A generalized quantum cluster algebra of Kronecker type

The cluster multiplication formulas for a generalized quantum cluster algebra of Kronecker type are explicitly given. Furthermore, a positive bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-basis of this algebra is constructed.


Introduction
Cluster algebras were introduced by Fomin and Zelevinsky [13,14] in order to set up an algebraic framework for studying the total positivity and Lusztig's canonical bases.Quantum cluster algebras, as the quantum deformations of cluster algebras, were later introduced by Berenstein and Zelevinsky [4] for studying the dual canonical bases in coordinate rings and their q-deformations.An important feature of (quantum) cluster algebras is the so-called Laurent phenomenon which says that all cluster variables belong to an intersection of certain (may be infinitely many) rings of Laurent polynomials.
Generalized cluster algebras were introduced by Chekhov and Shapiro [9] in order to understand the Teichmüller theory of hyperbolic orbifold surfaces.The exchange relations for cluster variables of generalized cluster algebras are polynomial exchange relations while the exchange relations for cluster algebras are binomial relations.Generalized cluster algebras also possess the Laurent phenomenon [9] and are studied by many people in a similar way as cluster algebras (see for example [18,6,7,17,2]).As a natural generalization of both quantum cluster algebras and generalized cluster algebras, we defined the generalized quantum cluster algebras [1].Not surprised that the Laurent phenomenon also holds in these algebras [3].
One of the most important problems in cluster theory is to construct cluster multiplication formulas.For acyclic cluster algebras, Sherman and Zelevinsky [22] firstly established the cluster multiplication formulas in rank 2 cluster algebras of finite and affine types.Cerulli [8] generalized this result to rank 3 cluster algebra of affine type A (1) 2 .Caldero and Keller [5] constructed the cluster multiplication formulas between two cluster characters for simply laced Dynkin quivers, which was generalized to affine types by Hubery in [15] and to acyclic types by Xiao and Xu in [23,24].In the quantum case, Ding and Xu [11] firstly gave the cluster multiplication formulas of the quantum cluster algebra of Kronecker type.Recently, Chen, Ding and Zhang [10] obtained the cluster multiplication formulas in the acyclic quantum cluster algebras with arbitrary coefficients through some quotients of derived Hall algebras of acyclic valued quivers.One of the most powerful tools in cluster theory is the cluster multiplication formulas which are very useful to construct bases of Key words and phrases.generalized quantum cluster algebra, cluster multiplication formula, positive basis.
(quantum) cluster algebras with nice properties (see for example [22,5,8,11,12]).In cluster theory, a basis is called positive if its structure constants are positive.Several positive bases such as the atomic bases and the triangular bases of some (quantum) cluster algebras have been found (see [20,16]).So far, no similar results have been obtained in generalized quantum cluster algebras.It becomes natural to think whether one can give an explicit treatment of the above mentioned problems for generalized quantum cluster algebras.
In this paper, we study a generalized quantum cluster algebra of Kronecker type denoted by A q (2, 2).We recall the definition of generalized quantum cluster algebras in Section 2, provide the cluster multiplication formulas of A q (2, 2) in Section 3, and explicitly construct a positive bar-invariant Z[q ± 1 2 ]-basis of A q (2, 2) in Section 4.

Preliminaries
In this section, we mainly review the definition of generalized quantum cluster algebras [1].Throughout this section, m and n are positive integers with m ≥ n.Let B = (b ij ) be an m × n integer matrix whose upper n × n submatrix is denoted by B and Λ = (λ ij ) a skew-symmetric m × m integer matrix.
for some positive integers d j (1 ≤ j ≤ n).
Note that the skew-symmetric matrix Λ gives a skew-symmetric bilinear form on Z m defined by Λ(a, b) = a T Λb for any column vectors a, b ∈ Z m .
Let q be a formal variable and Z[q 2 ) the ring of integer Laurent polynomials in q 1 2 .One can associate to (Λ, q) a quantum torus algebra as follows.
Definition 2.2.The quantum torus T over Z[q ± 1  2 ] is generated by the symbols {X(a) | a ∈ Z m } subject to the multiplication relations for any a, b ∈ Z m .
The skew-field of fractions of T is denoted by F. On the quantum torus T , the Z-linear bar-involution is defined by setting q r 2 X(a) = q − r 2 X(a) for any r ∈ Z and a ∈ Z m .Let e k be the k-th standard unit vector in Z m and set X k = X(e k ) for 1 ≤ k ≤ m.An easy computation shows that For any 1 ≤ i ≤ n, we say that B ′ = (b ′ kl ) is obtained from the matrix B = (b kl ) by the matrix mutation in the direction i if For any 1 ≤ i ≤ n and a sign ε ∈ {±1}, denote by E ε the m × m matrix associated to the matrix B = (b ij ) with entries as follows is also compatible and independent of the choice of ε, where We say that the compatible pair (Λ ′ , B ′ ) is obtained from the compatible pair (Λ, B) by mutation in the direction i and denoted by For each 1 ≤ i ≤ n, let d i be a positive integer such that b li d i are integers for all 1 ≤ l ≤ m and denote by , where b i is the i-th column of B. Denote by where h k,l (q 2 ) and h k,0 (q Definition 2.4.With the above notations, the quadruple (X, h, Λ, B) is called a quantum seed if the pair (Λ, B) be compatible.For a given quantum seed (X, h, Λ, B) and each is defined by ) We say that the quadruple µ i (X, h, Λ, B) is obtained from (X, h, Λ, B) by mutation in the direction i.
Note that µ i is an involution by [1,Proposition 3.7].Two quantum seeds are said to be mutation-equivalent if they can be obtained from each other by a sequence of seed mutations.Given the initial quantum seed (X, h, Λ, B), let (X ′ , h ′ , Λ ′ , B ′ ) is mutationequivalent to (X, h, Λ, B).Denote by X ′ = {X ′ 1 , . . ., X ′ m } which is called the extended cluster and the set {X ′ 1 , . . ., X ′ n } is called the cluster.The element X ′ i is called a cluster variable for any 1 ≤ i ≤ n and X ′ k a frozen variable for any n For convenience, let P denote the multiplicative group generated by X n+1 , . . ., X m and q 1 2 , and ZP the ring of the Laurent polynomials in X n+1 , . . ., X m with coefficients in Z[q ± 1  2 ].
Definition 2.6.Given the initial quantum seed (X, h, Λ, B), the associated generalized quantum cluster algebra A(X, h, Λ, B) is the ZP-subalgebra of F generated by all cluster variables from the quantum seeds which are mutation-equivalent to (X, h, Λ, B).
The following Laurent phenomenon is one of the most important results on generalized quantum cluster algebras.
Theorem 2.7.[3, Theorem 3.1] The generalized quantum cluster algebra A(X, h, Λ, B) is a subalgebra of the ring of Laurent polynomials in the cluster variables in any cluster over ZP.

The cluster multiplication formulas of
In the following, we will consider the generalized quantum cluster algebra associated with the initial seed (X, h, Λ, B), where d = (2, 2), Note that Λ T B = 2 0 0 2 , and the based quantum torus is . The quiver associated to the matrix B is the Kronecker quiver Q: We call this algebra a generalized quantum cluster algebra of Kronecker type, denoted by A q (2, 2).By the definition and the Laurent phenomenon, A q (2, 2) is the Z[q ± 1 2 ]-subalgebra of T generated by the cluster variables {X k | k ∈ Z} which are obtained from the following exchange relations: Recall that the n-th Chebyshev polynomials of the first kind F n (x) is defined by and Proof.An direct computation shows that thus X δ is a bar-invariant element in A q (2, 2).Then the proof follows from the definition of the n-th Chebyshev polynomials F n (x).
We define an automorphism denoted by σ on the generalized quantum cluster algebra A q (2, 2) as follows σ(X k ) = X k+1 and σ(q for any k ∈ Z. Then we have the following result which will be useful for us to prove the cluster multiplication formulas.
For a real number x, denote the floor function by ⌊x⌋ and the ceiling function by ⌈x⌉.
The following Theorem 3.3 and Remark 3.4 give the explicit cluster multiplication formulas for A q (2, 2).Theorem 3.3.Let m and n be integers.
(1) For any m > n ≥ 1, we have (2) For any n ≥ 1, we have (3) For any n ≥ 2, we have where The proof is immediately from the definition of the n-th Chebyshev polynomials F n (x).
(2) By using the automorphism σ repeatedly, it suffices to prove the following equation and (3) In order to prove (3.3), it suffices to show that for n ≥ 1.
When n = 2, it is the exchange relation.When n = 3, by (3.2), we have that Assume that When n is even and n ≥ 4, then It follows that Because It suffices to prove that q All elements in B are bar-invariant.
Proof.According to [1, Lemma 4.3, Proposition 4.6], the following equations hold for any m ∈ Z: Thus, for any m ∈ Z and (a 1 , a 2 ) ∈ Z 2 ≥0 , we have which assert that all elements in the set {q } are barinvariant.Together with Lemma 3.1, we know that any element in B is bar-invariant.
In order to prove that the elements in B are Z[q ± 1 2 , h]-independent, we need the following definition which gives a partial order ≤ on Z 2 .
Proof.According to Theorem 3.3 and Remark 3.4, we can deduce that the generalized quantum cluster algebra A q (2, 2) is Z[q ± 1 2 , h]-spanned by the elements in B. Note that X δ has the minimal non-zero term X (−1,−1) associated to the partial order in Definition 4.3, and thus by Theorem 3.3, we deduce that the element F n (X δ ) has the minimal non-zero term X (−n,−n) for each n ∈ Z >0 .According to Theorem 3.4, we have X n X δ = q 1 2 X n+1 + q − 1 2 X n−1 + h.Thus, for each n ≥ 2, we obtain that the cluster variable X n has the minimal non-zero term a n X −(n−2,n−3) where a n ∈ Z[q ± 1 2 ], and for each n ≥ −1, the cluster variable X −n has the minimal non-zero term b n X −(n,n+1) where b n ∈ Z[q ± 1  2 ].Hence, there exists a bijection between the set of all minimal non-zero terms in cluster variables and almost positive roots associated to the affine Lie algebra ŝl 2 .Using the same discussion as [22, Proposition 3.1], we have that there exists a bijection between the set of all minimal non-zero terms in the elements in B and Z 2 , which implies that the elements in B are Z[q ± 1 2 ]-independent.By using Theorem 3.3 and Remark 3.4 repeatedly, we can deduce that the structure constants of the basis elements in B belong to Z ≥0 [q ± 1 2 , h].Together with Lemma 4.2, the proof is completed.Remark 4.5.If we set h = 0 and q = 1, then the set B is exactly the canonical basis of the cluster algebra of Kronecker quiver obtained in [22].Definition 4.6.An element in A q (2, 2) is called positive if the coefficients of its Laurent expansion associated to any cluster belong to Z ≥0 [q ± 1 2 , h].
Remark 4.7.According to Theorem 3.3 and Remark 3.4 or using the same arguments as [19,Corollary 8.3.3], it is not difficult to see that every element in B is positive.In particular, we obtain that all cluster variables of A q (2, 2) are positive, which is a special case in [21].