Quantum Grothendieck rings as quantum cluster algebras

Abstract We define and construct a quantum Grothendieck ring for a certain monoidal subcategory of the category O of representations of the quantum loop algebra introduced by Hernandez–Jimbo. We use the cluster algebra structure of the Grothendieck ring of this category to define the quantum Grothendieck ring as a quantum cluster algebra. When the underlying simple Lie algebra is of type A, we prove that this quantum Grothendieck ring contains the quantum Grothendieck ring of the category of finite‐dimensional representations of the associated quantum affine algebra. In type A1, we identify remarkable relations in this quantum Grothendieck ring.


Introduction
Let g be a simple Lie algebra of Dynkin type A, D or E (also called simply laced types), and let Lg " g b Crt˘1s be the loop algebra of g. For q a generic complex number, Drinfeld [17] introduced a q-deformation of the universal enveloping algebra U pLgq of Lg called the quantum loop algebra U q pLgq. It is a Hopf algebra over C and therefore the category C of its finite-dimensional representations is monoidal. The category C was studied extensively, in particular to build solutions to the quantum Yang-Baxter equation with spectral parameter (see [1, 12-14, 24, 27, 38, 42, 46] to name but a few). Using the so-called 'Drinfeld-Jimbo' presentation of the quantum loop algebra, one can define a quantum Borel subalgebra U q pbq, which is a Hopf subalgebra of U q pLgq. We are here interested in studying a category O of representations of U q pbq introduced by Hernandez-Jimbo [32]. The category O is a monoidal category which contains all finite-dimensional U q pbq-modules, as well as some infinite-dimensional representations, however, with finite-dimensional weight spaces. In particular, this category O contains the prefundamental representations. These are a family of infinite-dimensional simple U q pbq-modules, which first appeared in the work of Bazhanov, Lukyanov, Zamolodchikov [3] for g " sl 2 under the name q-oscillator representations.
These prefundamental representations were also used by Frenkel-Hernandez [22] to prove Frenkel-Reshetikhin's conjecture on the spectra of quantum integrable systems [25]. More precisely, quantum integrable systems are studied via a partition function Z, which in turns can be scaled down to the study of the eigenvalues λ j of the transfer matrix T . For the 6-vertex (and 8-vertex) models, [2] showed that the eigenvalues of T have the following remarkable form: where q and z are parameters of the model, the rational functions Apzq, Dpzq are universal and Q j is a polynomial. This relation is called the Baxter relation. In the context of representation theory, relation (1.1) can be categorified as a relation in the Grothendieck ring of the category O. For g " sl 2 , if V is the two-dimensional simple representation of U q pLgq of highest loopweight Y aq´1 , then, in the Grothendieck ring K 0 pOq, rV b Là s " rω 1 srLà q´2 s`r´ω 1 srLà q 2 s, (1.2) where r˘ω 1 s are one-dimensional representations of weight˘ω 1 and Là denotes the positive prefundamental representation of quantum parameter a.
Frenkel-Reshetikhin's conjecture stated that for more general quantum integrable systems, constructed via finite-dimensional representations of the quantum affine algebra U q pĝq (of which the quantum loop algebra is a quotient) the spectra had a similar form as relation (1.1).
Let t be an indeterminate. The Grothendieck ring of the category C has an interesting t-deformation called the quantum Grothendieck ring, which is contained in some quantum torus Y t . The quantum Grothendieck ring was first studied by Nakajima [42] and Varagnolo-Vasserot [45] in relation with quiver varieties. Inside this ring, one can define for all simple modules L classes rLs t , called pq, tq-characters. Using these classes, and the knowledge of characters of fundamental modules, Nakajima was able to compute the characters of all simple modules L, thanks to a Kazhdan-Lusztig type algorithm.
One would want to extend these results to the context of the category O, with the ultimate goal of (algorithmically) computing characters of all simple modules in O. In order to do that, one first needs to build a quantum Grothendieck ring K t pOq inside which the classes rLs t can be defined.
Another interesting approach to this category O is its cluster algebra structure (see below). Hernandez-Leclerc [34] first noted that the Grothendieck ring of a certain monoidal subcategory C 1 of the category C of finite-dimensional U q pLgq-modules had the structure of a cluster algebra. Then, they proved [36] that the Grothendieck ring of a certain monoidal subcategory OZ of the category O had a cluster algebra structure, of infinite rank, for which one can take as initial seed the classes of the positive prefundamental representations (the category OZ contains the finite-dimensional representations and the positive prefundamental representations whose spectral parameter satisfy an integrality condition). Moreover, some exchange relations coming from cluster mutations appear naturally. For example, the Baxter relation (1.2) is an exchange relation in this cluster algebra.
In order to construct of quantum Grothendieck ring for the category O, the approaches used previously are not applicable anymore. The geometrical approach of Nakajima and Varagnolo-Vasserot (in which the t-graduation naturally comes from the graduation of cohomological complexes) requires a geometric interpretation of the objects in the category O, which has not yet been found. The more algebraic approach consisting of realizing the (quantum) Grothendieck ring as an invariant under a sort of Weyl symmetry, which allowed Hernandez to define a quantum Grothendieck ring of finite-dimensional representations in non-simply laced types, is again not relevant for the category O. Only the cluster algebra approach yields results in this context.
In this paper, we propose to build the quantum Grothendieck of the category OZ as a quantum cluster algebra. Quantum cluster algebras are non-commutative versions of cluster algebras, they live inside a quantum torus, generated by the initial variables, together with t-commutation relations: First of all, one has to build such a quantum torus T t , and check that it contains the quantum torus Y t of the quantum Grothendieck ring of the category C . This is proven as the first result of this paper (see Theorem 5.2.1) : Theorem. There is an injective homomorphism of Zptq-algebras : Next, one has to show that this quantum torus is compatible with a quantum cluster algebra structure based on the same quiver as the cluster algebra structure of the Grothendieck ring K 0 pOZ q. In order to do that, we exhibit a compatible pair (see Proposition 6.2.5) : Proposition. The quiver appearing in the cluster algebra structure of K 0 pOZ q and the quantum torus T t form a compatible pair, in the sense of quantum cluster algebras.
From then, the quantum Grothendieck ring K t pOZ q is defined as the quantum cluster algebra defined from this compatible pair.
We then conjecture (Conjecture 7.2.1) that this quantum Grothendieck ring K t pOZ q contains the quantum Grothendieck ring K t pC Z q. We propose to demonstrate this conjecture by proving that K t pOZ q contains the pq, tq-characters of the fundamental representations rLpY i,q r qs t , as they generate K t pC Z q. We state in Conjecture 7.2.6 that these objects can be obtained in K t pOZ q as quantum cluster variables, by following the same finite sequences of mutations used in the classical cluster algebra K 0 pOZ q to obtain the rLpY i,q r qs. Naturally, Conjecture 7.2.6 implies Conjecture 7.2.1. Finally, we prove Conjecture 7.2.6 (and thus Conjecture 7.2.1) in the case where the underlying simple Lie algebra g is of type A (see Theorem 8.1.1).
Theorem. When the underlying simple Lie algebra is of type A, the constructed quantum Grothendieck ring K t pOZ q contains the quantum Grothendieck ring K t pC Z q.
The proof is based on the thinness property of the fundamental representations in this case. When g " sl 2 , some explicit computations are possible. For example, we give a quantum version of the Baxter relation (8.5), for all r P Z, Additionally, we realize a part of the quantum cluster algebra we built as a quotient of the Drinfeld double of the full quantum group U q psl 2 q. This is a reminiscence of the result of Qin [43] who constructed U q pgq as a quotient of the Grothendieck ring arising from certain cyclic quiver varieties. The paper is organized as follows. The first three sections are mostly reminders. In Section 2 we recall some background on cluster algebras and quantum cluster algebras, including some recent and important results, such as the positivity theorem in Section 2.6, which we require later on. In Section 3 we introduce some notations, the usual notations for the Cartan data associated to a simple Lie algebra, as well as what we call quantum Cartan data, which is related to the quantum Cartan matrix and its inverse. In Section 4 we review some results for the category O, its subcategories O˘and OZ and their Grothendieck rings. In Section 5, after recalling the definition of the quantum torus Y t , we define the quantum torus T t in which K t pOZ q will be constructed and we prove the inclusion of the quantum tori. In Section 6 we prove that we have all the elements to build a quantum cluster algebra and we define the quantum Grothendieck ring K t pOZ q. In the concluding Section 7, we state some properties of the quantum Grothendieck ring. We present the two conjectures regarding the inclusion of the quantum Grothendieck rings in Section 7.2. Finally, in Section 8 we prove these conjectures in type A, and we prove finer properties specific to the case when g " sl 2 .

Cluster algebras and quantum cluster algebras
Cluster algebras were defined by Fomin and Zelevinsky in the 2000s in a series of fundamental papers [4,18,20,21]. They were first introduced to study total positivity and canonical bases in quantum groups but soon applications to many different fields of mathematics were found.
In [6], Berenstein and Zelevinsky introduced natural non-commutative deformations of cluster algebras called quantum cluster algebras.
In this section, we recall the definitions of these objects. The interested reader may refer to the aforementioned papers for more details, or to surveys such as [19].

Cluster algebras
Let m ě n be two positive integers and let F be the field of rational functions over Q in m independent commuting variables. Fix of subset ex Ă 1, m of cardinal n.
In what follows, we use the usual notation: rxs`" maxpx, 0q.
. , x m u is an algebraically independent subset of F which generates F. •B " pb i,j q ofB is a mˆn integer matrix with rows labeled by 1, m and columns labeled by ex such that (1) the nˆn submatrix B " pb ij q i,jPex is skew-symmetrizable.
(2)B has full rank n.
The matrix B is called the principal part ofB, x x x " tx j | j P exu Ăx x x is the cluster of the seed px x x,Bq, ex are the exchangeable indices and c c c "x x xzx x x is the set of frozen variables.
For all k P ex, define the seed mutation in direction k as the transformation from px x x,Bq to μ k px x x,Bq " px x x 1 ,B 1 q, with •B 1 " μ k pBq is the mˆn matrix whose entries are given by This operation is called matrix mutation in direction k. This matrix can also be obtained via the operationB where E k and F k are the mˆm and nˆn matrices with entries is also a seed in F and the seed mutation operation is involutive: μ k px x x 1 ,B 1 q " px x x,Bq. Thus, we have a equivalence relation: px x x,Bq is mutation-equivalent to px x x 1 ,B 1 q, denoted by px x x,Bq " px x x 1 ,B 1 q, if px x x 1 ,B 1 q can be obtained from px x x,Bq by a finite sequence of seed mutations.
Graphically, if the matrixB is skew-symmetric, it can be represented by a quiver and the matrix mutation by a simple operation on the quiver. FixB a skew-symmetric matrix. Define the quiver Q whose set of vertices is 1, m , where the vertices corresponding to c c c are usually denoted by a square˝and called frozen vertices. For all i P 1, m , j P ex, b ij is the number of arrows from i to j (can be negative if the arrows are from j to i).
In this context, the operation of matrix mutation can be translated naturally to an operation on the quiver Q. For k P ex, the quiver Q 1 " μ k pQq is obtained from Q by the following operations: • For each pair of arrows i Ñ k Ñ j in Q, create an arrow from i to j.
• Invert all arrows adjacent to k.
• Remove all 2-cycles that were possibly created.
Definition 2.1.3. Let S be a mutation-equivalence class of seeds in F. The cluster algebra ApSq associated to S is the Zrc c c˘s-subalgebra of F generated by all the clusters of all the seeds in S.

Compatible pairs
A quantum cluster algebra is a non-commutative version of a cluster algebra. Cluster variables will not commute anymore, but, if they are in the same cluster, commute up to some power of an indeterminate t. These powers can be encoded in a skew-symmetric matrix Λ. In order for the quantum cluster algebra to be well defined, one needs to check that these t-commutation relations behave well with the exchange relations. This is made explicit via the notion of compatible pairs. Definition 2.2.1. LetB be a mˆn integer matrix, with rows labeled by 1, m and columns labeled by ex. Let Λ " pλ ij q 1ďi,jďm be a skew-symmetric mˆm integer matrix. We say that pΛ,Bq forms a compatible pair if, for all i P ex and 1 ď j ď m, we have with pd i q iPex some integers, all positive or negative. Relation (2.5) is equivalent to saying that, up to reordering, if ex " 1, n , the matrixB T Λ consists of two blocks, a diagonal nˆn block, and a nˆpm´nq zero block:¨d Fix a compatible pair pΛ,Bq and fix k P ex. Define, in a similar way as in (2.2), Proposition 2.2.2 [6]. The pair pΛ 1 ,B 1 q is compatible.

Definition of quantum cluster algebras
We now introduce the last notions we need in order to define quantum cluster algebras.
Let t be a formal variable. Consider Zrt˘1 {2 s, the ring of Laurent polynomials in the variable t 1{2 .
Recall that any skew-symmetric integer matrix Λ of size mˆm determines a skew-symmetric Z-bilinear form on Z m , which will also be denoted by Λ: Definition 2.3.1. The (based) quantum torus T " pT pΛq,˚q associated with the skewsymmetric bilinear form Λ is the Zrt˘1 {2 s-algebra generated by the tX e | e P Z m u, together with the t-commuting relations: (2.9) The quantum torus T pΛq is an Ore domain (see details in [6]), thus it is contained in its skew-field a fractions F " pF,˚q. The field F is a Qpt 1{2 q-algebra. where φ : F Ñ F is a Qpt 1{2 q-algebra automorphism and η : Z m Ñ Z m is an isomorphism of Z-modules. Next, we need to define mutations of quantum seeds. Let pM,Bq be a quantum seed, and fix k P ex. Define M 1 : Z m Ñ Fzt0u by setting where E k is the matrix from (2.3), b k P Z m is the kth column ofB and the t-binomial coefficient is defined by " r p j t :" pt r´t´r qpt r´1´t´r`1 q¨¨¨pt r´p`1´t´r`p´1 q pt p´t´p qpt p´1´t´p`1 q¨¨¨pt´t´1q , @0 ď p ď r. (2.13) Recall the definition of the mutated matrixB 1 " μ k pBq from Section 2.1. Then the mutation in direction k of the quantum seed pM,Bq is the pair μ k pM,Bq " pM 1 ,B 1 q Proposition 2.3.4 [6]. (1) The pair pM 1 ,B 1 q is a quantum seed.
(2) The mutation in direction k of the compatible pair pΛ M ,Bq is the pair pΛ M 1 ,B 1 q.
For a quantum seed pM,Bq, letX " tX 1 , . . . , X m u be the free generating set of F, given by X i :" M pe i q. Let X " tX i | i P exu, we call it the cluster of the quantum seed pM,Bq, and let C "XzX.
For all k P ex, if pM 1 ,B 1 q " μ k pM,Bq, then the X 1 i " M 1 pe i q are obtained by: 14) The mutation of quantum seeds, as the mutation of compatible pairs, is an involutive process: μ k pM 1 ,B 1 q " pM,Bq. Thus, as before, we have an equivalence relation: two quantum seeds pM 1 ,B 1 q and pM 2 ,B 2 q are mutation equivalent if pM 2 ,B 2 q can be obtained from pM 1 ,B 1 q by a sequence of quantum seed mutations. From (2.14), the set C only depends on the mutationequivalence class of the quantum seed. The variables in C, pX i q iRex , are called the frozen variable of the mutation-equivalence class. Definition 2.3.5. Let S be a mutation-equivalence class of quantum seeds in F and C the set of its frozen variables. The quantum cluster algebra ApSq associated with S is the Zrt˘1 {2 s-subalgebra of the skew-field F generated by the union of all clusters in all seeds in S, together with the elements of C and their inverses.

Laurent phenomenon and quantum Laurent phenomenon
One of the main properties of cluster algebras is the so-called Laurent phenomenon which was formulated in [4]. Quantum cluster algebras present a counterpart to this result called the quantum Laurent phenomenon.
Here, we follow [6,Section 5]. In order to state this result, one needs the notion of upper cluster algebras.
Fix pM,Bq a quantum seed, andX " tX 1 , . . . , X m u given by X k " M pe k q. Let ZPrX˘1s denote the based quantum torus generated by the pX k q 1ďkďm ; it is a Zrt˘1 {2 s-subalgebra of F with basis tM pcq | c P Z m u, such that the ground ring ZP is the ring of integer Laurent polynomials in the variables t 1{2 and pX j q jRex . For k P ex, let pM k ,B k q be the quantum seed obtained from pM,Bq by mutation in direction k, and let X k denote its cluster, thus: . The cluster algebra ApSq is contained in U pSq. Equivalently, ApSq is contained in the quantum torus ZPrX˘1s for every quantum seed pM,Bq P S of cluster X.

Specializations of quantum cluster algebras
Fix a quantum seed pM,Bq and X its cluster. Using notations from Section 2.1, the based quantum torus ZPrX˘1s specializes naturally at t " 1, via the ring morphism: If we restrict this morphism to the quantum cluster algebra ApSq, it is not clear that we recover the (classical) cluster algebra ApBq. This question was tackled in a recent paper by Geiss, Leclerc and Schröer [26].
Remark 2.5.1. For a combinatorial point of view, the cluster algebras ApSq and ApBq are constructed on the same quiverB, and the mutations have the same effect on the quiver. Assume the initial seeds are fixed and identified, via the morphism (2.17). Then, each quantum cluster variable in ApSq is identified to a cluster variable in ApBq. Lemma 3.3]. The restriction of π to ApSq is surjective on ApBq, and quantum cluster variables are sent to the corresponding cluster variables.
They also conjectured that the specialization at t " 1 of the quantum cluster algebra is isomorphic to the classical cluster algebra, and gave a proof under some assumptions on the initial seed.
Nevertheless, by applying Proposition 2.5.2 to different seeds (while keeping the identification (2.17) of the initial seeds), one gets Corollary 2.5.3. The evaluation morphism π sends all quantum cluster monomials to the corresponding cluster monomials.

Positivity
Let us state a last general result on quantum cluster algebras: Davison's positivity theorem [15].
We have recalled in Section 2.4 that each (quantum) cluster variable can be written as a Laurent polynomial in the initial (quantum) cluster variables (and t 1{2 ). For classical cluster algebras, Fomin-Zelevinski conjectured that these Laurent polynomials have positive coefficients. The so-called positivity conjecture was proven by Lee-Schiffler in [40].
For quantum cluster algebras, the result is the following.
with a e P Zrt˘1 {2 s. Then a e " t´d egpbeq{2 b e for some b e P Nrts.

Cartan data and quantum Cartan data
We fix here some notations for the rest of the paper.

Root data
Let the g be a simple finite-dimensional Lie algebra of type A, D or E, and let I :" t1, . . . , nu be the indexed set of its Dynkin diagram. Fix simple coroots pα _ i q iPI of g and let pα i q iPI , pω i q iPI be the corresponding sets of simple roots (respectively, fundamental weights). We will use the usual lattices Q " À iPI Zα i , Q`" À iPI Nα i and P " À iPI Zω i . Let P Q " P b Q, endowed with the partial ordering : ω ď ω 1 if and only if ω 1´ω P Q`.
The Dynkin diagram of g is numbered as in [39], and let a 1 , a 2 , . . . , a n be the Kac labels (a 0 " 1).
The Cartan matrix of g is the nˆn matrix C such that C i,j " α j pα _ i q. As g is of simply laced type:

Quantum Cartan matrix
Let z be an indeterminate.
The evaluation Cp1q is the Cartan matrix of g. As detpCq ‰ 0, then detpCpzqq ‰ 0 and detpCpzqq is invertible as a formal Laurent series with degree bounded from below. Thus we can defineCpzq, the inverse of the matrix Cpzq. The entries of the matrix Cpzq belong to Qppzqq.

One can writexc
Cpzq " pz`z´1q Id´A, where A is the adjacency matrix of the Dynkin diagram of g. Hence, Therefore, we can write the entries ofCpzq as power series in z. For all i, j P I, We will need the following lemma: By writingCpzq as a formal power series, and using the definition of Cpzq, we obtain, for all pi, jq P I 2 ,`8 This is equivalent toC

Infinite quiver
Next, let us define an infinite quiver Γ as in [37]. LetΓ be the quiver with vertex set IˆZ and arrows This quiver has two isomorphic connected components (see [37]). Let Γ be one of them, and letÎ be the set of its vertices. Example 3.3.1. For g " sl 4 , fixÎ to bê and Γ is the following:

Category O of representations of quantum loop algebras
We now start with the more representation theoric notions of this paper. We first recall the definitions of the quantum loop algebra and its Borel subalgebra, before introducing the Hernandez-Jimbo category O of representations, as well as some known results on the subject. We will sporadically use concepts and notations from the two previous sections.

Quantum loop algebra and Borel subalgebra
Fix a non-zero complex number q, which is not a root of unity, and h P C such that q " e h . Then for all r P Q, q r :" e rh is well defined. Since q is not a root of unity, for r, s P Q, we have q r " q s if and only if r " s.
We will use the following standard notations: Definition 4.1.1. The quantum loop algebra U q pLgq corresponding to g is the associative C-algebra generated by e i , f i , k˘1 i , 0 ď i ď n, subject to the relations, for 0 ď i, j ď n : Both the quantum loop algebra and its Borel subalgebra are Hopf algebras. From now on, except when explicitly stated otherwise, we are going to consider representations of the Borel algebra U q pbq. Particularly, we consider the action of the -Cartan subalgebra U q pbq 0 : a commutative subalgebra of U q pbq generated by the so-called Drinfeld generators (generators appearing in the Drinfeld presentation [17] of U q pLgq):

Highest -weight modules
Let V be a U q pbq-module and ω P P Q a weight. One defines the weight space of V of weight ω by Definition 4.2.1. A series Ψ Ψ Ψ " pψ i,m q iPI,mě0 of complex numbers, such that ψ i,m P q Q for all i P I is called an -weight. The set of -weights is denoted by P . One identifies the -weight Ψ Ψ Ψ to its generating series : Let us define some particular -weights which are important in our context. For any ω P P Q , define rωs P P by The set P is a group with respect to the component-wise multiplication of formal power series, as the components Ψ i of -weights satisfy ψ i p0q P q Q . Moreover, for all Ψ Ψ Ψ P P , if one writes ψ i p0q " q bi , for all i P I, then the element ω " Let be the surjective group morphism : P Ñ P Q such that ψ i p0q " q pΨ Ψ Ψqpα _ i q , for all Ψ Ψ Ψ P P and all i P I.
Let V be U q pbq-module and Ψ Ψ Ψ P P an -weight. One defines the -weight space of V of -weight Ψ Ψ Ψ by In that case, the -weight Ψ Ψ Ψ is entirely determined by V , it is called the -weight of V , and v is the highest -weight vector of V .
Example 4.2.5. For ω P P Q , Lprωsq is a one-dimensional representation of weight ω. We also denote it by rωs (tensoring by this representation is equivalent to shifting the weights by ω).

Definition of the category O
As explained in the Introduction, our focus here is a category O of representations of the Borel algebra, which was first defined in [32], mimicking the usual definition of the BGG category O for Kac-Moody algebras. Here, we are going to use the definition in [36], which is slightly different.
(2) for all ω P P Q , one has dimpV ω q ă 8; (3) there is a finite number of λ 1 , . . . , λ s P P Q such that all the weights that appear in V are in the cone The category O is a monoidal category. Let P r be the set of -weights Ψ such that, for all i P I, Ψ i pzq is rational. We will use the following result.  Throughout this paper, we will use results already known for finite-dimensional representations of the quantum loop algebra U q pLgq with the purpose of generalizing some of them to the context of the category O of representations of the Borel subalgebra U q pbq. Let us first see why this approach is valid, as the representation theories of these two algebras are similar. Let C be the category of all (type 1) finite-dimensional U q pLgq-modules.
Using this result and the classification of finite-dimensional simple module of quantum loop algebras in [12], one has Then LpΨ Ψ Ψq is finite-dimensional. Moreover the action of U q pbq can be uniquely extended to an action of U q pLgq, and any simple object in the category C is of this form.
Hence, the category C is a subcategory of the category O and the inclusion functor preserves simple objects. In general, simple modules in C are indexed by monomials in the variables pY i,a q iPI,aPCˆ, called dominant monomials. Frenkel-Reshetikhin [25] defined a q-character morphism χ q (see Section 4.8) on the Grothendieck ring of C . It is an injective ring morphism . These categories are interesting to study for different reasons; here we use in particular the cluster algebra structure of their Grothendieck rings. Definition 4.5.1. Consider the submonoids P˘ of P r generated by the rωs, ω P P Q , the Y i,a and the Ψ Ψ Ψ˘1 i,a , i P I, a P Cˆ. An -weight of P` (respectively, P´ q is said to be positive (respectively, negative).
Definition 4.5.2. The category O˘is the full subcategory of O whose objects are modules whose simple constituents are LpΨ Ψ Ψq with Ψ Ψ Ψ P P˘ .

The category OZ
Recall the infinite quiver Γ from Section 3.3 and its set of verticesÎ.
In [33, Section 3.7], Hernandez and Leclerc defined a subcategory C Z of the category C . Let Y Y Y Z be the submonoid of P generated by the Y i,q r`1 for pi, rq PÎ. Then C Z is the full subcategory of C whose objects have simple constituents LpΨ Ψ Ψq such that This subcategory is interesting to study because each simple object in C can be written as a tensor product of simple objects which are essentially in C Z (see [33,Section 3.7]). Thus, the study of simple modules in C is equivalent to the study of simple modules in C Z .
Consider the same type of restriction on the category O. Let P˘ ,Z be the submonoids of P generated by the Ψ Ψ Ψ˘1 i,r for pi, rq PÎ, respectively. Let us note that our definition of C Z is slightly different from that of [33]. As the Y i,a are expressed in terms of Ψ Ψ Ψ˘1 i,aq˘1 , the definition of C Z is shifted for it to be a subcategory of OZ .
From now on, we will consider the Ψ Ψ Ψ i,q r and the Y i,q r`1 , for pi, rq PÎ.

The Grothendieck ring K 0 pOq
Hernandez and Leclerc showed that the Grothendieck rings of the categories OZ have some interesting cluster algebra structures.
First of all, define E as the additive group of maps c : P Q Ñ Z whose support is contained in a finite union of sets of the form Dpμq. For any ω P P Q , define rωs P E as the δ-function at ω (this is compatible with the notation in Example 4.2.5). The elements of E can be written as formal sums c " ÿ ωPsupppcq cpωqrωs. (4.10) E can be endowed with a ring structure, where the product is defined by rωs¨rω 1 s " rω`ω 1 s, @ω, ω 1 P P Q .
This product is well defined because the supports of the maps c in E are contained in a finite union of cones. If pc k q kPN is a countable family of elements of E such that for any ω P P Q , c k pωq " 0 except for finitely many k P N, then ř kPN c k is a well-defined map from P Q to Z. In that case, we say that ř kPN c k is a countable sum of elements in E. The Grothendieck ring of the category O can be viewed as a ring extension of E. Similar to the case of representations of a simple Lie algebra (see [39,Section 9.6]), every object in this category O has a (generalized) Jordan-Hölder series, thus the multiplicity of an irreducible representation in a given representation of the category O is well defined. The Grothendieck ring of the category O is formed of formal sums ÿ Ψ Ψ ΨPP r λ Ψ Ψ Ψ rLpΨ Ψ Ψqs, (4.11) such that the λ Ψ Ψ Ψ P Z satisfy ÿ In this context, E is identified with the Grothendieck ring of the category of representations of O with constant -weight.
A notion of countable sum of elements in K 0 pOq is defined exactly as for E. Now consider the cluster algebra ApΓq defined by the infinite quiver Γ of Section 3.3, with infinite set of coordinates denoted by . In particular, the map χ is defined on ApΓq b Z E, and for each A P ApΓq b Z E, one can write χpAq " ř ωPP Q A ω rωs P E. Define |χ| by |χ|pAq " ř ωPP Q |A ω |rωs. Consider the completed tensor product of countable sums ř kPN A k of elements A k P ApΓq b Z E, such that ř kPN |χ|pA k q is a countable sum of elements of E, as defined above.

The q-character morphism
Here we detail the notion of q-character on the category O. This notion extends the q-character morphism on the category of finite-dimensional U q pLgq-modules mentioned in Section 4.4.
Similar to Section 4.7, consider E , the additive group of maps c : P r Ñ Z such that the image by of its support is contained in a finite union of sets of the form Dpμq, and for any ω P P Q , the set supppcq X ´1 ptωuq is finite. The map extends naturally to a surjective morphism : E Ñ E. For Ψ Ψ Ψ P P r , define the delta function rΨ Ψ Ψs " δ Ψ Ψ Ψ P E .
The elements of E can be written as formal sums Endow E with a ring structure given by In particular, for Ψ Ψ Ψ, Ψ Ψ Ψ 1 P P r , rΨ Ψ Ψs¨rΨ Ψ Ψ 1 s " rΨ Ψ ΨΨ Ψ Ψ 1 s. (4.20) As in Section 4.7, this multiplication is well defined thanks to the support condition on E .
For V a module in the category O, define the q-character of V as in [25,32]: By definition of the category O, χ q pV q is an object of the ring E . The following result extends the one from [25] to the context of the category O.
Example 4.8.2. For any a P Cˆ, i P I, one has [22,32], where χ i " χpLì ,a q P E does not depend on a. For example, if g " sl 2 ,

Quantum tori
Let t be an indeterminate. The aim of this section is to build a non-commutative quantum torus T t which will contain the quantum Grothendieck ring for the category O. For the category C of finite-dimensional U q pLgq-modules, such a quantum torus already exists, denoted by Y t here. Thus one natural condition on T t is for it to contain Y t . We show it is the case in Theorem 5.2.1. We start this section by recalling the definition and some properties of Y t . Here we use the same quantum torus as in [31], which is slightly different from the one used in [42,45].

The torus Y t
In this section, we consider U q pLgq-modules and no longer U q pbq-modules. We have seen in Section 4.4 that for finite-dimensional representations, these settings were not too different.
As seen in (4.8), the Grothendieck ring of C can be seen as a subring of a ring of Laurent polynomials K 0 pC q ĎŶ " ZrY i,a | pi, rq PÎs.
In order to define a t-deformed non-commutative version of this Grothendieck ring, one first needs a non-commutative, t-deformed version ofŶ, denoted by Y t .
Following [31], we define Y :" ZrY˘1 i,q r | pi, rq PÎs, (5.1) the Laurent polynomial ring generated by the commuting variables Y i,q r . Let pY t ,˚q be the Zpt 1{2 q-algebra generated by the pYȋ ,q r q pi,rqPÎ , with the t-commutations relations: where N i,j : Z Ñ Z is the antisymmetrical map, defined by using the notations from Section 3.2.
Example 5.1.1. If we continue Example 3.2.3, for g " sl 2 , in this case,Î " p1, 2Zq, for r P Z, one has Y 1,2r˚Y1,2s " t 2p´1q s´r Y 1,2s˚Y1,2r , @s ą r ą 0. The Zpt 1{2 q-algebra Y t is viewed as a quantum torus of infinite rank. Consider the bar-involution , the antiautomorphism of Y t defined by: where on the right-hand side an order onÎ is chosen so as to give meaning to the sum, and the product˚is ordered by it (note that the result does not depend on the order chosen). The bar-invariant monomials form a basis of the free Zpt 1{2 q-module Y t .

The torus T t
We now want to extend the quantum torus Y t to a larger non-commutative algebra T t which would contain at least all the -weights, and possibly all the candidates for the pq, tq-characters of the modules in the category OZ .
In particular, T t contains the Ψ Ψ Ψ i,q r , for pi, rq PÎ, and these t-commutes with a relation compatible with the t-commutation relation between the Y i,q r`1 (5.2).
We start as in Section 5.1. First of all, define the Laurent polynomial ring generated by the commuting variables z i,r . Then, build a t-deformation T t of T , as the Zrt˘1s-algebra generated by the zȋ ,r , for pi, rq PÎ, with a non-commutative product˚, and the t-commutations relations z i,r˚zj,s " t Fij ps´rq z j,s˚zi,r ,´pi, rq, pj, sq PÎ¯, (5.8) where, for all i, j P I, F ij : Z Ñ Z is an antisymmetrical map such that, for all m ě 0, (5.9) Now, let This based quantum torus will be enough to define a structure of quantum cluster algebra, but for it to contain the quantum Grothendieck ring of the category OZ , one needs to extend it. In order to do that, we draw inspiration from Section 4.7. Recall the definition of χ from (4.13). We extend it to the E-algebra morphism χ : T t b Z E Ñ E defined by imposing χpt˘1 {2 q " 1, as well as As before, for z P T t b Z E, one writes χpzq " ř ωPP Q z ω rωs and |χ|pzq " ř ωPP Q |z ω |rωs. Define the completed tensor product of countable sums ř kPN z k of elements z k P T t b Z E, such that ř kPN |χ|pz k q is a countable sum of E, as in Section 4.7.
The bar involution defined on Y t has a counterpart on the larger quantum torus T t . There is unique E-algebra antiautomorphism of T t such that Consistently with the identification (4.15), and the character of the z˘1 i,r , we use the following notation, for pi, rq PÎ, where the products on the right-hand side are bar-invariant, defines an injective homomorphism J : Y t Ñ T t of Zptq-algebras.
Proof. One needs to check that the images of the Y i,q r`1 satisfy (5.2). Thus, we need to show that, for all pi, rq, pj, sq PÎ, Thus 2F i,j pmq´F i,j pm`2q´F i,j pm´2q " N i,j pmq, using (5.3).
If s " r`1, the left-hand side of (5.15) is equal to  (4.15) is between the element z i,r rrω i {2s and the class of the prefundamental representation rLì ,q r s. But this identification is not compatible with the character χ defined in (4.13), as the character of Lì ,q r is χ i , as in (4.23). Here, we choose to identify the variables z i,r with the highest -weights of the prefundamental representations (up to a shift of weight), in particular, this identification is compatible with the character morphism χ.

Quantum Grothendieck rings
The aim of this section is to build K t pOZ q, a t-deformed version of the Grothendieck ring of the category OZ . This ring will be built inside the quantum torus T t , as a quantum cluster algebra.
Let us summarize the existing objects in this context in a diagram: A natural idea to build a t-deformation of the Grothendieck ring K 0 pOZ q is to use its cluster algebra structure and define a t-deformed quantum cluster algebra, as in Section 2.3, with the same basis quiver. One has to make sure that the resulting object is indeed a subalgebra of the quantum torus T t .

The finite-dimensional case
We start this section with some reminders regarding the quantum Grothendieck ring of the category of finite-dimensional U q pLgq-modules.
This object was first discussed by Nakajima [42] and Varagnolo-Vasserot [45] in the study of perverse sheaves. Then Hernandez gave a more algebraic definition, using t-analogs of screening operators [30,31]. This is the version we consider here, with the restriction to some specific tensor subcategory C Z , as in [35]. 6.1.1. Definition of the quantum Grothendieck ring. As in Section 4.6, consider C Z the full subcategory of C whose simple components have highest -weights which are monomials in the Y i,q r , with pi, rq PÎ.
For pi, r´1q PÎ, define the bar-invariant monomials Finally, as in [31], define Remark 6.1.1. Frenkel-Mukhin's algorithm [23] allows for the computation of certain q-characters, in particular those of the fundamental representations. In [31], Hernandez introduced a t-deformed version of this algorithm to compute the pq, tq-characters of the fundamental representations, and thus to characterized the quantum Grothendieck ring as the subring of Y t generated for those pq, tq-characters: Let us recall some more detailed results about the theory of pq, tq-characters for the modules in the category C Z .
Let M be the set of monomials in the variables pY i,q r`1 q pi,rqPÎ , also called dominant monomials. From [31] we know that for all dominant monomial m, there is a unique element F t pmq in K t pC Z q such that m occurs in F t pmq with multiplicity 1, and no other dominant monomial occurs in F t pmq. These F t pmq form a Cpt 1{2 q-basis of K t pC Z q.
For all dominant monomial m " where αpmq P 1 2 Z is fixed such that m appears with coefficient 1 in the expansion of rM pmqs t on the basis of the bar-invariant monomials. The specialization at t " 1 of rM pmqs t recovers the q-character χ q pM pmqq of the standard module M pmq.
Remark 6.1.2. By definition, the subring K t pC Z q is invariant under the bar-involution defined in Section 5.
Using Lusztig's lemma [41, 7.10] and the triangularity of the basis prM pmqs t q mPM with respect to the bar-involution, there is a unique family trLpmqs t P K t pC Z q | m P Mu such that (i) rLpmqs t " rLpmqs t , (6.6) (ii) rLpmqs t P rM pmqs t`ÿ m 1 ăm t´1Zrt´1srM pm 1 qs t , (6.7) where m 1 ď m means that mpm 1 q´1 is a product of A i,r (see [42]).
Lastly, we recall this result from Nakajima, proven using the geometry of quiver varieties.
Theorem 6.1.3 [42]. For all dominant monomial m P M, the specialization at t " 1 of rLpmqs t is equal to χ q pLpmqq.
Moreover, the coefficients of the expansion of rLpmqs t as a linear combination of products of Y˘1 i,r belong to Nrt˘1s.
Thus to all simple modules LpΨ Ψ Ψq in C Z is associated an object rLpmqs t P K t pC Z q, called the pq, tq-character. It is compatible with the q-character of the representation.
Remark 6.1.4. With the cluster algebra approach, we shed a new light on this last positivity result. We interpret the pq, tq-characters of the fundamental modules (and actually all simple modules which are realized as cluster variables in K 0 pOZ q) as quantum cluster variables (Conjecture 7.2.6). Thus using Theorem 2.6.1, we recover the fact that the coefficients of their expansion on the commutative monomials in the pY˘1 i,r q belong to Nrt˘1s.
Remark 6.1.5. In order to fully extended this picture to the context of the category O, and implement a Kazhdan-Lusztig type algorithm to compute the pq, tq-characters of all simple modules, one would need an equivalent of the standard modules in this category. These do not exist in general. This question was tackled by the author in another paper [7], in which equivalent of standard modules where defined when g " sl 2 .

Compatible pairs
We now begin the construction of K t pOZ q.
First of all, to define a quantum cluster algebra, one needs a compatible pair, as in Section 2.2. The basis quiver we consider here is the same quiver Γ as before (see Section 3.3).
Explicitly, the corresponding exchange matrix is theÎˆÎ skew-symmetric matrixB such that, for all ppi, rq, pj, sqq PÎ 2 , or i " j and s " r´1, 1 i f i " j and s " r´2 or i " j and s " r`1, 0 otherwise.
(6.9) Remark 6.2.1. In [36], it is noted that one can use sufficiently large finite subseed of Γ instead of an infinite rank cluster algebra. For our purpose, the same statement stays true, but one has to check that the subquiver still forms a compatible pair with the torus structure. Hence, we have to give a more precise framework for the restriction to finite subseeds.
For all N P Z ą0 , define Γ N , which is a finite slice of Γ of length 2N`1, containing an upper and lower row of frozen vertices. More precisely, defineÎ N andĨ N aŝ ) , (6.10) . (6.11) Then Γ N is the subquiver of Γ with set of verticesĨ N , where the vertices isĨ N zÎ N are frozen (thus the vertices inÎ N are the exchangeable vertices). This way, all cluster variables of ApΓq obtained from the initial seed after a finite sequence of mutations are cluster variables of the finite rank cluster algebra ApΓ N q, for N large enough. With the same index restrict onB, we will be able to define a size increasing family of finite rank quantum cluster algebras. Example 6.2.2. Recall from Example 3.3.1 the infinite quiver Γ when g " sl 4 . Then the quiver Γ N is the following: where the boxed vertices are frozen.
For N P Z ą0 , letB N be the corresponding exchange matrix. It is theĨ NˆÎN submatrix of B, thus its coefficients are as in (6.8).
For all N P Z ą0 , let Λ N be theĨ NˆĨN submatrix of Λ. It is a finite pnp2N`1qq 2 skewsymmetric matrix, where n is the rank of the simple Lie algebra g.
Example 6.2.3. For g of type D 4 , let us exhibit a finite slice of Γ of length 4, containing an upper and lower row of frozen vertices (which is thus not Γ 1 , of length 3, nor Γ 2 , of length 5): If the setĨ " tpi, rq PÎ | i P 1, 4 ,´5 ď r ď 2u is ordered lexicographically by r then i (reading order), the quiver is represented by the following exchange matrix: The principal part B ofB is the square submatrix obtained by omitting the first four rows and the last four rows. One notes that B is skew-symmetric. Moreover, using Formula (6.9), one can compute the corresponding matrix Λ. We get the following 16ˆ16 skew-symmetric matrix (with the same order ofĨ as before): (6.13) From here, it is easy to check that the productB T Λ is of the form: (6.14) Thus, pΛ,Bq is a compatible pair.
Remark 6.2.4. Note the coefficients appearing of the diagonal here are all negative. From the definition of compatible pair from [6], for pΛ,Bq to be a compatible pair, these should be positive. That is why in Definition 2.2.1, the coefficients are of constant sign (the properties of quantum cluster algebras are still satisfied by taking negative coefficients in the compatibility condition). This allows for our t-commutation to be coherent with that of the quantum torus Y t .
We show that this result is true in general. Furthermore, the specific form we obtain in Equation (6.14) is what we get in general.
Proposition 6.2.5. We have In particular, pΛ,Bq and ppΛ N ,B N qq N ą0 are compatible pairs, in the sense of structure condition for quantum cluster algebras.
As in Definition 2.2.1, up to reordering of the variables (placing the mutable variable first), the resulting matrix is of the form rD 0s, with D diagonal with sign coherent entries.
This is a finite sum, as each vertex in Γ is adjacent to a finite number of other vertices. Suppose first that r ‰ s. Without loss of generality, we can assume that r ă s. Then, using the definition of the matrix Λ in (6.9) and the coefficients ofB in (6.8), we obtaińB As pi, rq PÎ N is not a frozen variable, the pj, sq PÎ such that b pk,uq,pi,rq ‰ 0 are all inĨ N . Hence the rest of the reasoning is still valid, and the result follows. l

Definition of K t pOZ q
Everything is now in place to define K t pOZ q. Recall the based quantum torus T t , defined in Section 5.2. By construction, the associated skew-symmetric bilinear form Λ identifies with the infinite skew-symmetricÎˆÎ-matrix from the previous section: Λpe pi,rq , e pj,sq q " Λ pi,rq,pj,sq " F ij ps´rq, pi, rq, pj, sq PÎ, (6.20) where pe pi,rq q pi,rqPÎ is the standard basis of Z pÎq .
Let F be the skew-field of fractions of T t . We define the toric frame M : Z pÎq Ñ Fzt0u by setting M pe pi,rq q " z i,r P F, @pi, rq PÎ. From the result of Proposition 6.2.5, S " pM,Bq is an infinite rank quantum seed, and one would want to define our quantum Grothendieck ring as the corresponding quantum cluster algebra. However, the definition of quantum cluster algebra recalled in Section 2 does not typically cover infinite rank quantum cluster algebras. Nonetheless, in [29], Grabowski-Gratz gave a construction of certain infinite rank quantum cluster algebra as (co)limits of sequences of finite rank quantum cluster algebras. We give here an explicit construction of this result.
Fix N P Z ą0 . Let m " p2N`1qˆn, where n is the rank of the simple Lie algebra g. Consider L N , the sublattice of T t generated by the z i,r , with pi, rq PĨ N (recall the definition ofĨ N in (6.11)). L N is of rank m. Consider the toric frame M N which is the restriction of M to L N . In that case, from the previous section.
Thus, from the result of Proposition 6.2.5 is a quantum seed.
Definition 6.3.2. We define A t pΓ N q to be the quantum cluster algebra associated to the mutation-equivalence class of the quantum seed S N .
Then, let us define A t pΓq to be the quantum cluster algebra associated to the mutationequivalence class of the infinite rank quantum seed S. As the mutation sequences are finite, one can always assume we are working in the quantum cluster algebra A t pΓ N q, with N large enough.
Definition 6.3.3. Define K t pOZ q :" A t pΓqbE, (6.24) where the tensor product is completed as in (5.11). The ring K t pOZ q is a Ert˘1 {2 s-subalgebra of T t . For N P Z ą0 , with the same completion of the tensor product, define Properties of K t pOZ q

The bar involution
Recall the bar-involution maps defined on Y t and T t in Section 5. We have seen in Section 6.1.2 that the pq, tq-character of simple modules in C Z are bar-invariant by definition. Thus it is natural for pq, tq-characters of simple modules in OZ to also be bar-invariant.
What is crucial to note here is that the definition of the bar-involution on T t is compatible with the bar-involution defined in general on the quantum torus of any quantum cluster algebra (see [6,Section 6]). However, this latter bar-involution has an important property: all cluster variables are invariant under the bar involution.

Inclusion of quantum Grothendieck rings
As stated earlier, one natural property we would want to be satisfied by the quantum Grothendieck ring K t pOZ q is to include the already-existing quantum Grothendieck K t pC Z q of the category C Z .
Note that those rings are contained in quantum tori, which are included in one another by the injective morphism J from Theorem 5.2.1: Thus it is natural to formulate the following conjecture: Conjecture 7.2.1. The injective morphism J restricts to an inclusion of the quantum Grothendieck rings Recall that the quantum Grothendieck ring K t pC Z q is generated by the classes of the fundamental representations rLpY i,q r`1 qs t , for pi, rq PÎ (see Section 6.1.1). Hence, in order to prove Conjecture 7.2.1, it is enough to show that the images of these rLpY i,q r`1 qs t belong to K t pOZ q.
In Example 4.7.2 we saw how, when the g " sl 2 , the class of the fundamental representation rLpY 1,q´1 qs could be obtained as a cluster variable in ApΓq after one mutation in direction (1,0). This fact is actually true in more generality, as seen in [36], in the proof of Proposition 6.1. Let us recall this process precisely.
Fix pi, rq PÎ. We first define a specific sequence of vertices in Γ, as in [37]. Recall the definition of the dual Coxeter number h _ .
Note also that z p1q 2,3 was already in the image of J and that z p1q 2,3 " J pχ q pLpY 2,q 2 qqq.
Thus, for each pi, rq PÎ, consider the quantum cluster variablesχ i,r P K t pOZ q obtained from the initial quantum seed pz z z, Λq via the sequence of mutations S.
Example 7.2.5. Suppose g " sl 2 . Consider the quiver Γ 1 a well as the skew-symmetric matrix Λ 1 , As seen in Example 4.7.2 (with a shift of quantum parameters), the fundamental representation rLpY 1,q´1 qs is obtained in K 0 pOZ q after one mutation at (1,0) (here S " p1, 0q).
In particular, Conjecture 7.2.1 is satisfied in this case.
This example incites us to formulate another conjecture.
Conjecture 7.2.6. For all pi, rq PÎ, the quantum cluster variableχ i,r recovers, via the morphism J , the pq, tq-character of the fundamental representation LpY i,q r`1 q: The author plans to prove this conjecture for all simply laced types in a follow-up work. (iI) the coefficients of its expansion as a Laurent polynomial in the initial quantum cluster variables tz i,r u are Laurent polynomials in t 1{2 with non-negative integers coefficients: with z z z u u u " ś pi,rqPÎ z ui,r i,r denoting the bar-invariant monomial; (iii) its evaluation at t " 1 (as seen in (5.18)), recovers the q-character of the fundamental representation LpY i,q r`1 q: Proof. The first property is a direct consequence of Proposition 7.1.1 and the second is a direct consequence of the positivity result of Theorem 2.6.1.
For the third property, note we have used two evaluation maps so far, with the same notation.
• The evaluation map defined in (2.17) on the bases quantum torus of a quantum cluster algebra: π : A t pM,Bq Ñ ZrX˘1s, • The evaluation map defined in (5.18) on T t : These notations are coherent because the map π from (5.18) is the evaluation map defined on a based quantum torus (of infinite rank) of a quantum cluster algebra, extended to a Emorphism on T t . In this case, the Laurent polynomial ring ZrX˘1s is Zrz˘1 i,r | pi, rq PÎs, which becomes ErΨ Ψ Ψ˘1 i,r s after extension to a E-morphism and via the identification (4.15).
Thus we can apply Corollary 2.5.3 to this map π. Asχ i,r is a quantum cluster variable, its evaluation by π is the cluster variable χ i,r , which is obtained from the initial seed z, via the same sequence of mutations S (the initial seed and quantum seeds are fixed and identified by the evaluation π on the quantum torus T t ). By Proposition 7.2.3, These two properties imply that theχ i,r are good candidates for the pq, tq-characters of the fundamental representations, as stated in Conjecture 7.2.1.

pq, tq-Characters for positive prefundamental representations
Recall the q-characters of the positive prefundamental representations in (4.23), for all i P I, a P Cˆ, where χ i P E is the (classical) character of Lì ,a . Definition 7.3.1. For pi, rq PÎ, define rLì ,q r s t :" rΨ Ψ Ψ i,q r s b χ i P K t pOZ q, (7.13) using the notation from (5.13).
Remark 7.3.2. It is the quantum cluster variable obtained from the initial quantum seed, via the same sequence of mutations used to obtain rLì ,q r s in K 0 pOZ q, which in this case, is no mutation at all.
In particular, the evaluation of rLì ,q r s t recovers the q-character of rLì ,q r s: πprLì ,q r s t q " rΨ Ψ Ψ i,q r s b χ i " χ q pLì ,a q P E . (7.14)

Results in type A
Throughout this section we suppose that g is of type A.

Proof of the conjectures
In this case, the situation of Example 7.2.5 generalizes. In this case, the key ingredient is the following well-known result (see, for example, [24,Section 11], and references therein). Proof. Fix pi, rq PÎ. From the second property of Proposition 7.2.8, we know thatχ i,r can be written asχ i,r " ÿ u u uPZ pÎq P u u u pt 1{2 qz z z u u u , (8.1) where the P u u u pt 1{2 q are Laurent polynomials with non-negative integer coefficients. Using the third property of Proposition 7.2.8, we deduce the evaluation at t " 1 of equality (8.1): From the above theorem, this decomposition is multiplicity-free. Thus, the non-zero coefficients P u u u pt 1{2 q are of the form t k{2 , with k P Z. Finally, as χ i,r is bar-invariant, from the first property of Proposition 7.2.8, and the z z z u u u are also bar-invariant, we know that the Laurent polynomials P u u u pt 1{2 q are even functions: P u u u p´t 1{2 q " P u u u pt 1{2 q. Thus the variable t 1{2 does not explicitly appear in the decomposition (8.1), and so: ÿ u u uPZ pÎq P u u u p1qz z z u u u , " J`χ q pLpY i,q r´1 qq˘.
Moreover, with the same arguments, as rLpY i,q r´1 qs t is bar-invariant by definition, rLpY i,q r´1 qs t " χ q pLpY i,q r´1 qq, (8.4) written in the basis of the bar-invariant monomials.
Hence we recover the fact that the quantum cluster variable χ i,r is equal, via the inclusion map J , to the pq, tq-character of LpY i,q r´1 q and Conjecture 7.2.6 is satisfied. l

A remarkable subalgebra in type A 1
When g " sl 2 , we can make explicit computations. Retain the notation of Example 5.2.2. For all r P Z, the pq, tq-character of the prefundamental representation L1 ,q 2r defined in (7.13) is rL1 ,q 2r s t " rΨ Ψ Ψ 1q 2r sχ 1 .
Remark 8.2.2. If we identify the variables Y 1,q 2r and their images through the injection J , this relation is actually the exchange relation related to the quantum mutation in Example 7.2.5 (for a generic quantum parameter q 2r ). Now consider the quantum cluster algebra ApΛ 1 , Γ 1 q, with notations from Section 6.2 (Λ 1 and Γ 1 are given explicitly in Example 7.2.5).
It is a quantum cluster algebra of finite type (if we remove the frozen vertices from the quiver, we get just one vertex, which is a quiver of type A 1 ). It has two quantum clusters, containing the two frozen variables z 1,2 , z 1,´2 and the mutable variables z 1,0 and z p1q 1,0 , respectively. Thus, it is generated as a Cpt 1{2 q-algebra by E :" rLpY 1,q´1 qs t p" z p1q 1,0 q, F :" rL1 ,1 s t p" z 1,0 q, K :" rω 1 srL1 ,q´2 s t p" z 1,´2 q, K 1 :" r´ω 1 srL1 ,q 2 s t p" z 1,2 q.

(8.6)
This algebra is a quotient of a well-known Cpt 1{2 q-algebra. Let q be a formal parameter. The quantum group U q psl 2 q can be seen as the quotient where D 2 is the Cpqq-algebra with generators E, F, K, K 1 and relations: KE " q 2 EK, K 1 E " q´2EK 1 KF " q´2F K, K 1 F " q 2 F K 1 KK 1 " K 1 K, and rE, F s " pq´q´1qpK´K 1 q. But both presentations are equivalent, given the change of variables E " pq´q´1qe, F " pq´q´1qf.
The presentation (8.8) also appeared in the work of Bridgeland [9], as it is more natural from the point of view of Hall algebras than the usual presentation.
• D 2 is the Drinfeld double [16] of the Borel subalgebra of U q psl 2 q (the subalgebra generated by K, E), see also [5, equation (1.2)].
We check that the other relations in (8.8) are also satisfied using the structure of the quantum torus T t (which is given explicitly in Example 5.2.2). Hence the map sending generators to generators is well defined and descends onto the quotient Moreover, from [10], the cluster monomials in a given cluster in a cluster algebra are linearly independent. In this case, the quantum cluster algebra ApΛ 1 , Γ 1 q is of type A 1 (without frozen variables), thus of finite-type. It has two (quantum) clusters : pE, K, K 1 q and pF, K, K 1 q. Thus, the set of bar-invariant quantum cluster monomials forms a Cpt 1{2 q-basis of ApΛ 1 , Γ 1 q. Consider the PBW basis of D 2 : From the expression of the Casimir element C´t1{2 (8.10), we deduce a Cpt 1{2 q-basis of D 2 {C´t1{2 , of the same form as (8.12): Hence, the map θ sends a basis to a basis, thus it is isomorphic. l Remark 8.2.5. The basis obtained in (8.14) is related to the double canonical basis of U q psl 2 q introduced by Berenstein-Greenstein [5] . Their work also adopts the presentation (8.8), moreover it uses crucially the quantum Heisenberg algebra, which is also related to Bridgeland's Hall algebra (see [28,Remark 4.8]).
This result should be compared with the recent work of Schrader and Shapiro [44], in which they recognize the same structure of D 2 in an algebra built on a quiver, with some quantum X -cluster algebra structure. In their work, they generalized this result in type A (Theorem 4.4).
Ultimately, they obtain an embedding of the whole quantum group U q psl n q into a quantum cluster algebra. The result of Proposition 8.2.4, together with their results, gives hope that one could find a realization of the quantum group U q pgq as a quantum cluster algebra, related to the representation theory of U q pLgq.
Furthermore, define in this case O1 , the subcategory of OZ of objects whose image in the Grothendieck ring K 0 pOZ q belongs to the subring generated by rL1 ,q´2 s, rL1 ,1 s, rL1 ,q 2 s and rLpY 1,q´1 qs. Then O1 is a monoidal category.
From the classification of simple modules when g " sl 2 in [36, Section 7], we know that the only prime simple modules in O1 are L1 ,q´2 , L1 ,1 , L1 ,q 2 , LpY 1,q´1 q. (8.15) Moreover, a tensor product of those modules is simple if and only if it does not contain both a factor L1 ,1 and a factor LpY 1,q´1 q (the others are in so-called pairwise general position). Thus, in this situation, the simple modules are in bijection with the cluster monomials: " simple modules in O1 * ÐÑ " bar-invariant quantum cluster monomials in ApΛ 1 ,