Greedy bases n rank quantum cluster algebras Kyungyong Lee a,ll b, Dylan Rupel c,, and Andre Zelensky c, a Department of Mathematcs, Wayne State Unersty, Detrot, MI 480; b Department of Mathematcs and Statstcs, Oakland Unersty, Rochester, MI 48309; and c Department of Mathematcs, Northeastern Unersty, Boston, MA 05 Edted by Bernard Leclerc, Unersty of Caen, France, and accepted by the Edtoral Board December 5, 03 (receed for reew August 6, 03) We dentfy a quantum lft of the greedy bass for rank coeffcentfree cluster algebras. Our man result s that our constructon does not depend on the choce of ntal cluster, that t bulds all cluster monomals, and that t produces bar-narant elements. We also present seeral conjectures related to ths quantum greedy bass and the trangular bass of Berensten and Zelensky. standard monomal bass trangular bass Cluster algebras were ntroduced by Fomn and Zelensky () as a combnatoral tool for understandng the canoncal bass and postty phenomena n the coordnate rng of an algebrac group. Ths was followed by the defnton of quantum cluster algebras by Berensten and Zelensky () as a smlar combnatoral tool for understandng canoncal bases of quantum groups. In connecton wth these foundatonal goals, t s mportant to understand the arous bases of cluster algebras and quantum cluster algebras. In ths paper, we confrm the exstence of a quantum greedy bass n rank quantum cluster algebras. It s ndependent of the choce of an ntal cluster, contans all cluster monomals, and specalzes at = to the greedy bass of a classcal rank cluster algebra ntroduced by Lee et al. (3). Another mportant bass n an acyclc quantum cluster algebra s the trangular bass of Berensten and Zelensky (4). Lke trangular bases, quantum greedy bases can be easly computed. Ths enables us to study and compare these bases computatonally. The structure of the paper s as follows. Secton begns wth a recollecton of rank commutate cluster algebras. We recall the constructon of greedy bases here and n the process we prode an axomatc descrpton. In Sec. we reew the defnton of rank quantum cluster algebras. In Sec. 3 we present our man result: the exstence of a quantum lft of the greedy bass usng the axomatc descrpton. Secton 4 recalls the defnton of the trangular bass and Sec. 5 prodes an oerew of results and open problems on the comparson between arous bases of rank quantum cluster algebras.. Rank Cluster Algebras and Ther Greedy Bases Fx poste ntegers b; c > 0. The commutate cluster algebra Aðb; cþ s the Z subalgebra of Qðx ; x Þ generated by the cluster arables fx m g m Z, where the x m are ratonal functons n x and x defned recursely by the exchange relatons x b x m x m = m f m s odd; x c m f m s een: It s a fundamental result of Fomn and Zelensky () that, although the exchange relatons appear to produce ratonal functons, one always obtans a Laurent polynomal whose denomnator s smply a monomal n x and x. They actually showed the followng slghtly stronger result. Theorem. (ref., theorem 3., Laurent phenomenon) For any m Z we hae Aðb; cþ Z½x m ± ; x ± m Š. The cluster algebra Aðb; cþ s of fnte type f the collecton of all cluster arables s a fnte set. Fomn and Zelensky (5) went on to classfy cluster algebras of fnte type. Theorem. (ref. 5, theorem.4) The cluster algebra Aðb; cþ s of fnte type f and only f bc 3. The proof of ths theorem n partcular establshes a connecton between denomnator ectors of cluster arables and almost poste roots n a root system Φ Z. Thus, we say Aðb; cþ s of affne (respectely, wld) type f bc = 4 (respectely, bc 5). An element x T m Z Z x m ± ; x m ± s called unersally Laurent because the expanson of x n eery cluster s Laurent. If for a nonzero element x the coeffcents of the Laurent expanson n each cluster are poste ntegers, then x s called unersally poste. A unersally poste element n Aðb; cþ s sad to be ndecomposable f t cannot be expressed as a sum of two unersally poste elements. Sherman and Zelensky (6) studed n great detal the collecton of ndecomposable unersally poste elements. One of ther man results s the followng: f the commutate cluster algebra Aðb; cþ s of fnte or affne type, then the ndecomposable unersally poste elements form a Z bass n Aðb; cþ, moreoer ths bass contans the set of all cluster monomals fx a m xa m : a ; a Z 0 ; m Zg. Howeer, the stuaton becomes much more complcated n wld types. In partcular, t was shown by Lee et al. (7) that for bc 5 the ndecomposable unersally poste elements of Aðb; cþ are not lnearly ndependent. The greedy bass of Aðb; cþ ntroduced by Lee et al. (3) s a subset of the ndecomposable unersally poste elements whch admts a beautful combnatoral descrpton. In ths secton we wll descrbe an axomatc characterzaton of the greedy bass. The elements of the greedy bass take on a partcular form whch s motated by a well-known pattern n the ntal cluster expanson of cluster monomals. An element x Aðb; cþ s ponted at ða ; a Þ Z f t can be wrtten n the form x = x a x a p;q 0 eð p; qþx bp xcq wth eð0; 0Þ = and eðp; qþ = eða ; a ; p; qþ Z for all p; q 0. It s well-known that x a xa s the denomnator of a cluster monomal f and only f ða ; a Þ Z Φ m (cf. 8, 9), where Φm s the set of poste magnary roots,.e., Sgnfcance The quantum cluster algebras are a famly of noncommutate rngs ntroduced by Berensten and Zelensky as the quantum deformaton of the commutate cluster algebras. At the heart of ther defnton s a desre to understand bases of quantum algebras arsng from the representaton theory of nonassocate algebras. Thus a natural and mportant problem n the study of quantum cluster algebras s to study ther bases wth good propertes. In ths paper, we lay out a framework for understandng the nterrelatonshps between arous bases of rank two quantum cluster algebras. Author contrbutons: K.L., L.L., D.R., and A.Z. desgned research; K.L., L.L., D.R., and A.Z. performed research; K.L., L.L., D.R., and A.Z. analyzed data; and K.L., L.L., and D.R. wrote the paper. The authors declare no conflct of nterest. Ths artcle s a PNAS Drect Submsson. B.L. s a guest edtor nted by the Edtoral Board. To whom correspondence should be addressed. Emal: d.rupel@neu.edu. Deceased Aprl 0, 03. 97 976 PNAS July 8, 04 ol. no. 7 www.pnas.org/cg/do/0.073/pnas.33078
Φ m d ða ; a Þ Z >0 : ca bca a ba 0 : For ða ; a Þ Φ m, defne the regon R greedy = ðp; qþ R 0 j q b ba p < a or ca p c ca Sfð0; q < a a Þ; ða ; 0Þg: ba In other words, R greedy s the regon bounded by the broken lne ð0; 0Þ; ða ; 0Þ; ða =b; a =cþ; ð0; a Þ; ð0; 0Þ; wth the conenton that ths regon ncludes the closed segments ½ð0; 0Þ; ða ; 0ÞŠ and ½ð0; a Þ; ð0; 0ÞŠ but excludes the rest of the boundary (Fg. ). We can defne greedy elements n two dfferent ways, ether by axoms or by recurrence relatons. Here we choose to defne them usng recurrence relatons as follows. Theorem 3. (ref. 3, proposton.6) For each ða ; a Þ Z, there exsts a unque element n Aðb; cþ ponted at ða ; a Þ whose coeffcents eð p; qþ satsfy the followng recurrence relaton: eð0; 0Þ =, eð p; qþ = 8 p ð Þ >< k ½a cqš eðp k; qþ k f ca k q ba p; >: k= q l= ð Þ l ½a bpš eðp; q lþ l l f ca q ba p; where we use the standard notaton ½aŠ = maxða; 0Þ. We defne the greedy element ponted at ða ; a Þ, denoted x½a ; a Š, to be the unque element determned by Theorem 3. For ða ; a Þ Φ m, we ge a drect characterzaton for the greedy element. Theorem 4. For each ða ; a Þ Φ m, the greedy element x½a ; a Š s the unque element n Aðb; cþ ponted at ða ; a Þ whose coeffcents satsfy the followng two axoms: (Support) eðp; qþ = 0 for ð p; qþ R greedy ; (Dsblty) f a > cq; then ð tþ a cq eð; qþt ; f a > bp; then ð tþ a bp eðp; Þt : The key step n the proof of Theorem 4 s based on an obseraton made n ref. 3, sec. : the frst (respectely, second) recurrence relaton n Theorem 3 s equalent to the anshng of the (p, q)th coeffcent of the Laurent expanson of x½a ; a Š wth respect to the cluster fx 0 ; x g (respectely, fx ; x 3 g). It s straghtforward to check that such anshng mples the Support and Dsblty axoms n Theorem 4. The followng theorem summarzes the results from ref. 3. Theorem 5. (ref. 3, theorem.7) (a) The greedy elements x½a ; a Š for ða ; a Þ Z form a Z bass n Aðb; cþ, whch we refer to as the greedy bass. (b) The greedy bass s ndependent of the choce of an ntal cluster. (c) The greedy bass contans all cluster monomals. (d) Greedy elements are unersally poste and ndecomposable. Our goal n ths work s to generalze the aboe theorem to the settng of rank quantum cluster algebras. The proof of Theorem 5 gen n ref. 3 uses combnatoral objects called compatble pars n an essental way (cf. ref. 3, theorem.). Unfortunately ths method has dffcultes lmtatons n generalzng to the study of quantum cluster algebras. More precsely, f one could smply assgn a power of to each compatble par then the quantum greedy elements would hae agan been unersally poste, whch s unfortunately false n general (see the example at the end of Sec. 3). Ths justfes our approach to Theorem 5, wth the excepton of (d), whch can easly be generalzed to the quantum settng.. Rank Quantum Cluster Algebras In ths secton we defne our man objects of study, namely quantum cluster algebras, and recall mportant fundamental facts related to these algebras. We restrct attenton to rank quantum cluster algebras where we can descrbe the setup n ery concrete terms. We follow (as much as possble) the notaton and conentons of refs. 3, 4. Consder the followng quantum torus: T = Z ± D E ± ; ± : = (ths setup s related to the one n ref. 0, whch uses the formal arable q nstead of by settng q = ). There are many choces for quantzng cluster algebras; to rgdfy the stuaton we requre the quantum cluster algebra to be narant under a certan noluton. The bar noluton s the Z-lnear antautomorphsm of T determned by f ðþ = f ð Þ for f Z½ ± Š and f a a = fa a = a a f a a ða ; a ZÞ: An element whch s narant under the bar noluton s sad to be bar-narant. Let F be the skew feld of fractons of T. The quantum cluster algebra A ðb; cþ s the Z½ ± Š subalgebra of F generated by the quantum cluster arables f m g m Z defned recursely from the quantum exchange relatons MATHEMATICS SPECIAL FEATURE m m = b m b c m c f m s odd; f m s een: Fg.. Support regon of (quantum) greedy elements. By a smple nducton one can easly check the followng (quas-) commutaton relatons between neghborng cluster arables m ; m n A ðb; cþ: m m = m m ðm ZÞ: [] It then follows that all cluster arables are bar-narant, therefore A ðb; cþ s also stable under the bar noluton. Moreoer, Eq. mples that each cluster f m ; m g generates a quantum torus Lee et al. PNAS July 8, 04 ol. no. 7 973
T m = Z ± ± m ; ± m : m m = m m : It s easy to see that the bar noluton does not depend on the choce of an ntal quantum torus T m. The approprate quantum analogs of cluster monomals for A ðb; cþ are the (bar-narant) quantum cluster monomals whch are certan elements of a quantum torus T m,moreprecsely they are ða;aþ m = aa a m a m ða ; a Z 0 ; m ZÞ: The followng quantum analog of the Laurent phenomenon was proen by Berensten and Zelensky (). Theorem 6 (). For any m Z we hae A ðb; cþ T m. Moreoer, A ðb; cþ = T T m : m Z A nonzero element of A ðb; cþ s called unersally poste f t les n T m Z Z 0 h ± m ± ; ± m. A unersally poste element n A ðb; cþ s ndecomposable f t cannot be expressed as a sum of two unersally poste elements. It s known as a ery specal case of the results of refs. 5 that cluster monomals are unersally poste Laurent polynomals. Explct combnatoral expressons for these poste coeffcents can be obtaned from the results of refs. 6 8. 3. Quantum Greedy Bases Ths secton contans the man result of the paper. Here we ntroduce the quantum greedy bass and present ts nce propertes. Analogous to the constructon of greedy elements, the elements of the quantum greedy bass take on the followng partcular form. An element A ðb; cþ s sad to be ponted at ða ; a Þ Z f t has the form = eð p; qþ ðbp a ;cq a Þ p;q 0 wth eð0; 0Þ = and eð p; qþ Z½ ± Š for all p and q. The next theorem s an analog of Theorem 3; to state our result precsely we need more notaton. Let w denote a formal nertble arable. For n Z, k Z 0 defne the bar-narant quantum numbers and bar-narant quantum bnomal coeffcents by ½nŠ w = wn w n w w = wn w n 3 w n ; n = ½nŠ w ½n Š w ½n k Š w: k ½kŠ w w ½k Š w ½Š w h n Recall that wll always be a Laurent polynomal n w. Hence k w takng w = b or c we obtan Laurent polynomals n as well. Theorem 7. For each ða ; a Þ Z, there exsts a unque element n A ðb; cþ ponted at ða ; a Þ whose coeffcents eðp; qþ satsfy the followng recurrence relaton: eð0; 0Þ =, eðp; 8 qþ = p ½a ð Þ k cqš k eðp k; qþ f ca q ba p; >< k= k b q ½a ð Þ l bpš l >: eðp; q lþ f ca q ba p: l= l c We defne the quantum greedy element ponted at ða ; a Þ, denoted ½a ; a Š, to be the unque element determned by Theorem 7. The followng theorem s analogous to Theorem 4. Theorem 8. For each ða ; a Þ Φ m, the quantum greedy element ½a ; a Š s the unque element n A ðb; cþ ponted at ða ; a Þ whose coeffcents satsfy the followng two axoms: (Support) eðp; qþ = 0 for ðp; qþ R greedy ; (Dsblty) Let t denote a formal arable whch commutes wth. Then a cq bða cq jþ t eð; qþt ; j= a bp cða bp jþ t eðp; Þt : j= Theorem 8 follows from an argument smlar to the proof of Theorem 4. Our man theorem below states that quantum greedy elements possess all of the desred propertes descrbed n Theorem 5 except the postty (d). Theorem 9. (Man Theorem) (a) The quantum greedy elements ½a ; a Š for ða ; a Þ Z form a Z½ ± Š bass n A ðb; cþ, whch we refer to as the quantum greedy bass. (b) The quantum greedy bass s bar-narant and ndependent of the choce of an ntal cluster. (c) The quantum greedy bass contans all cluster monomals. (d) If ½a ; a Š s unersally poste, then t s ndecomposable. (e) The quantum greedy bass specalzes to the commutate greedy bass by the substtuton =. Full proofs of Theorems 7, 8, and 9 wll appear elsewhere. The hardest part s to show the exstence of quantum greedy elements,.e., that the recursons n Theorem 7 termnate. The man dea n the proof s to realze ½a ; a Š as a degeneraton of certan elements n A ðb; cþ whose supports are contaned n the closure of R greedy. We end ths secton wth an example where postty of quantum greedy elements fals. Let ðb; cþ = ð; 3Þ and consder the ponted coeffcent eð; Þ n the Laurent expanson of ½3; 4Š. Usng Theorem 7, we see that eð; Þ s not n Z 0 ½ ± Š: eð; Þ = eð; Þ eð0; Þ = eð; Þ eð0; Þ 4 3 = eð; 0Þ eð0; Þ = eð0; 0Þ eð0; 0Þ 3 3 = 6 6 6 6 = : Further computaton shows that the greedy elements ½3; 5Š; ½5; 4Š; ½5; 7Š; ½5; 8Š; ½7; 5Š; ½7; 0Š; ½7; Š; etc: are not poste. Smlarly, postty can be seen to fal for a large class of quantum greedy elements when ðb; cþ = ð; 5Þ; ð3; 4Þ; ð4; 6Þ: 4. Trangular Bases The constructon of the trangular bass begns wth the standard monomal bass. For eery ða ; a Þ Z, we defne the standard monomal M½a ; a Š (whch s denoted E ð a ; a Þ n ref. 4) by settng M½a ; a Š = aa ½a Š 3 ½ a Š ½ aš ½aŠ 0 : [] It s known () that the elements M½a ; a Š form a Z½ ± Š bass of A ðb; cþ. 974 www.pnas.org/cg/do/0.073/pnas.33078 Lee et al.
The mportance of ths bass comes from ts computablty; howeer, ths bass wll not sere n our goals of understandng quantum cluster algebras. Indeed, t s easy to see that the standard monomals are not bar-narant and do not contan all of the cluster monomals, moreoer they are nherently dependent on the choce of an ntal cluster. These drawbacks prode a motaton to consder the trangular bass (as defned below) constructed from the standard monomal bass wth a bult-n bar-narance property. As the name suggests, the trangular bass s defned by a trangularty property relatng t to the standard monomal bass. To descrbe ths trangular relatonshp we ntroduce the -poste lattce L = Z½ŠM½a ; a Š, where the summaton runs oer all ða ; a Þ Z. We now defne a bass fc½a ; a Š : ða ; a Þ Z g by specfyng how t relates to the standard monomal bass: (P) Each C½a ; a Š s bar-narant. (P) For each ða ; a Þ Z, we hae C½a ; a Š M½a ; a Š L: Usng the unersal acyclcty of rank quantum cluster algebras, we apply the followng theorem. Theorem 0. (ref. 4, theorem.6]) The trangular bass does not depend on the choce of ntal cluster and t contans all cluster monomals. Smlar to the support condton for (quantum) greedy elements, we make the followng conjecture about the supports of trangular elements. Conjecture. Let ða ; a Þ Φ m. For 0 p a,0 q a, the ponted coeffcent eðp; qþ of ðbp a ;cq a Þ n C½a ; a Š s nonzero f and only f (Fg. ). bp bcpq cq ca q ba p; 5. Open Problems In ths secton we present open problems and conjectures relatng to the trangular bass and quantum greedy bass of a rank quantum cluster algebra. Our am s to fnd Z½ ± Š bases of A ðb; cþ satsfyng the propertes lad out n the followng defnton. Fg.. Conjectured support regon of trangular bass elements s the closed regon OAC wth a cured edge AC, whereo = ð0,0þ, A = ða,0þ, and C = ð0,a Þ. The support regon of the (quantum or nonquantum) greedy element ponted at ða,a Þ s the polygon OABC, whereb = ða =b,a =cþ. Note that the lne BA (respectely, BC) s tangent to the cured edge AC at pont A (respectely, C). Fg. 3. Propertes of the quantum greedy bass for bjc or cjb. A bass B for A ðb; cþ s sad to be strongly poste f the followng hold: () each element of B s bar-narant; () B s ndependent of the choce of an ntal cluster; (3) B contans all cluster monomals; (4) any product of elements from B can be expanded as a lnear combnaton of elements of B wth coeffcents n Z 0 ½ ± Š. Note that all elements n a strongly poste bass are unersally poste. Indeed, consder multplyng a bass element B B by a cluster monomal ða;aþ wth a ; a suffcently large to clear the denomnator n the ntal cluster expanson of B. By propertes (3) and (4) of a strongly poste bass the product B ða ;a Þ s n Z 0 ½ ± Š½ ; Š and thus we see that a strongly poste bass element has nonnegate coeffcents n ts ntal cluster expanson. It then follows that B s unersally poste because t s ndependent of the choce of an ntal cluster. Kmura and Qn (5) showed the exstence of bases for acyclc skew-symmetrc quantum cluster algebras [hence for A ðb; cþ wth b = c] whch satsfy (), (3), and (4). In the rank case, ther bases are exactly the trangular bases, and hence satsfy () as well (see ref. 4, remark.8]). Conjecture. Gen any strongly poste bass B, there exsts a unque bass element ponted at ða ; a Þ for each ða ; a Þ Z. It s easy to see that property (3) of strongly poste bases together wth Conjecture mples that eery element of B s ponted. We note that the concluson of Conjecture holds for both the quantum greedy bass and the trangular bass. In what follows, we wrte Y Z for Y; Z A ðb; cþ f Z Y s a Laurent polynomal wth coeffcents n Z 0 ½ ± Š. Conjecture 3. (a) There exsts a unque strongly poste bass B upper satsfyng the bass element n the bass element n B ponted at ða ; a Þ B upper ponted at ða ; a Þ for eery strongly poste bass B and eery ða ; a Þ Z. (b) The trangular bass s B upper. Conjecture 4. Suppose that bjcorcjb. (a) There exsts a unque strongly poste bass B lower satsfyng the bass element n the bass element n B lower ponted at ða ; a Þ B ponted at ða ; a Þ for eery strongly poste bass B and eery ða ; a Þ Z. (b) The quantum greedy bass s B lower. In contrast wth the example at the end of Sec. 3, we do not obsere a falure of postty when ðb; cþ = ð; 5Þ; ð; 6Þ; ð; 4Þ; MATHEMATICS SPECIAL FEATURE Lee et al. PNAS July 8, 04 ol. no. 7 975
Proposton 5. h r a ;a a ;a = 0 ða ;a Þ>ða ;a Þ h r a ;a a ;a q a ;a a ;a 0 : [5] Fg. 4. ð; 6Þ; ð3; 3Þ; ð3; 6Þ. Ths prodes the motaton for assumng the condton bjc orcjb n Conjecture 4. Conjectures, 3, and 4 are trally true for fnte types ðbc < 4Þ, and are also known for affne types ðbc = 4Þ (9, 0). Next we study the expanson coeffcents relatng the arous bases. Let us begn by ntroducng notaton. For ða ; a Þ; ða ; a Þ Z we defne expanson coeffcents q a;a a ;a ; r a;a a ;a Z½ ± Š as follows: ½a ; a Š = M½a ; a Š C½a ; a Š = ½a ; a Š Propertes of the trangular bass. ða ;a Þ<ða ;a Þ ða ;a Þ<ða ;a Þ q a ;a a ;a M½a ; a Š; [3] r a ;a a ;a ½a ; a Š; [4] where we wrte ða ; a Þ < ða ; a Þ when a < a and a < a. We now dere a recurson for the expanson coeffcents n Eq. 4 relatng the trangular bass to the quantum greedy bass. Our man ngredents wll be the defnng propertes (P) and (P) of the trangular bass. Note that because each ½a ; a Š s bar-narant, we hae r a ;a a ;a = r a ;a a ;a by (P). For f Z½ ± Š let ½f Š 0 denote the nonposte part of f,.e., n f we ealuate = 0 n all terms for whch ths makes sense. Wth ths notaton, we note that because r a ;a a ;a s bar-narant, t s determned by ½r a ;a a ;a Š 0. Ths says that we hae an easy recurse way to compute the decomposton n Eq. 4 f we know the decomposton n Eq. 3. Ths proposton can be proed by reducng Eq. 3 mod L to get r a ;a a ;a ½a ; a Š h r a ;a a ;a 0 M½a ; a Š P ða ;a Þ<ða ;a Þ h r a;a a ;a q a ;a a ;a M½a ; a [6] Š mod L: 0 Keepng n mnd (P), we reduce the equalty 4 mod L and apply Eq. 6 to arre at the desred recurson. Based on extense computatons usng Proposton 5 we make the followng postty conjecture. Conjecture 6. For any ða ; a Þ Z the expanson coeffcents of the trangular bass element C½a ; a Š n terms of the quantum greedy bass are poste; more precsely, we hae r a ;a a ;a Z 0 ± for all ða ; a Þ Z : More generally, the expanson coeffcents of any strongly poste bass n terms of the quantum greedy bass are poste. We end ths artcle by summarzng the aforementoned proen or conjectured propertes of the quantum greedy bass (Fg. 3) and the trangular bass (Fg. 4). ACKNOWLEDGMENTS. Most of the deas toward ths work from D.R. and A.Z. were had durng ther stay at the Mathematcal Scences Research Insttute (MSRI) as part of the Cluster Algebras Program. They thank the MSRI for ther hosptalty and support. Sadly Andre Zelensky passed away n the early stages of wrtng. The authors offer the sncerest grattude to Sergey Fomn for hs careful readng of seeral drafts of ths note. We hope to hae acheed the clarty of exposton that Andre s artful eye would hae proded. The authors thank F. Qn for aluable dscussons. Research supported n part by Natonal Scence Foundaton Grants DMS-090367 (to K.L.) and DMS-0383 (to A.Z.), and by Oakland Unersty s Unersty Research Commttee Faculty Research Fellowshp Award (to L.L.).. Fomn S, Zelensky A (00) Cluster algebras I: Foundatons. J Am Math Soc 5(): 497 59.. Berensten A, Zelensky A (005) Quantum cluster algebras. Ad Math 95():405 455. 3. Lee K, L L, Zelensky A (0) Greedy elements n rank cluster algebras. Sel Math, 0.007/s0009-0-05-. 4. Berensten A, Zelensky A (0) Trangular bases n quantum cluster algebras. Int Math Res Notces, 0.093/mrn/rns68. 5. Fomn S, Zelensky A (003) Cluster algebras II: Fnte type classfcaton. Inent Math 54():63. 6. Sherman P, Zelensky A (004) Postty and canoncal bases n rank cluster algebras of fnte and affne types. Mosc Math J 4(4):947 974. 7. Lee K, L L, Zelensky A (03) Postty and tameness n rank cluster algebras. ar:303.5806. 8. Caldero P, Chapoton F (006) Cluster algebras as Hall algebras of quer representatons. Comment Math Hel 8(3):595 66. 9. Caldero P, Keller B (006) From trangulated categores to cluster algebras. II. Ann Sc École Norm Sup 39(6):983 009. 0. Rupel D (0) On a quantum analog of the Caldero-Chapoton formula. Int Math Res Notces 0(4):307 336.. Qn F, Keller B (0) Quantum cluster arables a Serre polynomals. J Rene Angew Math 0(668):49 90.. Efmo A (0) Quantum cluster arables a anshng cycles. ar:.360. 3. Dason B, Maulk D, Schürmann J, Szendr}o B (03) Purty for graded potentals and quantum cluster postty. ar:307.3379. 4. Nakajma H (0) Quer aretes and cluster algebras. Kyoto J Math 5():7 6. 5. Kmura Y, Qn F (0) Graded quer aretes, quantum cluster algebras and dual canoncal bass. ar:05.066. 6. Lee K, Schffler R (0) Proof of a postty conjecture of M. Kontsech on noncommutate cluster arables. Compos Math 48(6):8 83. 7. Rupel D (0) Proof of the Kontsech non-commutate cluster postty conjecture. C R Math Acad Sc Pars 350(-):99 93. 8. Lee K, L L (03) On natural maps from strata of quer Grassmannans to ordnary Grassmannans. AMS Contemp Math 59:99 4. 9. Chen, Dng M, Sheng J (0) Bar-narant bases of the quantum cluster algebra of type A ðþ. Czech Math J 6(4):077 090. 0. Dng M, u F (0) Bases of the quantum cluster algebra of the Kronecker quer. Acta Math Sn (Engl Ser) 8(6):69 78. 976 www.pnas.org/cg/do/0.073/pnas.33078 Lee et al.