moodygroup of g as the unital subalgebra of U q ( g ) ∗ consisting of those f ∈ U q ( g ) ∗ such that U q ( g ) · f ∈ O int ( g ) and f · U q ( g ) ∈ O int ( g op ) . Kashiwara also proved [25, Proposition 7.2.2] a quantum version of the Peter-Weyl theoremthat there is an isomorphism of U q ( g )-bimodules(5.22) A q ( g ) ∼ = M µ ∈ P + V r ( µ ) ⊗ V ( µ ) . For M ∈ O int ( g ) and v ∈ M , ξ ∈ D ϕ M define the matrix coefficient(5.23) c ξv ∈ U q ( g ) ∗ given by h c ξv , x i := h ξ, x · v i ∀ x ∈ U q ( g ) . It follows from (5.22) that A q ( g ) = { c ξv | M ∈ O int ( g ) , v ∈ M, ξ ∈ D ϕ M } = ⊕ µ ∈ P + { c ξv | v ∈ V ( µ ) , ξ ∈ D ϕ V ( µ ) } . NTEGRAL QUANTUM CLUSTER STRUCTURES 29 This is the form in which quantum function algebras were defined in the finite dimensionalcase [32]. The algebra A q ( g ) is P × P -graded by(5.24) A q ( g ) µ,ν = { c ξv | ξ ∈ ( V µ ) ∗ ⊂ D ϕ V, v ∈ V ν , V ∈ O int ( g ) } , ∀ µ, ν ∈ P. Homogeneous prime ideals of A q ( n + ( w ))6.1. The algebras A q ( n + ) and A q ( n + ( w )) . It follows from the first identity in (5.5) andthe nondegeneracy of the form that the map(6.1) ι : U q ( n − ) → U q ( b + ) ∗ given by h ι ( x ) , y i = ( x, y ) RT , ∀ x ∈ U q ( n − ) , y ∈ U q ( b + )is an injective algebra homomorphism. Here U q ( b + ) ∗ denotes the unital algebra which isthe full dual of the Hopf algebra U q ( b + ) over Q ( q ).Following Geiß–Leclerc–Schr¨oer [9, § A q ( n + ) ⊂ U q ( b + ) ∗ con-sisting of those f ∈ U q ( b + ) ∗ such that(i) f ( xq h ) = f ( x ) for all x ∈ U q ( n + ), h ∈ P ∨ and(ii) f ( x ) = 0 for all x ∈ U q ( n + ) γ and γ ∈ Q + \ S for a finite subset S of Q + .The properties( xq h , yq h ′ ) RT = ( x, y ) RT q − ( h,h ′ ) , ( U q ( n − ) − γ , U q ( n + ) δ ) RT = 0for x ∈ U q ( n − ), y ∈ U q ( n + ), h, h ′ ∈ P ∨ , γ = δ in Q + (see [20, Eq 6.13(1)]) and thenondegeneracy of ( − , − ) RT imply that A q ( n + ) := ι ( U q ( n − )). Thus(6.2) ι : U q ( n − ) ∼ = −→ A q ( n + )is an algebra isomorphism. Following [9, § A q ( n + ( w )) := ι ( U q ( n − ( w ))). Hence, ι restricts to the algebra isomorphsim(6.3) ι : U q ( n − ( w )) ∼ = −→ A q ( n + ( w )) . Using the isomorphism ι , transport the isomorphisms T w : U q ( n − ) ∩ T − w ( U q ( n − )) → T w ( U q ( n − )) ∩ U q ( n − ) to such maps on A q ( n + ). Denote the integral forms over A A q ( n + ) A := ι ( U q ( n − ) ∨A ) and A q ( n + ( w )) A := ι ( U q ( n − ( w )) ∨A )of A q ( n + ) and A q ( n + ( w )). The algebra A q ( n + ( w )) is Q + -graded by A q ( n + ( w )) γ := ι ( U q ( n − ( w )) − γ ) , ∀ γ ∈ Q + . In other words, the isomorphism (6.3) is not H -equivariant, but satisfies ι ( t · u ) = t − · ι ( u )for t ∈ H , u ∈ U q ( n − ( w )). Remark 6.1. Using the bilinear form (5.16), in [9] the algebra A q ( n + ) is identified with U q ( n + ) via the isomorphismΨ : U q ( n + ) ∼ = −→ A q ( n + ) , h Ψ( y ) , y ′ q h i := ( y, y ′ ) KL , ∀ y, y ′ ∈ U q ( n + ) , h ∈ P ∨ . Ψ fits in the commutative diagram A q ( n + ) U q ( n − ) U q ( n + ) ι Ψ( t · ) ◦ ϕ ∗ in terms of t ∈ H given by (5.15). This and Remark 5.3 imply that A q ( n + ) A and ι ( B up )are precisely the integral form of A q ( n + ) and the dual canonical basis of A q ( n + ) consideredin [9]. However the braid group action of [9] on A q ( n + ) is a conjugate of ours by an elementof the torus H , and involves extra scalars compared to our formulas. An algebra isomorphism. For µ ∈ P + , fix a highest weight vector v µ of V ( µ ). For w ∈ W , define the extremal weight vector v wµ := T − w − v µ ∈ V ( µ ) wµ . Denote the associated Demazure modules V ± w ( µ ) := U q ( b ± ) v wµ ⊆ V ( µ ) . Let ξ wµ ∈ V ( µ ) ∗ wµ ⊂ D ϕ ( V ( µ )) be such that h ξ wµ , v wµ i = 1 . For u, w ∈ W and µ ∈ P + , using the notation (5.23), define the quantum minors∆ uµ,wµ := c ξ uµ ,v wµ ∈ A q ( g ) , which are equivalently given by [2, Eq. (9.10)], [9, Eq. (3.5)]. It is well known that(6.4) T − w − ( v µ ⊗ v ν ) = T − w − v µ ⊗ T − w − v ν for all µ, ν ∈ P + . This implies that(6.5) ∆ uµ,wµ ∆ uν,wν = ∆ u ( µ + ν ) ,w ( µ + ν ) , ∀ µ, ν ∈ P + . Following Joseph [21, § A + q ( g ) := ⊕ µ ∈ P + { c ξv µ | ξ ∈ D ϕ ( V ( µ )) } of A q ( g ). By [8, Lemma 2.1(i)], the multiplicative set E w := { ∆ wµ,µ | µ ∈ P + } is a denominator set in A + q ( g ). Denote the subsets J ± w := ⊕ µ ∈ P + { c ξv µ | ξ ∈ D ϕ ( V ( µ )) , ξ ⊥ V ± w ( µ ) } ⊂ A + q ( g ) . By the proofs of Theorems 6.2 and 6.4 below, they are completely prime ideals of A + q ( g ).The P × P -grading (5.24) of A q ( g ) extends to a P × P -grading of the localization A + q ( g )[ E − w ]. For a graded subalgebra R ⊆ A + q ( g )[ E − w ], denote the subalgebra R := ⊕ ν ∈ P R ν, , noting that R is naturally P -graded. It is easy to show that every element of ( A + q ( g )[ E − w ]) has the form c ξ,v µ ∆ − wµ,µ for some µ ∈ P + , ξ ∈ D ϕ ( V ( µ )); in particular, this algebra is Q -graded. The following theorem was proved in the finite dimensional case in [40]. Theorem 6.2. For all symmetrizable Kac–Moody algebras g and w ∈ W , there exists a Q -graded surjective homomorphism ψ w : ( A + q ( g )[ E − w ]) → A q ( n + ( w )) such that (6.6) h ψ w ( c ξ,v µ ∆ − wµ,µ ) , yq h i = h ξ, yv wµ i for µ ∈ P + , ξ ∈ D ϕ ( V ( µ )) , y ∈ U q ( b + ) , h ∈ P ∨ . Its kernel equals ( J + w [ E − w ]) . We will need the following lemma. Lemma 6.3. [40, Lemma 3.2] Let H be a Hopf algebra over K and A be an H -modulealgebra equipped with a right H -action. For every algebra homomorphism θ : A → K , themap ψ : A → H ∗ , given by ψ ( a )( h ) = θ ( a · h ) , is an algebra homomorphism. NTEGRAL QUANTUM CLUSTER STRUCTURES 31 Proof of Theorem . Eq. (6.4) implies that θ w : A + q ( g ) → Q ( q ) given by θ w ( c ξv µ ) := h ξ, v wµ i , ∀ µ ∈ P + , ξ ∈ D ϕ ( V ( µ ))is an algebra homomorphism. We apply the lemma to it and to the right action (5.21) of U q ( b + ) on A + q ( g ). It shows that the map ψ w : A + q ( g ) → U q ( b + ) ∗ , given by h ψ w ( c ξv µ ) , y i := h ξ, yv wµ i , ∀ µ ∈ P + , ξ ∈ D ϕ ( V ( µ )) , y ∈ U q ( b + ) , is an algebra homomorphism. The element ψ w (∆ wµ,µ ) is a unit of U q ( b + ) ∗ because h ψ w (∆ wµ,µ ) , yq h i = ǫ ( y ) q h h,µ i , ∀ y ∈ U q ( n + ) , h ∈ P ∨ . Hence, ψ w extends to A + q ( g )[ E − w ], ψ w (( A + q ( g )[ E − w ]) ) ⊂ A q ( n + ), and the restriction of ψ w to ( A + q ( g )[ E − w ]) is given by (6.6). From now on we will denote by ψ w this restriction.The formula (6.6) implies at once that the kernel of ψ w equals ( J + w [ E − w ]) and h Im ψ w , U q ( n + ( w )) y i = 0 , ∀ y ∈ (cid:0) U q ( n + ) ∩ T − w − ( U q ( n + )) (cid:1) γ , γ ∈ Q + \{ } . For each γ ∈ Q + such that U q ( n + ( w )) γ = 0, there exists µ ∈ P + such that the pairing( V w ( µ ) γ + wµ ) ∗ × U q ( n + ( w )) γ given by ξ, y 7→ h ξ, yv wµ i is nondegenerate. This, the second equality in (5.10) and the fact that( U q ( n − ( w )) , U q ( n + ( w )) y ) RT = 0 , ∀ y ∈ (cid:0) U q ( n + ) ∩ T − w − ( U q ( n + )) (cid:1) γ , γ ∈ Q + \{ } imply that Im ψ w = A q ( n + ( w )). (cid:3) Theorem 6.4. In the setting of Theorem , there exists a ( Q -graded ) homomorphism ψ − w : ( A + q ( g )[ E − w ]) → U q ( b − ) ∗ such that h ψ w ( c ξ,v µ ∆ − wµ,µ ) , yq h i = h ξ, y ∗ v wµ i for µ ∈ P + , ξ ∈ D ϕ ( V ( µ )) , y ∈ U q ( b − ) , h ∈ P ∨ . Its kernel equals ( J − w [ E − w ]) . Its imageis contained in the image of the antiembedding U q ( n + ( w )) → ( U q ( b − )) ∗ coming from thesecond component of the Rosso-Tanisaki form. The proof of the theorem is analogous to that of Theorem 6.2.6.3. The prime spectrum of A q ( n + ( w )) . The fact that O int ( g ) is a braided monoidalcategory gives rise to R -matrix commutation relations in A q ( g ), [21, Proposition 9.1.5].Particular cases of those are the relations(6.7) ∆ wµ,µ x = q ± (( wµ,ν ) − ( µ,γ )) x ∆ wµ,µ mod J ± w , ∀ x ∈ A + q ( g ) ν,γ , µ ∈ P + , ν, γ ∈ P. For u ∈ W , µ ∈ P + , denote the unipotent quantum minors D uµ,wµ := ψ w (∆ uµ,µ ∆ − wµ,µ ) ∈ A q ( n + ( w )) . They are alternatively defined as the elements of A q ( n + ( w )) ( u − w ) µ ⊂ A q ( n ) such that(6.8) h D uµ,wµ , xq h i = h ξ uµ , xv wµ i , ∀ x ∈ U + q ( g ) , h ∈ P ∨ , which implies that they are precisely the elements of A q ( n + ( w )) defined in [9, Eqs. (5.3)-(5.4)]. Set W ≤ w = { u ∈ W | u ≤ w } . For u ∈ W ≤ w , denote the ideals I w ( u ) := ψ w (cid:0) ( J − u [ E − w ]) (cid:1) of A q ( n + ( w )) . It follows from (6.5) and (6.7) that D uµ,wµ D uν,wν = q ( wµ,uν ) − ( µ,ν ) D u ( µ + ν ) ,w ( µ + ν ) , ∀ µ, ν ∈ P +2 K. R. GOODEARL AND M. T. YAKIMOV and that D uµ,wµ x = q (( w + u ) µ, wt x ) xD uµ,wµ mod I w ( u ) , ∀ µ ∈ P + , homogeneous x ∈ A q ( n + ( w )) . We have I w (1) = 0, thus(6.9) D µ,wµ x = q (( w +1) µ, wt x ) xD µ,wµ , ∀ µ ∈ P + , homogeneous x ∈ A q ( n + ( w )) . Denote the multiplicative sets E w ( u ) := q Z { D uµ,wµ | µ ∈ P + } in A q ( n + ( w )) . Analogously to (5.3), we use the Q + -grading of A q ( n + ( w )) to construct an action of thetorus H on it. Theorem 6.5. For all symmetrizable Kac–Moody algebras g and w ∈ W , the followinghold: (a) The graded prime ideals of A q ( n + ( w )) are the ideals I w ( u ) for u ∈ W ≤ w . The map u I w ( u ) is an isomorphism of posets from W ≤ w with the Bruhat order to the setof graded prime ideals of A q ( n + ( w )) with the inclusion order. (b) All prime ideals of A q ( n + ( w )) are completely prime and Spec A q ( n + ( w )) = G u ∈ W ≤ w Spec u A q ( n + ( w )) , where Spec u A q ( n + ( w )) := {J ∈ Spec A q ( n + ( w )) | ∩ t ∈H ( t · J ) = I w ( u ) } . The following hold for u ∈ W ≤ w : (c) I w ( u ) ∩ E w ( u ) = ∅ and the localization R u,w = (cid:0) A q ( n + ( w )) /I w ( u ) (cid:1) [ E w ( u ) − ] is an H -simple domain. (d) For u ∈ W ≤ w , the center Z ( R u,w ) is a Laurent polynomial ring over Q ( q ) andthere is a homeomorphism η u : Spec Z ( R u,w ) ∼ = −→ Spec u A q ( n + ( w )) where for J ∈ Spec Z ( R u,w ) , η u ( J ) is the ideal of A q ( n + ( w )) containing I w ( u ) such that η u ( J ) /I w ( u ) = J R u,w ∩ ( A q ( n + ( w )) /I w ( u )) . Denote for brevity the algebra A + w := ( A + q ( g )[ E − w ]) ⊆ A + q ( g )[ E − w ] . It is Q -graded by ( A + w ) ν := ( A + q ( g )[ E − w ]) ν, for ν ∈ Q in terms of the P × P -grading (5.24) of A + q ( g )[ E − w ]. Define the commuting (inner) auto-morphisms τ µw ∈ Aut( A + w ) for µ ∈ P + by τ µw ( c ) := ∆ − wµ,µ c ∆ wµ,µ . For each i ∈ I , define the automorphism κ i ∈ Aut( A + q ( g )) by κ i ( c ) := c · q d i h i and thelocally nilpotent (right skew) κ i -derivation ∂ i of A + q ( g ) by ∂ i ( c ) := c · f i in terms of thesecond action in (5.21). It easy to check that κ i ∂ i κ − i = q i ∂ i . Following Joseph [21, § A.2.9],for c ∈ A + q ( g ) \{ } set deg i ( c ) := max { n ∈ Z > | ∂ ni ( c ) = 0 } and ∂ ∗ w ( c ) := ∂ n i . . . ∂ n N i N ( c ) = 0where n N , . . . , n ∈ Z ≥ are recursively defined by n k := deg i k ( ∂ n k +1 i k +1 . . . ∂ n N i N ( c )) in termsof the reduced expression (5.8). Set ∂ ∗ w (0) := 0. NTEGRAL QUANTUM CLUSTER STRUCTURES 33 Proof. We carry out the proof in four steps as follows: Step 1. For all u ∈ W ≤ w , the ideals I w ( u ) of A q ( n + ( w )) are completely prime. The image of ψ w is an iterated skew polynomial extension, and thus is a domain. Simi-larly one shows that the image of ψ − w is also a domain. Therefore ( J ± w [ E − w ]) are completelyprime ideals of A + w . By direct extension and contraction arguments one gets that J ± u arecompletely prime ideals of A + q ( g ) for u ∈ W , and that the same is true for the ideals( J ± u [ E − w ]) of A + w . The remaining part of the proof of the statement of step 1 uses ele-ments of Gorelik’s and Joseph’s proofs [18, 21] of related facts in the finite dimensionalcase. We prove the stronger fact that there exists an embedding A q ( n + ( w )) /I w ( u ) ֒ → A + w / ( J − w [ E − w ]) which we construct next. For a linear map τ on a Q ( q )-vector space V and t ∈ Q ( q ),denote by E τ ( t ) the generalized t -eigenspace of τ . Using the first action (5.21), one showsthat for all i ∈ I , w ∈ W such that ℓ ( s i w ) < ℓ ( w ) and ν ∈ P , λ ∈ P + , t ∈ Q ( q ):If c ∈ E τ µw ( t ) ∩ A + q ( g ) ν,λ , then ∂ ni ( c ) ∈ E τ µsiw ( tq ( wµ,ν ) − ( s i wµ,ν + nα i ) )where n := deg i ( c ); the proof of this is analogous to [18, Lemma 6.3.1]. By induction onthe length of w , this implies that(6.10) A + w = ⊕ γ ∈ Q + ( A + w )[2 γ ] where( A + w )[2 γ ] := M ν ∈ Q { c ∈ ( A + w ) ν | c ∈ E τ µw ( q − ( w − ν +2 γ,µ ) ) , ∀ µ ∈ P + } , and that for γ ∈ Q + , λ ∈ P + ,(6.11) c ξ,v λ ∆ − wλ,λ ∈ ( A + w )[2 γ ] ⇒ ( ∂ ∗ w − ( c ξ,v λ ))∆ − λ,λ ∈ ( A +1 )[2 γ ] . The base of the induction for w = 1 follows from (6.7) applied to J − = 0, which gives that τ µ ( c ) = q ( µ,ν ) c for all c ∈ ( A + q ( g )[ E − ]) ν, , ν ∈ − Q + ; that is(6.12) ( A +1 )[2 γ ] = M λ ∈ P + { c ξ,v λ ∆ − λ,λ | ξ ∈ ( V ( λ ) λ − γ )) ∗ ⊂ D ϕ ( V ( λ )) } , ∀ γ ∈ Q + . Furthermore, we have(6.13) ( J + w [ E − w ]) = ⊕ γ ∈ Q + \{ } A + w [2 γ ] . By (6.7), applied to J + w , the right hand side is contained in the left one. Because of (6.10)it remains to show that ( J + w [ E − w ]) ∩ A + w [0] = 0. Assume the opposite, that ( J + w [ E − w ]) ∩ A + w [0] = 0. By putting elements over a common denominator, each element of A + w can berepresented in the form c ξ,v λ ∆ − wλ,λ for some λ ∈ P + , ξ ∈ D ϕ ( V ( λ )). Choose a nonzeroelement of this form in ( J + w [ E − w ]) ∩ A + w [0]. By (6.11), ( ∂ ∗ w − ( c ξ,v λ ))∆ − λ,λ ∈ ( A +1 )[0]. Hence,(6.12) implies that ∂ ∗ w − ( c ξ,v λ ) = rc ξ λ ,v λ for some r ∈ Q ( t ) ∗ . The definition of ∂ ∗ w − givesthat h ξ, f n i . . . f n N i N v λ i 6 = 0 for some n , . . . , n N ∈ Z ≥ in terms of the reduced expression (5.8). However, f n i . . . f n N i N v λ ∈ V + w ( λ ) by the standardpresentation of Demazure modules [21, Lemma 4.4.3(v)]. This contradicts with c ξ,v λ ∆ − λ,λ ∈ ( J + w [ E − w ]) and proves (6.13).Since τ µw ∈ Aut( A + w ), A + w [0] is a subalgebra of A + w . Theorem 6.2 and (6.10), (6.13) imply A q ( n + ( w )) /I w ( u ) ∼ = A + w / (( J + w [ E − w ]) + ( J − u [ E − w ]) ) ∼ = A + w [0] / ( A + w [0] ∩ ( J − u [ E − w ]) ) ֒ → A + w / ( J − u [ E − w ]) . Step 2. For all u ∈ W ≤ w , I w ( u ) ∩ E w ( u ) = ∅ . Denote by G min the minimal Kac–Moody group associated to g , see [30, § H be the Cartan subgroup of G min , and N min+ and N − the subgroups of G min generatedby its one-parameter unipotent subgroups for positive and negative roots, respectively.Denote by B min+ and B − the associated Borel subgroups of G min . Denote by N + ( w ) theunipotent subgroup of N min+ corresponding to n + ( w ). By [27, Theorem 4.44], we have thespecialization isomorphism(6.14) A q ( n + ( w )) A ⊗ C ∼ = C [ N + ( w )]for the map A → C given by q 1. By [39, Proposition 9.7], I w ( u ) ∩ A q ( n + ( w )) A is an A -form of I w ( u ). The definitions of J − u and I w ( u ) in terms of Demazure modules imply thatunder the specialization isomorphism (6.14), I w ( u ) ∩ A q ( n + ( w )) A is mapped to functionsthat vanish on the nonempty set(6.15) N ( w ) ∩ B − u B min+ w − , which is isomorphic to the open Richardson variety in the flag scheme of G min correspond-ing to the pair u ≤ w ∈ W . Let µ ∈ P + . Analogously to the quantum situation, usingspecial representatives of w ∈ W in the normalizer of H in G min , one defines the gener-alized minor ∆ uµ,wµ which is a strongly regular function on G min . It is well known thatunder the specialization isomorphism (6.14), the element D uµ,wµ ∈ E w ( u ) corresponds tothe restriction of ∆ uµ,wµ to N min+ . This function is nowhere vanishing on the set (6.15).Therefore, the specializations I w ( u ) and E w ( u ) are disjoint, so I w ( u ) ∩ E w ( u ) = ∅ .For the next step, we denote for brevity c ξ := ψ w ( c ξ,v λ ∆ − wλ,λ ) ∈ A q ( n + ( w )) for ξ ∈ D ϕ ( V ( λ )) , λ ∈ P + . For J ∈ Spec A q ( n + ( w )) and λ ∈ P + , denote C J ( λ ) = { ν ∈ P | ∃ ξ ∈ ( V ( λ ) ν ) ∗ ⊂ D ϕ ( V ( λ )) such that c ξ / ∈ J } . Since c ξ wλ = 1 / ∈ J , wλ ∈ C J ( λ ). Denote by M J ( λ ) the set of maximal elements of C J ( λ )with respect to the partial order ν (cid:22) ν ′ if ν ′ − ν ∈ Q + . Step 3. For every J ∈ Spec A q ( n + ( w )) , there exists a unique u ∈ W ≤ w such that M J ( λ ) = { uλ } for all λ ∈ P + . This step is similar to [21, Proposition 9.3.8]. Let λ ∈ P + and ν ∈ M J ( λ ), so thereexists ξ ∈ ( V ( λ ) ∗ ) ν such that c ξ / ∈ J . The R -matrix commutation relations in A q ( g ) (seee.g. [21, Proposition 9.1.5]) and the homomorphism from Theorem 6.2 imply that c ξ x ≡ q − ( ν + wλ,γ ) xc ξ mod J , ∀ x ∈ A q ( n + ( w )) γ , γ ∈ Q + . Take any other pair λ ′ ∈ P ++ and ν ′ ∈ M J ( λ ′ ) going with ξ ′ ∈ ( V ( λ ′ ) ∗ ) ν ′ such that c ξ ′ / ∈ J . Applying the last relation twice gives c ξ c ξ ′ ≡ q − ( ν + wλ,ν ′ − wλ ′ ) − ( ν − wλ,ν ′ + wλ ′ ) c ξ ′ c ξ mod J . Since A q ( n + ( w )) / J is a prime ideal and the images of c ξ , c ξ ′ are nonzero normal elements,they are regular. Therefore the power of q above must equal 0, and thus,(6.16) ( λ, λ ′ ) − ( ν, ν ′ ) = 0 . It follows from [21, Lemma A.1.17] that ν = u λ ( λ ) for some u λ ∈ W ; that is M J ( λ ) = { u λ λ } (note that u λ is non-unique for λ ∈ P + \ P ++ ). It follows from the inclusion relationsfor Demazure modules [21, Proposition 4.4.5] and the definition of J + w that u λ ∈ W ≤ w for λ ∈ P ++ . Applying one more time (6.16) gives that u λ = u λ ′ for λ, λ ′ ∈ P ++ and that for λ ∈ P + , λ ′ ∈ P ++ , the element u λ can be chosen so that u λ = u λ ′ . NTEGRAL QUANTUM CLUSTER STRUCTURES 35 Step 4. Completion of proof. By step 3,Spec A q ( n + ( w )) = G u ∈ W ≤ w Spec ′ u A q ( n + ( w )) , where(6.17) Spec ′ u A q ( n + ( w )) := {J ∈ Spec A q ( n + ( w )) | M J ( λ ) = { uλ } , ∀ λ ∈ P + } . Steps 1, 2 and 3 and the fact that dim V ( λ ) wλ = 1 imply the following:(*) For all u ∈ W ≤ w , we have I w ( u ) ∈ Spec ′ u A q ( n + ( w )), all ideals in Spec ′ u A q ( n + ( w ))contain I w ( u ) and the stratum Spec ′ u A q ( n + ( w )) contains no other Q + -graded prime ideals.Therefore { I u ( w ) | u ∈ W ≤ w } exhaust all Q + -graded prime ideals of A + q ( n + ( w )). For u ≤ u in W ≤ w , we have I u ( w ) ⊆ I u ( w ) because V − u ( λ ) ⊇ V − u ( λ ). Step 2 and theinclusion relations between Demazure modules [21, Proposition 4.4.5] imply that there areno other inclusions between these ideals. This proves part (a).All prime ideals of A q ( n + ( w )) are completely prime by [12, Theorem 2.3]. It followsfrom (*) and the definition of M J ( λ ) that the stratum Spec u A q ( n + ( w )), defined in part(b) of the theorem, coincides with Spec ′ u A q ( n + ( w )) and equals {J ∈ Spec A q ( n + ( w )) | J ⊇ I w ( u ) , J ∩ E w ( u ) = ∅ } . The second statement in part (b) follows from (6.17), or equivalently, from [3, § II.2.1].The properties (*) imply that the ring ( A q ( n + ( w )) /I u ( w ))[ E u ( w ) − ] is H -simple sincethe stratum Spec ′ u A q ( n + ( w )) has a unique Q + -graded ideal. This and step 2 prove part(c). Part (d) now follows from [3, Lemma II.3.7, Proposition II.3.8, Theorem II.6.4]. (cid:3) The homogeneous prime elements of A q ( n + ( w )) . Denote the support of w : S ( w ) := { i ∈ I | s i ≤ w } = { i ∈ I | i = i k for some k ∈ [1 , N ] } where the second formula is in terms of a reduced expression (5.8). Corollary 6.6. The homogeneous prime elements of A q ( n + ( w )) up to scalar multiples are (6.18) D ̟ i ,w̟ i for i ∈ S ( w ) . Proof. Theorem 6.5(i) implies that the height one Q + -graded prime ideals of A q ( n + ( w )) are I w ( s i ) for i ∈ S ( w ). Since A q ( n + ( w )) ∼ = U q ( n − ( w )) is a CGL extension (Lemma 5.4), it isan H -UFD; thus, its height one Q + -graded prime ideals are principal and their generatorsare precisely the homogeneous prime elements of A q ( n + ( w )). Applying Theorem 6.5(c)for u = 1 and taking into account that I w (1) = 0 gives that I w ( s i ) ∩ E w (1) = ∅ for i ∈ S ( w ). However, E w (1) consists of monomials in the elements (6.18). Hence each of the(completely prime) ideals I w ( s i ), i ∈ S ( w ) is generated by one of the elements in (6.18).The two sets have the same number of elements and D ̟ i ,w̟ i ∈ I w ( s i ). Hence, I w ( s i ) = D ̟ i ,w̟ i A q ( n + ( w )) , ∀ i ∈ S ( w ) , and the set (6.18) exhausts all homogeneous prime elements of A q ( n + ( w )) up to scalarmultiples. (cid:3) Integral cluster structures on A q ( n + ( w ))7.1. Statements of main results. Recall the notation (1.1). Throughout the section, g denotes an arbitrary symmetrizable Kac–Moody algebra and w a Weyl group element.We fix a reduced expression (5.8). Set U q ( n − ( w )) ∨A / := U q ( n − ( w )) ∨A ⊗ A A / , A q ( n + ( w )) A / := A q ( n + ( w )) A ⊗ A A / 26 K. R. GOODEARL AND M. T. YAKIMOV and extend ι to an algebra isomorphism(7.1) ι : U q ( n − ( w )) ∨A / ∼ = −→ A q ( n + ( w )) A / . For k ∈ [1 , N ], denote(7.2) x k := q / i k ι ( f ∗ β k ) = q / i k ( q − i k − q i k ) ι ( f β k ) ∈ A q ( n + ( w )) A / , recall (5.11). For j < k ∈ [1 , N ], set(7.3) a [ j, k ] := k ( w [ j,k ] − ̟ i k k / ∈ Z / . By applying ι to (5.18) and extending the scalars from Q ( q ) to Q ( q / ), we see that A q ( n + ( w )) ⊗ Q ( q ) Q ( q / ) is a symmetric CGL extension on the generators ι ( f ∗ β ) , . . . , ι ( f ∗ β N ).It follows from Lemma 5.4(a) that the scalars λ l , λ ∗ k of the CGL extension are given by(7.4) λ k = q i k , λ ∗ k = q − i k , ∀ k ∈ [1 , N ] . Lemma 5.4(b) implies that A q ( n + ( w )) A / with the generators x , . . . , x N is an A / -formof the symmetric CGL extension A q ( n + ( w )) ⊗ Q ( q ) Q ( q / ). It follows from (5.14) that thescalars ν kj := q ( β k ,β j ) / , ∀ ≤ j < k ≤ N satisfy Condition (A) in § A q ( n + ( w )) A / are(7.5) (cid:0) A q ( n + ( w )) A / (cid:1) [ j,k ] = T − w − ≤ j − (cid:0) A q ( n + ( w [ j,k ] ])) A / (cid:1) , ∀ j ≤ k in [1 , N ] . Our first main theorem on quantum Schubert cells is: Theorem 7.1. Let g be a symmetrizable Kac–Moody algebra and w ∈ W with a reducedexpression (5.8) . Consider the A / -form A q ( n + ( w )) A / of the symmetric CGL extension A q ( n + ( w )) ⊗ Q ( q ) Q ( q / ) with the generators x , . . . , x N given by (7.2) . (a) The sequence of prime elements from Theorem of A q ( n + ( w )) ⊗ Q ( q ) Q ( q / ) withrespect to the generators x , . . . , x N is y k = q ( O − ( k )+1) / i k D ̟ ik ,w ≤ k ̟ ik , k = 1 , . . . , N. The corresponding sequence of normalized prime elements is y k = q a [1 ,k ] D ̟ ik ,w ≤ k ̟ ik , k = 1 , . . . , N. Moreover, y , . . . , y N , y , . . . , y N ∈ A q ( n + ( w )) A / . (b) The η -function η : [1 , N ] → Z of A q ( n + ( w )) ⊗ Q ( q ) Q ( q / ) from Theorem 3.2 isgiven by (7.6) η ( k ) := i k , ∀ k ∈ [1 , N ] . (c) The normalized interval prime elements of A q ( n + ( w )) A / are y [ j,k ] = q a [ j,k ] D w ≤ j − ̟ ik ,w ≤ k ̟ ik = q a [ j,k ] T w ≤ k − D ̟ ik ,w [ j,k ] ̟ ik for all j < k in [1 , N ] such that i j = i k . In the rest of this section we will use the notation (3.4) for the predecessor and successorfunctions p , s and the notation (3.5) for the functions O ± associated to the η -function (7.6).Eq. (6.9) and Theorem 7.1 imply that for k > j , D ̟ ik ,w ≤ k ̟ ik D ̟ ij ,w ≤ j ̟ ij = q − (( w ≤ k +1) ̟ ik , ( w ≤ j − ̟ ij ) D ̟ ij ,w ≤ j ̟ ij D ̟ ik ,w ≤ k ̟ ik , NTEGRAL QUANTUM CLUSTER STRUCTURES 37 and thus there is a unique toric frame M w : Z N → Fract( A q ( n + ( w )) ∨A / ) with clustervariables M w ( e k ) = q a [1 ,k ] D ̟ ik ,w ≤ k ̟ ik , ∀ k ∈ [1 , N ]and matrix r w with(7.7) ( r w ) kj := q − (( w ≤ k +1) ̟ ik , ( w ≤ j − ̟ ij ) / , ∀ ≤ j < k ≤ N. We will use a quantum cluster algebra in which the exchangeable variables are(7.8) ex ( w ) := { k ∈ [1 , N ] | s ( k ) = ∞} . The number of elements of this set is N − |S ( w ) | . We will index the columns of theexchange matrices of this quantum cluster algebra (which have sizes N × ( N − |S ( w ) | )) bythe elements of the set ex ( w ). Proposition 7.2. The matrix e B w of size N × ( N − |S ( w ) | ) with entries ( e B w ) jk = , if j = p ( k ) − , if j = s ( k ) a i j i k , if j < k < s ( j ) < s ( k ) − a i j i k , if k < j < s ( k ) < s ( j )0 , otherwiseis compatible with r w , and more precisely, its columns ( e B w ) k , k ∈ ex ( w ) satisfy (7.9) Ω r w (( e B w ) k , e l ) = q − δ kl i k = ( λ ∗ k ) δ kl / and X j ( e B w ) jk ( w ≤ j − ̟ i j = 0 for all k ∈ ex ( w ) , l ∈ [1 , N ] , recall (7.4) . The next theorem relates the integral quantum cluster algebra and upper quantum clus-ter algebra with initial seed ( M w , e B w ) (both defined over A / ) to the algebra A q ( n + ( w )) A / . Theorem 7.3. In the setting of Theorem the following hold: (a) A q ( n + ( w )) A / = A ( M w , e B w , ∅ ) A / = U ( M w , e B w , ∅ ) A / . (b) For each σ ∈ Ξ N ⊂ S N , the quantum cluster algebra A ( M w , e B w , ∅ ) A / has a seedwith cluster variables M wσ ( e l ) = q a [ j,k ] D w ≤ j − ̟ ij ,w ≤ k ̟ ij = q a [ j,k ] T w ≤ j − D ̟ ij ,w [ j,k ] ̟ ij for j := min σ ([1 , l ]) and k := max { m ∈ σ ([1 , l ]) | i m = i j } . The initial seed ( M w , e B w ) equals the seed corresponding to σ = id N ∈ Ξ N . (c) The seeds in (b) are linked by sequences of one-step mutations of the followingkind:Suppose that σ, σ ′ ∈ Ξ N are such that σ ′ = ( σ ( k ) , σ ( k + 1)) ◦ σ = σ ◦ ( k, k + 1) forsome k ∈ [1 , N − . If η ( σ ( k )) = η ( σ ( k + 1)) , then M wσ ′ = M wσ · ( k, k + 1) in termsof the action (3.29) . If η ( σ ( k )) = η ( σ ( k + 1)) , then M wσ ′ = µ k ( M wσ ) · ( k, k + 1) . We illustrate Theorem 7.3 and the constructions in §§ g : a finite dimensionalone and an affine one. Example 7.4. Let g be of type B and w be the longest Weyl group element s s s s .The corresponding root sequence (5.9) is β = α , β = s ( α ) = α + α , β = s s ( α ) = α + 2 α , and β = s s s ( α ) = α . The root vectors f β k , 1 ≤ k ≤ 4, satisfy f β f β = q f β f β , f β f β = f β f β + 1 − q − q − + q f β , f β f β = q f β f β ,f β f β = q − f β f β − q − f β , f β f β = f β f β − ( q − + q ) f β , f β f β = q f β f β . Note that the scalar in the right hand side of the second equation is not in A . The CGLextension U q ( n − ( w )) = U q ( n − ) is the C ( q )-algebra with these generators and relations. Its η -function from Theorem 3.2 is given by η (1) = η (3) = 1, η (2) = η (4) = 2. The generatorsof the integral form U q ( n − ) ∨A of the CGL extension U q ( n − ) (cf. (5.11) and Lemma 5.4(b))are f ∗ β k = c k f β k , where c = c = q − − q , c = c = q − − q. They satisfy f ∗ β f ∗ β = q f β f β , f ∗ β f ∗ β = f ∗ β f ∗ β − q − ( q − − q )( f ∗ β ) ,f ∗ β f ∗ β = q f ∗ β f ∗ β , f ∗ β f ∗ β = q − f ∗ β f ∗ β − q − ( q − − q ) f ∗ β ,f ∗ β f ∗ β = f ∗ β f ∗ β − ( q − − q ) f ∗ β , f ∗ β f ∗ β = q f ∗ β f ∗ β . Recall the isomorphism (7.1). The rescaled generators of A q ( n + ( w )) A / = A q ( n + ) A / are x k = c ′ k ι ( f ∗ β k ) , where c ′ = c ′ = q, c ′ = c ′ = q / . The algebra A q ( n + ) A / is the A / -algebra with generators x , . . . , x and relations x x = q x x , x x = x x − ( q − − q ) x , x x = q x x ,x x = q − x x − q − ( q − − q ) x , x x = x x − ( q − − q ) x , x x = q x x . By Theorem 7.3, A q ( n + ) A / has the structure of a quantum cluster algebra over A / withinitial cluster variables y = x , y = x , y = x x − q − x , y = x x − q − x (where the 3rd and 4th variables are frozen) and mutation matrix e B = − 12 0 − − . Note that this is the quantum cluster algebra of type B with principal coefficients. (cid:3) Example 7.5. Let g be the twisted affine Kac–Moody algebra of type A (2)2 , whose Dynkindiagram is α α Following the standard convention [22, Ch. 6,8], we label its simple roots by { , } insteadof { , } . We have ( α , α ) = 2, ( α , α ) = 8, d = 1, d = 4 and q = q , q = q . Considerthe Weyl group element w = s s s s s . The corresponding root sequence (5.9) is β = α , β = 4 α + α , β = 3 α + α ,β = 8 α + 3 α , β = 5 α + 2 α . NTEGRAL QUANTUM CLUSTER STRUCTURES 39 The root vectors f β k , 1 ≤ k ≤ 5, given by (5.9), satisfy the relations(7.10) z z = q z z , z z = q z z + az , z z = q z z ,z z = q z z + abq − q z , z z = q z z + bz z z = q z z ,z z = q z z + abc ( q − q ( q − ( q − z , z z = q z z + bcq − q z ,z z = q z z + cz , z z = q z z with a = c = − q [4] q and b = − q ( q − − q ) / [4] q , where [ n ] q = ( q n − q − n ) / ( q − q − ).Note that b / ∈ A . The CGL extension U q ( n − ( w )) is the C ( q )-algebra with these generatorsand relations. Its η -function from Theorem 3.2 is given by η (1) = η (3) = η (5) = 0, η (2) = η (4) = 1. The generators of the integral form U q ( n − ( w )) ∨A of the CGL extension U q ( n − ( w )) (cf. (5.11) and Lemma 5.4(b)) are f ∗ β k = c k f β k , where c = c = c = q − − q, c = c = q − − q . They satisfy the relations (7.10) for a = c = q ( q − , b = q − ( q − ∈ A , andfurthermore, U q ( n − ( w )) ∨A is the A -algebra with these generators and relations. Recall theisomorphism (7.1). The rescaled generators of A q ( n + ( w )) A / are x k = c ′ k ι ( f ∗ β k ) , where c ′ = c ′ = c ′ = q / , c ′ = c ′ = q . They satisfy the relations (7.10) for a = c = q − , b = q − ∈ A ⊂ A / , and furthermore, A q ( n + ) A / is the A / -algebra with these generators and relations. By Theorem 7.3, A q ( n + ) A / has the structure of a quantum cluster algebra over A / with initial clustervariables y = x , y = x , y = qx x − q − x , y = q x x − q − x ,y = q x x x − qx x − q − [3] q x − qx x . (where the 4th and 5th variables are frozen) and mutation matrix e B = − − − − − . (cid:3) Proof of Theorem 7.1. If u , u ∈ W are such that ℓ ( u u ) = ℓ ( u ) + ℓ ( u ), thenwe have the decomposition A q ( n + ( u u )) A = A q ( n + ( u )) A T − u − ( A q ( n + ( u ))) A . This follows by applying the isomorphism ι to the dual PBW basis (5.13) of U q ( n − ( u u )) ∨A .The next lemma shows the equality of the unipotent quantum minors in Theorem 7.1(c)and that they belong to the correct integral forms. Lemma 7.6. If u , u ∈ W are such that ℓ ( u u ) = ℓ ( u ) + ℓ ( u ) , then (7.11) D u µ,u u µ = T − u − D µ,u µ ∈ T − u − A q ( n + ( u )) A ⊂ A q ( n + ( u u )) , ∀ µ ∈ P + . Proof. It was proved in [9, Proposition 6.3] that Ψ − ( D µ,u µ ) ∈ t · ϕ ∗ ( B up ) in the notationof Remarks 5.3 and 6.1. Theorem 5.2 and the commutative diagram in Remark 6.1 implythat D µ,u µ ∈ ι ( B up ) ⊂ A q ( n + ( u )) A .The equality (7.11) can be derived from [9, Proposition 7.1] and Remark 6.1, but it alsohas a direct proof as follows. For all y k ∈ U q ( n + ( u k )), k = 1 , h ∈ P ∨ , we have h D u µ,u u µ , y T − u − ( y ) q h i = h ξ u µ , y T − u − ( y ) v u u µ i = h ξ µ , T u − ( y ) y v u µ i = h ξ µ , y v u µ i ǫ ( y ) = h D µ,u µ , y i ǫ ( y ) = ( ι − ( D µ,u µ ) , y ) RT ǫ ( y )= ( T − u − ι − ( D µ,u µ ) , T − u − y ) RT ǫ ( y ) = h T − u − D µ,u µ , y T − u − ( y ) q h i , where the sixth equality uses (5.12). (cid:3) Proof of Theorem . We have e β k v w ≤ k − ̟ ik = T − w − ≤ k − ( e i k T − i k v ̟ ik ) = T − w − ≤ k − v ̟ ik = v w ≤ p ( k ) ̟ ik and e mβ k v w ≤ k − ̟ i = 0 for m > 1. Hence h D ̟ ik ,w ≤ k ̟ ik , y e mβ k i = δ m h D ̟ ik ,w ≤ p ( k ) ̟ ik , y i for all y ∈ U q ( n + ( w ≤ k − )), m ∈ Z ≥ . It follows from (5.12) and (7.5) that in A q ( n + ( w ≤ k )) A = (cid:0) A q ( n + ( w )) A / (cid:1) [1 ,k ] ⊂ A q ( n + ( w )) A we have(7.12) D ̟ ik ,w ≤ k ̟ ik ≡ D ̟ ik ,w ≤ p ( k ) ̟ ik ι ( f ∗ β k ) mod A q ( n + ( w ≤ k − )) A . Therefore, q ( O − ( k )+1) / i k D ̟ ik ,w ≤ k ̟ ik ≡ q ( O − ( p ( k ))+1) / i k D ̟ ik ,w ≤ p ( k ) ̟ ik x k mod A q ( n + ( w ≤ k − )) A / for all k ∈ [1 , N ]. Part (b) and the first statement in part (a) now follow from Corollary6.6.We have w ≤ k ̟ i k = w ≤ k − ( ̟ i k − α i k ) = w ≤ p ( k ) ̟ i k − β k . Iterating this gives a [1 , k ] = k w ≤ k ̟ i k − ̟ i k k / k β p O − ( k ) ( k ) + · · · + β k k = ( O − ( k ) + 1) k α i k k / X ≤ l ≤ m ≤ O − ( k ) ( β p l ( k ) , β p m ( k ) ) / . Therefore, y k = (cid:16) Y ≤ l ≤ m ≤ O − ( k ) ν − p l ( k ) p m ( k ) (cid:17) y k = (cid:16) Y ≤ l ≤ m ≤ O − ( k ) q ( β pl ( k ) ,β pm ( k ) ) / (cid:17) q ( O − ( k )+1) k α ik k / D ̟ ik ,w ≤ k = q a [1 ,k ] D ̟ ik ,w ≤ k which proves the second statement in part (a).It follows from Lemma 7.6 that y , . . . , y N , y , . . . , y N ∈ A q ( n + ( w )) A / . Part (c) followsfrom (7.5) and part (b). (cid:3) Proof of Theorem 7.3. Proof of Proposition . Extend e B w to an ( N + r ) × ( N − S ( w )) matrix whose rows areindexed by [ − r, − ⊔ [1 , N ] and columns by ex ( w ) by setting( e B w ) − i,k := ( , if i k = i and p ( k ) = −∞ , otherwise NTEGRAL QUANTUM CLUSTER STRUCTURES 41 for i ∈ [1 , r ], k ∈ ex ( w ).Denote for simplicity b jk := ( e B ) jk . We apply [2, Theorem 8.3 and § , . . . , r, − i , . . . , i N , which gives(7.13) N X j =1 b jk sign( j − l ) (cid:0) ( w ≤ j ̟ i j , w ≤ l ̟ i l ) − ( ̟ i j , ̟ i l ) (cid:1) + r X i =1 b − i,k (( w ≤ j − ̟ i j , ̟ i ) = 2 δ kl d k for all k ∈ ex ( w ), l ∈ [1 , N ]. The graded nature of the seed corresponding to the doubleword (cf. [2, Definition 6.5]) means that N X j =1 b jk w ≤ j ̟ i j + r X i =1 b − i,k ̟ i = 0 , (7.14) N X j =1 b jk ̟ i j + r X i =1 b − i,k ̟ i = 0(7.15)for all k ∈ ex ( w ). Subtracting (7.14) from (7.13) gives the second identity in (7.9). Thelinear combination (7.13) + ((7.14) , ̟ i l ) − ((7.15) , w ≤ l ̟ i l ) yields the identity N X j =1 b jk sign( j − l ) (cid:0) ( w ≤ j + 1) ̟ i j , ( w ≤ l − ̟ i l ) = 2 δ kl d k for all k ∈ ex ( w ), l ∈ [1 , N ], which is precisely the first identity in (7.9) in view of (7.7). (cid:3) Proposition 7.7. In the setting of Theorem , the A / -form A q ( n + ( w )) A / of thesymmetric CGL extension A q ( n + ( w )) ⊗ Q ( q ) Q ( q / ) with the generators x , . . . , x N from (7.2) satisfies all conditions in Theorem .Proof. The scalars ν kl are integral powers of q / and thus are units of A / . Obviouslycondition (A) is satisfied for the base field K = Q ( q / ). Recall from Lemma 5.4(a) and(7.4) that λ k = q i k = q d ik = q k α ik k for k ∈ [1 , N ] , and from Theorem 7.1(b) that η ( k ) = i k for k ∈ [1 , N ]. Therefore, Condition (B) issatisfied for the positive integers { d i | i ∈ I } from (5.1).The homogenous prime elements y , . . . , y N belong to A q ( n + ( w )) A / by Theorem 7.1(a).It remains to show that the condition (3.28) holds. Because of (7.5) and Lemma 7.6it is sufficient to consider the case when i = 1 and s ( i ) = N . Since the η -function of theCGL extension A q ( n + ( w )) is given by Theorem 7.1(b), this means that i = i N = i and i k = i for k ∈ [2 , N − g = sl and l ≥ n ∈ Z > e ( l )1 · T − v n̟ = δ ln v n̟ . For k ∈ [2 , N − T − i v ̟ i is a highest weight vector for the copy of U q ( sl ) inside U q ( g )generated by e i k , f i k , h i k of highest weight h s i ̟ i , h i k i = − a i k i ̟ i k . Hence, for l ≥ − a i k i , e ( l ) β k · T − w − ≤ k T − i v ̟ i = T − w − ≤ k − ( e ( l ) i k · T − i k T − i v ̟ i ) = δ l, − a iki T − i v ̟ i . Set a := ( − a i i , . . . , − a i N − i ) ∈ Z N − ≥ . Iterating this and using (7.12) and the identity T − i v ̟ i = − q − i v ̟ i gives (cid:10) D ̟ i ,w̟ i − q − i x x N , e ( l ) β . . . e ( l N − ) β N − (cid:11) = ( − q − i , if a = ( l , . . . , l N − )0 , if a (cid:22) ( l , . . . , l N − )with respect to the the reverse lexicographic order (3.9). 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Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A. E-mail address : [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. E-mail address ::