On Homomorphisms from Ringel-Hall Algebras to Quantum Cluster Algebras

(1)

DOI 10.1007/s10468-015-9568-1

On Homomorphisms from Ringel-Hall Algebras to Quantum Cluster Algebras

Xueqing Chen¹·Ming Ding²·Fan Xu³

Received: 11 February 2015 / Accepted: 11 August 2015 / Published online: 27 August 2015

Abstract In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian categoryAto an appropriate q-polynomial algebra. In the case thatAis the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math.55(10) 2045–2066,2012). Moreover, if the underlying graph ofQasso- ciated withAis bipartite and the matrixBassociated to the quiverQis of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

Keywords Ringel-Hall algebra·Quantum cluster algebra·Cluster variable·Bipartite graph

Mathematics Subject Classification (2010) 16G20·17B67·17B35·18E30

Presented by Vlastimil Dlab.

Ming Ding was supported by NSF of China (No. 11301282) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130031120004) and Fan Xu was supported by NSF of China (No. 11071133).

Ming Ding

m-ding04@mails.tsinghua.edu.cn Xueqing Chen

chenx@uww.edu Fan Xu

fanxu@mail.tsinghua.edu.cn

1 Department of Mathematics, University of Wisconsin–Whitewater, 800 W. Main Street, Whitewater, WI 53190, USA

2 School of Mathematical Sciences and LPMC, Nankai University, Tianjin, People’s Republic of China

3 Department of Mathematics, Tsinghua University, Beijing, 100875, People’s Republic of China

(2)

1 Background

The Ringel-Hall algebraH(A)of a (small) finitary abelian categoryAwas introduced by Ringel ([14]). When Ais the category Rep_F_qQof finite dimensional representations of a simply-laced quiverQover a finite fieldFq, the Ringel-Hall algebraH(A)is isomor- phic to the positive partU_q(n)of the corresponding quantum groupU_q(g)([14]). Lusztig ([12]) constructed the canonical basis of the quantum groupU_q(n)under the context of Ringel-Hall algebras. In order to study the canonical basis algebraically and combinatori- ally, Berenstein and Zelevinsky ([2]) defined quantum cluster algebras as a noncommutative analogue of cluster algebras (see [9,10]). A quantum cluster algebra is a subalgebra of a skew field of rational functions in q-commuting variables and generated by a set of generators called thecluster variables.

A natural question is to study the relations between Ringel-Hall algebras and quantum cluster algebras. Geiss, Leclerc and Schr¨oer ([11]) showed that quantum groups of type A, DandEhave quantum cluster structures. Recently, Berenstein and Rupel [1] constructed algebra homomorphisms from Ringel-Hall algebras to quantum cluster algebras. LetAbe a finitary hereditary abelian category andi=(i₁,· · ·, i_m)be a sequence of simple objects in A. Berenstein and Rupel [1] showed that, under certain co-finiteness conditions, the assignment[V]^∗→XV ,idefines a homomorphism of algebras

i:H^∗(A)→Pi

whereH^∗(A)is the dual Ringel-Hall algebra andX_{V ,i} is the quantum clusteri-character ofV in an appropriateq-polynomial algebraP_i. Moreover, for an appropriatei, the image restricting to the composition algebra of H^∗(A) is in the corresponding upper cluster algebra.

The aim of this note is to give an alternative proof of the above result whenAis the representation category of an acyclic quiver. Different from [1], a key ingredient of our proof is to apply the cluster multiplication formulas proved in [8] (see also Theorem 3.3). We show that if the underlying graph ofQis bipartite (i.e, we can associate this graph an orientation such that every vertex is a sink or a source) and the matrixBassociated to the quiverQis of full rank, then the algebraAH|k|(Q)generated by all quantum cluster characters is exactly the quantum cluster algebraA|k|(Q)(see Theorem 4.5). As a corollary, the image of the algebra homomorphism is in the quantum cluster algebraA|k|(Q)(see Corollary 4.6). We expect that the approach in this note can be extended to construct algebra homomorphisms from derived Hall algebras to quantum cluster algebras.

2 Quantum Cluster Algebras and Caldero-Chapoton Maps 2.1 Quantum Cluster Algebras

We briefly recall the definition of quantum cluster algebras. LetLbe a lattice of rankm and : L×L →Za skew-symmetric bilinear form. We will need a formal variableq and consider the ring of integeral Laurent polynomialsZ[q^±1/2]. Define thebased quan- tum torusassociated to the pair(L, )to be theZ[q^±1/2]-algebraT with a distinguished Z[q^±^1/2]-basis{X^e:e∈L}and the multiplication given by

X^eX^f =q^{(e,f )/2}X^e+f.

(3)

It is easy to see thatT is associative and the basis elements satisfy the following relations:

XêX^f =q^{(e,f )}X^fXê, X⁰=1 and(Xê)⁻¹=X^−e.

It is known thatT is an Ore domain, i.e., is contained in its skew-field of fractionsF. The quantum cluster algebra will be defined as aZ[q^±^1/2]-subalgebra ofF.

Atoric frameinFis a mapM:Z^m→F\ {0}of the form M(c)=ϕ(X^η(c))

whereϕ is an automorphism ofF andη : Z^m → Lis an isomorphism of lattices. By definition, the elementsM(c)form aZ[q^±1/2]-basis of the based quantum torusTM :=

ϕ(T)and satisfy the following relations:

M(c)M(d)=q^M^(c,d)/2M(c+d), M(c)M(d)=q^M^(c,d)M(d)M(c), M(0)=1 andM(c)⁻¹=M(−c),

where _M is the skew-symmetric bilinear form on Z^m obtained from the lattice isomorphism η. LetM also denote the skew-symmetric m×mmatrix defined by λij = M(ei, ej)where {e1, . . . , em} is the standard basis of Z^m. Given a toric frame M, let Xi =M(ei). Then we have

TM=Z

q^±^1/2 X₁^±1, . . . , X_m^±¹:XiXj =q^λ^ijXjXi

.

Letbe anm×mskew-symmetric matrix and letB˜ be anm×nmatrix withm≥n, whose principal part is denoted byB. We call the pair(,B)˜ compatibleifB˜^tr=(D|0) is ann×mmatrix withD = diag(d1,· · ·, dn)wheredi ∈ Nfor 1 ≤i ≤n. The pair (M,B)˜ is called aquantum seedif the pair(M,B)˜ is compatible. Define them×mmatrix E=(e_ij)by

e_ij=

⎧⎨

⎩

δ_ij ifj =k;

−1 ifi=j =k;

max(0,−b_ik) ifi=j=k.

Forn, k ∈ Z,k ≥ 0, denote ⁿ_k

q = ^(qⁿ^−q_(q⁻ⁿr−q^)···(q^−r^n−r+1)···(q−q^−q⁻¹^−n+r−1) ⁾. Let k ∈ [1, n]andc = (c₁, . . . , c_m)∈Z^mwithc_k≥0. Define the toric frameM:Z^m →F\ {0}as follows:

M(c)=

ck

p=0

c_k p

q^{dk /}²

M(Ec+pb^k)andM(-c)=M(c)⁻¹ (1) where the vectorb^k∈Z^mis thek-th column ofB.˜

Define them×nmatrixB˜=(b_ij )by b_ij =

−b_ij ifi=korj=k;

bij+^|b^ik^|b^kj^+b₂ ^ik^|b^kj^| otherwise.

Then the quantum seed (M,B˜) is defined to be the mutation of (M,B)˜ in direction k. Two quantum seeds (M,B)˜ and(M,B˜) are mutation-equivalent if they can be obtained from each other by a sequence of mutations, denoted by(M,B)˜ ∼(M,B˜). Let C:= {M(ei):(M,B)˜ ∼(M,B˜), i∈ [1, n]}. LetZPbe the ring of integral Laurent polynomials in the (quasi-commuting) variables in{q^1/2, Xn+1,· · ·, Xm}. Thequantum cluster algebraAq(M,B)˜ is theZP-subalgebra ofFgenerated byC.

The following proposition demonstrates the mutation of quantum cluster variables which can be viewed as a quantum analogue of cluster mutation.

(4)

Proposition 2.1 (Mutation of cluster variables)[2]The toric frameXis determined by X_k = X

⎧⎨

⎩

1≤i≤m

[b_ik]+e_i−e_k

⎫⎬

⎭+X

⎧⎨

⎩

1≤j≤m

[−b_{j k}]+e_j−e_k

⎫⎬

⎭, X_i = Xi, 1≤i≤m, i=k.

The quantum Laurent phenomenon proved by Berenstein and Zelevinsky is an important result concerning quantum cluster algebras.

Theorem 2.2 (Quantum Laurent phenomenon)[2] The quantum cluster algebra Aq(_M,B)˜ is a subalgebra ofTM.

SetX = {X₁,· · ·, X_n}andX_k = X− {X_k} ∪ {X_k}for any k ∈ [1, n]. Denote by U(M,B)˜ theZP-subalgebra ofFgiven by

U(M,B)˜ =ZP[X^±1] ∩ZP X^±1₁

∩ · · · ∩ZP X^±1_n

.

The algebraU(M,B)˜ is called thequantum upper cluster algebra. The following result shows that the acyclicity condition closes the gap between the upper bounds and the corresponding quantum cluster algebras.

Theorem 2.3 [2]If the principal matrixBis acyclic, thenU(M,B)˜ =Aq(M,B).˜ 2.2 Quantum Caldero-Chapoton Maps

Letkbe a finite field with cardinality|k| = q andm ≥ nbe two positive integers. Let Qbe an acyclic quiver with vertex set{1, . . . , m}. DenoteC := {n+1, . . . , m}. The full subquiverQon the vertices{1, . . . , n}is called theprincipal partofQ. For 1 ≤i≤m, let S_ibe theith simple module of the path algebrakQ.

LetBbe them×nmatrix associated to the quiverQwhose entry in position(i, j )is given by

b_ij = |{arrowsi−→j}| − |{arrowsj −→i}|

for 1≤i ≤mand 1≤j ≤n. Denote byIthe leftm×nsubmatrix of the identity matrix of sizem×m. Assume that there exists some antisymmetricm×minteger matrixsuch that

(−B) = I_n

0

, (2)

where Inis the identity matrix of sizen×n. LetR=R_Qbe them×nmatrix with its entry in position(i, j )given by

r_ij:=dim_kExt¹

kQ(S_j, S_i)= |{arrowsj−→i}|.

for 1≤i≤mand 1≤j ≤n. SetR^tr =R_Qop.Denote the principaln×nsubmatrices of BandRbyBandR, respectively. Note thatB=R^tr−RandB=R^tr−R.

LetCQbe the cluster category of kQ, i.e., the orbit category of the derived category D^b(Q) under the action of the functorF =τ◦ [−1](see [3]). LetIibe the indecomposable injectivekQmodule for 1≤i ≤m.Then the indecomposablekQ-modules and Ii[−1]for

(5)

1≤i ≤mexhaust all indecomposable objects of the cluster categoryCQ. Each objectM inCQcan be uniquely decomposed as

M=M0⊕IM[−1] whereM₀is a module andI_Mis an injective module.

The Euler form onkQ-modules MandNis given by

M, N =dim_kHom(M, N )−dim_kExt¹(M, N ).

Note that the Euler form only depends on the dimension vectors ofMandN.

The quantum Caldero-Chapoton map of the quiverQhas been defined in [6,8,13,15]

as

X_?:objCQ−→T by the following rules:

(1) IfMis akQ-module, then XM =

e

|GreM|q⁻¹²^e,m−eX⁻^Be−( ^I⁻^R^tr^)m; (2) IfMis akQ-module andIis an injectivekQ-module, then

X_M⊕I_[−1]=

e

|Gr_eM|q⁻¹²ê,m−e−iX⁻^Be−( Î−^R^tr^{)m+dim soc}Î,

where dimI =i,dimM=mand GreMdenotes the set of all submodulesV ofMwith dimV =e. We note that

XP[1]=Xτ P =X^{dimP /radP} =X^{dim soc}^I =XI[−1]=X_τ−1I. for any projectivekQ-module P and injectivekQ-module Iwith socI=P /radP .

In the following, for convenience, we always use the underlined lower letterxto denote the corresponding dimension vector of akQ-moduleXand viewxas a column vector in Zⁿ.

3 The Dual Ringel-Hall Algebras and the Cluster Multiplication Formulas

LetAbe the representation category of an acyclic quiverQover a finite field andK(A) the Grothendieck group ofA. For an objectV ∈A, we will write[V]for the isomorphism class of V. LetH(A)=

k[V]be thek-vector space spanned by the isomorphism classes of objects ofAwith the natural grading via class inK(A). ForU, V , W ∈Adefine the Hall number

g^V_{U W} = |{R⊂V|R∼=W, V /R∼=U}|. The assignment [U][W] =

[V]g^V_{U W}[V] defines an associative multiplication on H(A). The algebra H(A) is known as the Ringel-Hall algebra. Denote by H^∗(A) the dual Ringel-Hall algebra, which is the space of linear functions H(A) → k with a basis of all delta-functions δ_V labeled by isomorphism classes [V] of objects ofA.

(6)

Proposition 3.1 LetMandNbekQ-modules, then the product δ_M∗δ_N =q¹²⁽⁽^I⁻^R^)m,(^I⁻^R^)n)+m,n

E

h^MN_E δ_E

defines an associative multiplication on H^∗(A), where h^MN_E = ^|Ext_|_Hom¹^kQ_kQ^{(M,N )}_{(M,N )}^E^|_| and Ext¹_kQ(M, N )_E is the subset of Ext¹_kQ(M, N )consisting of those equivalence classes of short exact sequences with middle termE.

Proof Note that

E

g_MN^E g_EL^F =

G

g_{N L}^G g_MG^F ,and the relation betweenh^MN_E andg^E_MNis given by the Riedtmann-Peng’s formula

h^MN_E =g^E_MN|Aut (M)||Aut (N )||Aut (E)|⁻¹. Thus we have

Eh^E_MNh^F_EL =

Gh^G_{N L}h^F_MG.It is easy to see thatφ(m, n):= ¹₂((I − R)m, (I−R)n)+ m, nis a bilinear form onZⁿ. Hence the associativity can be deduced.

For any kQ−modules M, N and E, denote by ε_MN^E the cardinality of the set Ext¹

kQ(M, N )_Ewhich is the subset of Ext¹

kQ(M, N )consisting of those equivalence classes of short exact sequences with middle termE. Define

Hom_k_Q(M, I )_BI := {f :M−→I|kerf ∼=B, cokerf ∼=I}. Denote

[M, N] =dimkHom_k_Q(M, N )and[M, N]¹=dimkExt¹

kQ(M, N ).

We have the following cluster multiplication formulas.

Theorem 3.2 [8][6]LetM andN be anykQ-modules, andI any injectivekQ-module, then

(1) q^[M,N^]¹X_MX_N =q¹²⁽⁽^I−^R^)m,(^I⁻^R⁾ⁿ⁾

Eε^E_MNX_E; (2) q^[M,I]X_MX_I[−1]=q¹²⁽⁽^I⁻^R^)m,⁻^{dimSocI )}

B,I|Hom_k_Q(M, I )_BI|X_B_⊕_I[−1]. Note that Theorem 3.2(1) implies the following result which has been proved by Berenstein and Rupel using generalities on bialgebras in braided monoidal categories.

Theorem 3.3 [1] The assignment δV → XV defines an algebra homomorphism : H^∗(A)→T.

An alternative proof.

Note that the first cluster multiplication formula in Theorem 3.2 can be rewritten as XMXN =q¹²⁽⁽^I⁻^R^)m,(^I⁻^R⁾ⁿ⁾⁺<m,n>

E

h^MN_E XE.

(7)

Thus we have

(δ_M∗δ_N) =

q¹²⁽⁽^I−^R^)m,(^I⁻^R^)n)+m,n

E

h^MN_E δ_E

= q¹²⁽⁽^I⁻^R^)m,(^I−^R^)n)+m,n

E

h^MN_E X_E

= X_MX_N =(δ_M)(δ_N).

This completes the proof.

4 Quantum Cluster Algebras for Bipartite Graphs

In this section, we assume that Qis an acyclic quiver whose underlying graph is bipartite and the matrixBassociated to the quiverQis of full rank. Note that in this case the corresponding quantum cluster algebras are coefficient-free.

Definition 4.1 With respect to the quantum Caldero-Chapoton map, XL is called the quantum cluster characterifL∈CQ.

Definition 4.2 For a quiverQ, denote byAH|k|(Q)theZ-subalgebra ofFgenerated by all the quantum cluster characters.

We will show that the algebraAH|k|(Q)is equal to the quantum cluster algebraA|k|(Q).

Let Q be an acyclic quiver andi a sink or a source in Q. We define the reflected quiverσi(Q)by reversing all the arrows ending ati. Anadmissible sequence of sinks (resp.

sources) is a sequence(i₁, . . . , i_l)such that i₁ is a sink (resp. source) inQ andi_k is a sink (resp source) inσ_i_k−1· · ·σ_i₁(Q)for any 2 ≤k ≤l. A quiverQis calledreflection- equivalenttoQif there exists an admissible sequence of sinks or sources(i₁, . . . , i_l)such that Q = σi_l· · ·σi₁(Q). Note that mutations can be viewed as generalizations of reflec- tions, i.e, ifi is a sink or a source in a quiverQ, thenμi(Q)= σi(Q)whereμi denotes the quiver mutation in the directioni. From now on, we assume thatQis quiver mutation- equivalent toQ. Denote by_i :A|k|(Q)→A|k|(Q)the natural canonical isomorphism of quantum cluster algebras associated to sink or source for any 1≤i ≤n, which sends each initial cluster variable ofA|k|(Q)to its Laurent expansion in the initial cluster ofA|k|(Q).

Let_i⁺ : rep(kQ) −→ rep(kQ)be the standard BGP-reflection functor andR_i⁺ : CQ−→CQbe the extended BGP-reflection functor defined in [16]:

R_i⁺:

⎧⎪

⎪⎨

⎪⎪

⎩

X → _i⁺(X) ifXSiis a module Si → Pi[1]

Pj[1] → Pj[1] ifj=i P_i[1] → S_i

Rupel proved the following result.

Theorem 4.3 [15] For any indecomposable objectM inCQ, we have that _i(X^Q_M) = X^Q

R⁺_iM.

(8)

The following lemma is well-known.

Lemma 4.4 [4, Lemma 8(b)]Let

M−→E−→N−→M[1] be a non-split triangle inCQ.Then

dimkExt¹_C

Q(E, E) <dimkExt¹_C

Q(M⊕N, M⊕N ).

Theorem 4.5 Assume thatQis an acyclic quiver whose underlying graph is bipartite and the matrixBassociated to the quiverQis of full rank, thenAH_|k|(Q)=A_|k|(Q).

Proof Firstly, we prove that for any indecomposable objectM∈CQ,X_Mis in the quantum cluster algebraA|k|(Q).

Case 1: IfQis an alternating quiver (i.e, whose vertex is either a sink or a source).

By Theorem 4.3, we have that_i(X^Q_M) =X^Q

R_i^±M for any indecomposable objectM ∈ CQ. It is easy to see thatQis again an acyclic quiver. Then we obtain that

X_M∈Z X^±¹

∩Z X^±1₁

∩ · · · ∩Z X^±_n¹

.

Note that the quiverQis acyclic, thus the corresponding quantum upper cluster algebra associated toQcoincides with the quantum cluster algebraA|k|(Q)(see Theorem 2.3).

HenceX_Mis in the quantum cluster algebraA|k|(Q).

Case 2: IfQis an acyclic quiver whose underlying graph is bipartite.

Note thatQis reflection–equivalent to some alternating quiverQand in theCase 1we have showed that for any indecomposable objectM ∈ CQ,X_M is in the corresponding quantum cluster algebraA|k|(Q). Thus the rest of the proof immediately follows from Theorem 4.3.

Now we need to prove that for any quantum cluster characterXL∈AH|k|(Q), we have thatX_L ∈A|k|(Q). LetL∼=

l i=1

L^⊕_i ⁿⁱ, n_i ∈NwhereL_i (1≤i ≤l)are indecomposable objects inCQ. According to Theorem 3.2, we obtain the following equality:

X_Lⁿ¹

1X_Lⁿ²

2· · ·Xⁿ_L^l

l =q¹²ⁿ^LXL+

dimkExt¹_CQ(E,E)<dimkExt¹_CQ(L,L)

fnE(q^±¹²)XE

wheren_L∈Zandf_n_E(q^±¹²)∈Z[q^±¹²].Using Lemma 4.4 and proceeding by induction, it is straightforward to verify thatXL∈A|k|(Q).

This completes the proof.

Corollary 4.6 Assume thatQis an acyclic quiver whose underlying graph is bipartite, and the matrixBassociated to the quiverQis of full rank, then(H^∗(A))⊆A|k|(Q).

Proof By Theorem 3.3, we have that (H^∗(A)) ⊆ AH|k|(Q). Hence the proof immediately follows from Theorem 4.5.

(9)

Remark 4.7 It is natural to ask when (H^∗(A)) is equal to A|k|(Q). The key point of this problem is to prove that the initial cluster variables can be written as a Z[q^±1/2]−combination of some product of cluster characters associated tokQ-modules. In the following, we give an example in this direction.

Example 4.8 Set= 0 1

−1 0

andB = 0 2

−2 0

. Thus the quiverQassociated to this pair is the Kronecker quiver:

Letkbe a finite field andq = √|k|. The category rep(kQ)of finite-dimensional representations can be identified with the category of mod-kQof finite-dimensional modules over the path algebrakQ.It is well-known (see [5]) that up to isomorphism the indecompos- ablekQ-module contains three families: the preprojective modules with dimension vector (n−1, n)(denoted byM(n)), the indecomposable regular modules with dimension vector (nd_p, nd_p)forp∈P¹_kof degreed_p(in particular, denoted byR_p(n)ford_p =1) and the preinjective modules with dimension vector(n, n−1)(denoted byN (n)) for anyn∈N.

Form∈Z\ {1,2}, set

V (m)=

N (m−2) ifm≥3;

M(−m+1) ifm≤0.

Now, let T = Z[q^±1/2]X₁^±1, X₂^±1 : X1X2 = qX2X1 and F be the skew field of fractions of T. The quantum cluster algebra of the Kronecker quiver Aq(2,2) is the Z[q^±^1/2]-subalgebra of F generated by the cluster variables in {X_k|k ∈ Z} defined recursively by

X_m−1X_m+1=qX²_m+1.

With the above notation, we have the following results.

Lemma 4.9 [15]For anym∈Z\ {1,2}, them-th cluster variableX_mofAq(2,2)is equal toXV (m).

Lemma 4.10 [7]For anyn∈Z,we have that

X_nX_R_p₍₁₎=q⁻¹²X_n−1+q¹²X_n+1.

For the Kronecker quiver, we show that the image of the dual Ringel-Hall algebra under the homomorphismcoincides with the quantum cluster algebra.

Theorem 4.11 Assume thatQis the Kronecker quiver, we have that (H^∗(A))=A|k|(Q).

Proof By Corollary 4.6, we know that(H^∗(A))⊆A|k|(Q).Note thatis an algebra homomorphism according to Theorem 3.3, thus it is enough to prove thatX1andX2have preimages. By Lemma 4.10, we haveX₀X_R_p₍₁₎ = q⁻¹²X₋₁ +q¹²X₁.This givesX₁ = q⁻¹²X0XR_p(1)−q⁻¹X₋1which can be rewritten asX1 =q⁻¹²XV (0)XR_p(1)−q⁻¹XV (−1)

according to Lemma 4.9. Hence we haveX1=q⁻¹²(δV (0))(δRp(1))−q⁻¹(δ_{V (−1)})= (q⁻¹²δ_{V (0)}∗δ_R_p₍₁₎−q⁻¹δ_{V (−1)}).Similarly we haveX₃X_R_p₍₁₎=q⁻¹²X₂+q¹²X₄,and

(10)

using the same method we deduce thatX₂=(q¹²δ_{V (3)}∗δ_R_p₍₁₎−qδ_{V (4)}).This completes the proof.

Conflict of Interest Ming Ding was supported by NSF of China (No. 11301282) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130031120004) and Fan Xu was supported by NSF of China (No. 11071133).

References

1. Berenstein, A., Rupel, D.: Quantum cluster characters of Hall algebras. Selecta Mathematica (2015).

doi:10.1007/s00029-014-0177-3

2. Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math.195, 405–455 (2005)

3. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv.

Math.204, 572–618 (2006)

4. Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Invent. math.172(1), 169–211 (2008)

5. Dlab, V., Ringel, C.: Indecomposable Representations of Graphs and Algebras. Mem. Amer. Math. Soc., 173 (1976)

6. Ding, M.: On quantum cluster algebras of finite type. Front. Math. China6(2), 231–240 (2011) 7. Ding, M., Xu, F.: Bases of the quantum cluster algebra of the Kronecker quiver. Acta Math. Sin.28,

1169–1178 (2012)

8. Ding, M., Xu, F.: A quantum analogue of generic bases for affine cluster algebras. Sci. China Math.

55(10), 2045–2066 (2012)

9. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations, J. Amer. Math. Soc.15(2), 497–529 (2002) 10. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math.154(1), 63–121

(2003)

11. Geiss, C., Leclerc, B., Schr¨oer, J.: Cluster structures on quantum coordinate rings. Sel. Math. N. Ser.19, 337–397 (2013)

12. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc.3, 447–

498 (1990)

13. Qin, F.: Quantum cluster variables via Serre polynomials. J. reine angew. Math.668, 149–190 (2012) 14. Ringel, C.M.: Hall algebras and quantum groups. Invent. Math.101, 583–592 (1990)

15. Rupel, D.: On a quantum analog of the Caldero-Chapoton formula. Int. Math. Res. Notices14, 3207–

3236 (2011)

16. Zhu, B.: Equivalence between cluster categories. J. Algebra304, 832–850 (2006)