CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS

Transcription

1 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE Abstract. We generalse the noton of cluster structures from the work of Buan-Iyama-Reten-Scott to nclude stuatons where the endomorphsm rngs of the clusters may have loops. We show that n a Hom-fnte 2-Calab-Yau category, the set of maxmal rgd objects satsfes these axoms whenever there are no 2-cycles n the quvers of ther endomorphsm rngs. We apply ths result to the cluster category of a tube, and show that ths category forms a good model for the combnatorcs of a type B cluster algebra. Introducton Snce the ntroducton of cluster algebras by Fomn and Zelevnsky [FZ1], relatonshps between such algebras and nterestng topcs n several branches of mathematcs have emerged. The project of modellng cluster algebras n a representaton theoretc settng was ntated n [MRZ]. Inspred by ths, cluster categores were defned n [BMRRT] to be certan orbt categores obtaned from the derved category of Hom-fnte heredtary abelan categores. These categores have been wdely studed for the case when the ntal heredtary category s the category of fnte dmensonal representatons of an acyclc quver Q. (When Q s a quver wth underlyng graph A n, the cluster category was ndependently defned n [CCS].) It has then been shown that the ndecomposable rgd objects are n bjecton wth the cluster varables n the cluster algebra A Q assocated wth the same quver, and under ths bjecton the clusters correspond to the maxmal rgd objects ([CK] based on [CC], see also [BCKMRT]). Moreover, by [BMR], the quver of the endomorphsm rng of a maxmal rgd object s the same as the quver of the correspondng cluster. The last phenomenon also appears for maxmal rgd modules n the stable module category of preprojectve algebras of smply laced Dynkn type [GLS], whch s another example of a Hom-fnte 2-Calab-Yau trangulated category. Inspred by ths and [IY], [KR], an axomatc framework for mutaton n 2-CY categores was defned n [BIRS]. The essental features were consdered to be: the unque exchange of ndecomposable summands; the fact that exchange pars were related by approxmaton trangles; and the fact that on the level of endomorphsm rngs, exchange of ndecomposable summands led to Fomn-Zelevnsky quver mutaton on the Gabrel quvers. For the thrd of these features to make sense, one must requre that the endomorphsm rngs have no loops or 2-cycles n ther quvers. In [BIRS] t was also shown that the collecton of maxmal rgd objects n any Hom-fnte 2-CY trangulated category fulfls these axoms for cluster structures whenever the quvers of ther endomorphsm rngs do not have loops or 2-cycles. The cluster structures from [BIRS] have two lmtatons: Frstly, there exst Hom-fnte 2-CY trangulated categores where the endomorphsm rngs of maxmal rgd objects do have loops and 2-cycles n ther quvers. The unque exchange property holds also n these categores [IY], but FZ quver mutaton does not make sense n ths settng. Secondly, the cluster algebras whch can be modelled from the cases studed n [BIRS] are the ones defned from quvers (equvalently, skew-symmetrc matrces), whle cluster algebras can be defned also from more general matrces. The am of ths paper s to extend the noton of cluster structures from [BIRS]. We show that the set of maxmal rgd objects n a Hom-fnte 2-CY trangulated category satsfes ths new defnton of cluster structure regardless of whether the endomorphsm rngs have loops or not. (We must however assume that the quvers do not have 2-cycles.) One effect of ths s that we wll also relax the second lmtaton, snce the cluster algebras whch can be modelled n ths new settng but not n the settng of [BIRS] are defned from matrces whch are not necessarly skew-symmetrc. Whle prevous nvestgatons of cluster structures have regarded the quver of the endomorphsm rng as the essental combnatoral data, our approach s to emphasze the exchange trangles nstead. 1

2 2 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE We collect the nformaton from the exchange trangles n a matrx and requre for our cluster structure that exchange of ndecomposable objects leads to FZ matrx mutaton of ths matrx. In the no-loop stuaton consdered n [BIRS], the matrx defned here s the same as the one descrbng the quver of the endomorphsm rng, so our defnton s an extenson of the defnton n [BIRS]. In the cluster category defned from the module category of a heredtary algebra, the maxmal rgd objects are the same as the cluster-tltng objects. In general, however, the cluster-tltng condton s stronger, and there exst categores where the maxmal rgd objects are not cluster-tltng. Examples of ths n a geometrcal settng are gven n [BIKR]. In ths paper we present a class of examples from a purely representaton-theoretcal settng, namely the cluster categores C Tn defned from tubes T n. Here, the tube T n s the category of nlpotent representatons of a quver wth underlyng graph Ãn 1 and wth cyclc orentaton. The category C Tn has also been studed n [BKL1, BKL2]. An nterestng aspect of cluster categores from tubes s that the endomorphsm rngs of the maxmal rgd objects have quvers wth loops, but not 2-cycles. Thus they are covered by the defnton of cluster structures n ths paper, but not by the one n [BIRS]. We wll show that the set of rgd objects n C Tn s a model of the exchange combnatorcs of a type B cluster algebra. To ths end, we gve a bjecton between the ndecomposable rgd objects n C Tn and the cluster varables n a type B n 1 cluster algebra such that the maxmal rgd objects correspond to the clusters. Ths bjecton s va the cyclohedron, or Bott-Taubes polytope [BT]. Also, we wll see that the matrx defned from the exchange trangles assocated to a maxmal rgd object s the same as the one belongng to the correspondng cluster n the cluster algebra. The artcle s organsed as follows. In Secton 1 we gve the new defnton of cluster structures and show that the set of maxmal rgd objects n a Hom-fnte 2-CY category satsfes ths defnton whenever there are no 2-cycles n the quvers of ther endomorphsm rngs. In Secton 2 we gve a complete descrpton of the maxmal rgd objects n the cluster category of a tube. Fnally, n Secton 3, we show that the cluster structure n a cluster tube forms a good model of the combnatorcs of a type B cluster algebra. 1. Cluster structures In ths secton we generalse the noton of cluster structures from [BIRS] to nclude stuatons where the quvers of the clusters may have loops. We then proceed to show that the set of maxmal rgd objects n a Hom-fnte 2-Calab-Yau category admts a cluster structure wth ths new defnton, under the assumpton that the Gabrel quvers of ther endomorphsm rngs do not have 2-cycles. Let k be some algebracally closed feld. By a Hom-fnte k-category we wll mean a category C where Hom C (X, Y ) s a fnte dmensonal k-vector space for all pars of ndecomposable objects X and Y. We wll normally suppress the feld k. A trangulated category C s sad to be Calab-Yau of dmenson 2, or 2-CY, f D Ext C (X, Y ) Ext2 C (Y, X) for all objects X, Y of C and all n Z. For the remander of ths secton, C wll denote a Hom-fnte 2-Calab-Yau trangulated category. We now recall the defnton of a weak cluster structure from [BIRS]. Let T be a non-empty collecton of sets of non-somorphc ndecomposable objects of C. Each set T of ndecomposables whch s an element of T s called a precluster. The collecton T s sad to have a weak cluster structure f the followng two condtons are met: (a) For each precluster T = T {M} wth M ndecomposable, there exsts a unque ndecomposable M M such that the dsjont unon T = T {M } s a precluster. (b) M and M are related by trangles M f U M,T g M and M s U M,T t M where f and s are mnmal left addt-approxmatons and g and t are mnmal rght addtapproxmatons. These trangles are called the exchange trangles of M (equvalently, of M ) wth respect to T. We wll not dstngush between a precluster and the object obtaned by takng the drect sum of the ndecomposable objects whch form the precluster. The same goes for subsets of preclusters.

3 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 3 Assume now that T has a weak cluster structure. For each T = {T } I n T we defne a (possble nfnte) matrx B T = (b j ) by b j = α U T,T/T T j α UT,T/T T j where α Y X denotes the multplcty of X as a drect summand of Y. So the matrx B T records, for each T, T j T, the dfference between the multplctes of T j n the two exchange trangles for T. Note that b = 0 for all, snce T does not appear as a summand n the target (resp. source) of a left (resp. rght) add(t/t )-approxmaton. Recall Fomn-Zelevnsky matrx mutaton, defned n [FZ1]: For a matrx B = (b j ) the mutaton at k s gven by µ k (B) = (b j ), where { b j = bj = k or j = k b j + b k b kj +b k b kj 2, j k Note n partcular that for, j k, we have b j b j f and only f b k and b kj are both postve or both negatve. We say that T has a cluster structure f T has a weak cluster structure, and n addton the followng condtons are satsfed: (c) For each T T and each T T, the objects U T,T/T and U T,T/T have no common drect summands. (d) If T k s an ndecomposable object and T = T {T k } and T = T {Tk } are preclusters, then B T and B T are related by Fomn-Zelevnsky matrx mutaton at k. In ths case, we call the elements of T clusters. An nterpretaton of condton (c) above s that the endomorphsm rngs of the clusters have Gabrel quvers whch do not have 2-cycles. It should also be noted that f there are no loops at the vertces correspondng to T and T j n these quvers, then the multplcty of T j as a summand n the exchange trangle for T wll equal the number of arrows between these two vertces. In partcular, f there are no loops at any of the vertces, B T s skew-symmetrc and can be consdered as a record of the quver. Condton (d) then reduces to FZ quver mutaton, so our defnton concdes wth the defnton n [BIRS] n ths case. An object T n a trangulated category K s sad to be rgd f Ext 1 K (T, T) = 0. It s called maxmal rgd f t s maxmal wth ths property, that s, Ext 1 K(T X, T X) = 0 mples that X addt. If there exst maxmal rgd objects n a Hom-fnte 2-CY trangulated category, then the collecton of such objects has a weak cluster structure. Ths follows from [IY], as stated n Theorem I.1.10 (a) of [BIRS]. We can now prove a stronger verson of part (b) of the same theorem: Theorem 1.1. Let C be a Hom-fnte 2-Calab-Yau trangulated category, and let T be the collecton of maxmal rgd objects n C. Assume that T s nonempty, and that t satsfes condton (c) above. Then T has a cluster structure. The proof of Theorem 1.1 follows the same lnes as the proof of Theorem I.1.6. n [BIRS]. We need to show that when an ndecomposable summand of a maxmal rgd object s exchanged, the change n the matrx s gven by Fomn-Zelevnsky matrx mutaton. The fact that the matrces B T for maxmal rgd T are not necessarly skew-symmetrc forces us to prove dfferent cases separately. For the proof we wll need the followng lemma: Lemma 1.2. In the stuaton of the theorem, for any maxmal rgd object T, B T s sgn skew symmetrc,.e., for all, j, we have b j < 0 f and only f b j > 0. Proof. As remarked after the defnton of the matrx B T, the dagonal entres b vansh, so the statement n the lemma s clearly true for these. Now assume b j < 0 for some j. Then T j s a summand of the mddle term of the exchange trangle T U T,T/T T Snce the second map s a mnmal rght add(t/t )-approxmaton, ths means that there exsts a map T j T whch does not factor through any other object n add(t/(t T j )). In other words, there s

4 4 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE an arrow T j T n the quver of End C (T). Ths n turn mples that T s a summand of the mddle term n the exchange trangle T j U T j,t/t j T j and thus b j > 0. Proof of Theorem 1.1. Let T = n =1 T be a maxmal rgd object n C. Suppose we want to exchange the ndecomposable summand T k. Consder T = k T and the maxmal rgd object T = T Tk. We want to show that the matrces B T = (b j ) and B T = (b j ) are related by FZ matrx mutaton. Pck two ndecomposable summands T T j of T, so, j and k are all dstnct. For the purposes of ths proof we ntroduce some notaton. Let α a (T b ) = α UTa,T/Ta T b that s, the multplcty of T b as a drect summand of the mddle term n the exchange trangle endng n T a wth respect to T/T a. Smlarly, we denote the multplcty of T b n the other trangle by α a(t b ) = α U Ta,T/TaT b Note that under the assumpton of condton (c), at least one of these two numbers wll be zero for any choce of a, b. The exchange trangles for T k wth respect to T are (1) Tk T α k(t ) T α k(t j) j V k T k (2) T k T α k (T) T α k (Tj) j V k T k Note that V k and V k do not have T or T j as drect summand. These are also the exchange trangles for Tk wth respect to T, but the roles of the mddle terms are nterchanged. It follows mmedately that b k = b k, and the FZ formula holds for row k of the matrx. In the rest of the proof we study the changes n row, where k. For ths we also need the exchange trangles for T wth respect to T/T, whch are (3) T φ 1 T α (T j) j T α(t k) k V φ 2 T φ 3 α (4) T T (Tj) j T α (T k) k V φ 4 T Agan, note that V and V do not have T j or T k as drect summand. From these trangles we must collect nformaton about the exchange trangles of T wth respect to T = T/(T T k ) Tk, snce these determne the entres n row of B T. We wll consder three dfferent cases, dependng on whether b k s postve, negatve or zero. Case I: Assume b k = 0. Because of (c) ths means that T k does not appear n any of the exchange trangles for T. Then, by Lemma 1.2, we have b k = 0 as well, and by the above, b k = 0. By appealng once more to Lemma 1.2, we see that Tk does not appear n the exchange trangles for T wth respect to T. Ths s enough to establsh that the map φ 1 n trangle (3) s also a mnmal left add T-approxmaton. Smlarly, the map φ 4 n trangle (4) s a mnmal rght add T-approxmaton. Ths means that the trangles (3) and (4) are also the exchange trangles for T wth respect to T. Thus the entres n row reman unchanged and behave accordng to the FZ rule n ths stuaton. (Note also that ths proves that T s the complement of T both before and after we have exchanged a summand T k wth b k = 0.) Case II: Suppose now that b k < 0, whch means that T k appears as a summand n (3), whle α (T k) = 0. By Lemma 1.2, b k > 0, whch means that T appears n (2), not n (1). Our strategy s to construct redundant versons of the exchange trangles of T wth respect to T. We wll use the trangle (5) (T k) α(t k) ( T α(tj) ) ( ) α(t T α k(t j) k ) j V k φ 1 ( T α(tj) ) T α(t k) k

5 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 5 whch s the drect sum of α (T k ) copes of (1) and the dentty map of T α(tj). Applyng the octahedral axom to the composton of the map φ 1 n (5) and the map φ 2 n (3) yelds the followng commutatve dagram n whch the mddle two rows and mddle two columns are trangles. (5) T [ 1] (T k )α(t k) [1] (T k )α(t k) [1] χ 1 T ( ) T α(tj) T α(t k) k φ 2 T (3) φ 1 T [ 1] X ( ) ( T α(tj) T α k(t j) j V k ) α(t k ) φ T (T k )α(t k) (T k )α(t k) We now want to show that the map φ = φ 2 φ 1 s a (not necessarly mnmal) rght add T-approxmaton. Any map f : T t T where t wll factor through φ 2 snce ths s a rght add(t/t )-approxmaton, so f = φ 2 f 1. But snce φ 1 s a rght add(t)-approxmaton, f 1 factors through φ 1. Thus f factors through φ 2 φ 1 = φ. Suppose nstead that we have a map f : Tk T. Let h : Tk T α k(t j) j V k be the mnmal left addt-approxmaton for Tk. Then f = gh for some map g : T α k(t j) j V k T. Snce T s not a summand of V k, we have that T α k(t j) j V k s n add( T/T k ), and by the above, g factors through φ. We conclude that φ s a rght add T-approxmaton. Smlarly we now construct a second commutatve dagram. We use the octahedral axom on the composton of the map φ 4 n (4) and the map χ 1 n the trangle (6) (T k) α(t k) X T χ 1 (T k ) α(t k) [1] from the second column of the prevous dagram. (Note that by our assumpton, T k does not appear n (4).) (T k )α(t k) [1] (T k )α(t k) [1] χ χ 1 T T α (Tj) j V φ 4 T T [1] T Y ψ X ψ 1 T [1] (T k )α(t k) (T k )α(t k) We notce that n ths second dagram, the map χ must be zero, snce Ext 1 C(T j, Tk ) = Ext1 C(V, T k ) = 0. Therefore, the trangle splts, and ( ) Y T α (Tj) (Tk) α(t k)

6 6 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE We see that ψ n the dagram s a (not necessarly mnmal) rght add T-approxmaton as well: For any map f : U X where U add T, the composton ψ 1 f s zero snce Ext 1 C(U, T ) = 0, whch agan mples that f factors through ψ. Snce φ s a rght add T-approxmaton, X = Z T!, where T! T s the unque other ndecomposable that completes T to a maxmal rgd object. T! exsts and s unque snce T has a weak cluster structure and so satsfes (a). Consder the trangles we have constructed: ( ) ( ) α(t (7) Z T! T α(tj) T α k(t j) k ) j V k and (8) T φ T ( ) T α (Tj) (Tk )α(t k) ψ Z T! Snce φ and ψ are rght add( T)-approxmatons, we see that an automorphsm of Z splts off n both trangles, and the remanng parts are the exchange trangles for T and T! wth respect to T. So to fnd the entry b j we calculate the dfference of the multplctes of T j n the two trangles (7) and (8), snce the dfference s not affected when Z s splt off from both trangles. So f we denote by α (7) (T j ) the multplcty of T j n the mddle term of trangle (7) and smlarly for trangle (8) we get b j = α (8) (T j ) α (7) (T j ) = α (T j) (α (T j ) + α k (T j )α (T k )) = (α (T j) α (T j )) α k (T j )α (T k ) { bj when α = k (T j ) = 0,.e. when b kj 0 b j ( b kj )( b k ) = b j b kj b k when α k (T j ) > 0,.e. when b kj < 0 Also, t s clear from (7) and (8) that b k = α (T k ) = b k. Summarsng, we see that the entres n the th row change as requred by the FZ rule n ths case. Case III: Fnally we consder the case where b k > 0. Ths means that T k appears n (4), but not n (3). Furthermore, by Lemma 1.2, T appears n (1), but not n (2). The argument follows the same lnes as n Case II. Instead of (5), we use the followng trangle: (9) (T k) α (T k) [ 1] ( ) T α (Tj) T α (T k) k φ 2 ) ( T α (Tj) ( T α k (Tj) j V k ) α (T k) (T k) α (T k) whch s the drect sum of α (T k) copes of (2) and the dentty map of T α (Tj) j V. By the octahedral axom, appled to the composton of the map φ 3 n (4) and the map φ 2 n (9), we get ths commutatve dagram: (T k )α (T k) (T k )α (T k) T φ T φ 3 ( ) ( T α (Tj) φ 2 ( T α (Tj) T α k (Tj) j V k ) T α (T k) k ) α (T k ) X T T [1] T [1] (T k )α (T k) [ 1] (T k )α (T k) [ 1] By arguments dual to those n Case II, we can check that φ s a left (not necessarly mnmal) add Tapproxmaton. Detals are left to the reader.

7 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 7 Now the octahedral axom appled to the composton n the left square below gves a new dagram and a new object Y, where the second column s the same trangle as the thrd column n the prevous dagram: (T k )α (T k) (T k )α (T k) T [ 1] X ψ Y T T [ 1] T T α(tj) j χ V T (T k )α (T k) [ 1] (T k )α (T k) [ 1] Once agan, by the fact that T k s not a summand of V, and the vanshng of Ext 1 C-groups, χ n ths dagram s zero, the trangle splts, and Y (T k )α (T k) T α(tj) j and we may also conclude that ψ s a left add T-approxmaton as n the prevous case. As n Case II, we can fnd the entry b j by subtractng the multplcty of T j n the trangle nvolvng ψ from ts multplcty n the trangle nvolvng φ : b j = (α (T j ) + α k(t j )α (T k )) α (T j ) V = (α (T j) α (T j )) + α k (T j)α (T k) { bj when α = k (T j) = 0,.e. when b kj 0 b j + b kj b k when α k (T j) > 0,.e. when b kj > 0 Also, argung n a smlar way to Case II, we obtan b k = α (T k) = b k. We have shown that b j s obtaned from b j usng the FZ mutaton rule, so the proof s complete. Recall that a collecton R of non-somorphc ndecomposable objects n a trangulated category K s sad to be rgd f Ext 1 K(U, V ) = 0 for all objects U and V n R. It s called maxmal rgd f t s maxmal wth respect to ths property, that s, whenever X s an ndecomposable object of K such that Ext 1 K (U X, U X) = 0 for all objects U n R, then X s somorphc to an object of R. We say that R s functorally fnte f the full addtve subcategory wth objects gven by drect sums of objects of R s functorally fnte n the sense of [AS]. The followng more general statement can be proved usng the same proof as that for Theorem 1.1. Theorem 1.3. Let C be a Hom-fnte 2-Calab-Yau trangulated category, and let T be the collecton of functorally fnte maxmal rgd collectons n C. Assume T s non-empty and satsfes condton (c) above. Then T has a cluster structure. 2. Maxmal rgd objects n cluster categores of tubes In ths secton we wll gve a complete descrpton of the maxmal rgd objects n the cluster category of a tube, as defned n [BMRRT]. It turns out that none of these are cluster-tltng objects. In Secton 3, we wll apply the man result n Secton 1 to show that ths category provdes a model for the combnatorcs of a type B cluster algebra. We wll denote by T n the tube of rank n, where n s always understood to be at least 2. One realzaton of ths category s as the category of nlpotent representatons of a quver wth underlyng graph Ãn 1 and cyclc orentaton. We wll also thnk of of T n as the full exact subcategory generated by the tube of rank n n the module category modh, where H s the path algebra of the quver wth underlyng graph Ãn and n arrows orented n clockwse drecton and one arrow orented n antclockwse drecton. We then know that for nstance the AR-formula holds n T n : Ext 1 T n (X, Y ) D Hom Tn (Y, τx)

8 8 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE T n [ 1] T n T n [1] Fgure 1. The AR-quver of the derved category of T n ; a countable set of dsconnected tubes. There exst maps from ndecomposables n each copy to the next copy on the rght, correspondng to extensons n the tube tself. (a,b+1) (a,b) (a+1,b) (a+1,b 1) (n 1,3) (n,3) (1,3) (a,2) (n,2) (1,2) (a,1) (a+1,1) (n,1) (1,1) (2,1) Fgure 2. Coordnate system for the ndecomposable objects n the tube. We wll wrte just T for ths category f the actual value of n s not mportant. The category T s a Hom-fnte heredtary abelan category, and we can therefore apply the defnton from [BMRRT] to form ts cluster category. The AR-quver of the bounded derved category D b (T ) of T s a countable collecton of copes of the tube, one for each shft. See Fgure 1. The only maps n D b (T ) whch are not vsble as a composton of fntely many maps n the AR-quver are the maps from each T [] to T [ + 1] whch correspond to the extensons n T. The cluster category s now defned as the orbt category C Tn = D b (T n )/τ 1 [1] where τ s the AR translaton and [1] s the shft functor. Agan, we wll sometmes wrte just C T. There s a 1-1 correspondence between the ndecomposable objects n C Tn and those of T n, snce nd T n s tself a fundamental doman for the acton of τ 1 [1]. We wll often denote both an object n T n and ts orbt as an object n C Tn by the same symbol, and we wll sometmes refer to the category C Tn as a cluster tube. Snce T does not have tltng objects, t does not follow drectly from Keller s theorem [K] that C T s trangulated. However, C T s a thck subcategory of C H, or, as n [BKL1], a subcategory of the category of sheaves over a weghted projectve lne. It follows that C T s trangulated, and that the canoncal functor D b (T n ) C Tn s a trangle functor. It also follows that C T s a Hom-fnte 2-Calab- Yau category, snce cluster categores of heredtary algebras are, and that we have an AR-formula for the cluster tube as well. We wll use a coordnate system on the ndecomposable objects. We wll let (a, b) be the unque object wth socle (a, 1) and quas-length b, where the smples are arranged such that τ(a, 1) = (a 1, 1) for 1 a n. Throughout, when we wrte equatons and nequaltes whch nvolve frst coordnates outsde the doman 1,..., n, we wll mplctly assume dentfcaton modulo n. See Fgure 2. Lemma 2.1. If X and Y are ndecomposables n T, we have Hom CT (X, Y ) D Hom T (Y, τ 2 X) Hom T (X, Y ) where D denotes the k-vector space dualty Hom k (, k).

9 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 9 (a 2,b) (a 1,b) (a,b) (a+1,b) Fgure 3. For an ndecomposable X = (a, b), the Hom-hammock s llustrated by the full lnes. Shftng t one to the rght, we get the Ext-hammock. The backwards and forwards hammocks wll overlap, dependng on the rank of the tube. Proof. By the defnton of orbt categores, Hom CT (X, Y ) = Z Hom D b (T )(τ X[], Y ) Snce T s heredtary, the only possble contrbuton can be for = 1, 0: Hom CT (X, Y ) = Hom D b (T )(τx[ 1], Y ) Hom D b (T )(X, Y ) Hom D b (T )(τx, Y [1]) Hom D b (T )(X, Y ) = Ext 1 T (τx, Y ) Hom T (X, Y ) D Hom T (Y, τ 2 X) Hom T (X, Y ). In the last step, we have used the AR-formula for T. The Hom- and Ext-hammocks of an ndecomposable object X (that s, the supports of Hom CT (X, ) and Ext 1 C T (X, )) are llustrated n Fgure 3. For two ndecomposables X and Y n C T, let X and Ŷ be ther premages n T. Then the mages n C T of maps X Ŷ wll be called T -maps and the mages of maps X τ 1 Ŷ [1] wll be called D-maps. We wll repeatedly use the fact that the exstence of a D-map X Y s equvalent to the exstence of a T -map Y τ 2 X. The followng lemma s necessary for understandng endomorphsm rngs of objects n C T. For any ndecomposable X C T the ray startng n X = (a, b) s the nfnte sequence of rreducble T -maps R X : (a, b) (a, b + 1) (a, b + ). Smlarly, the coray endng n X s the nfnte sequence of rreducble T -maps C X : (a, b + ) (a 1, b + 1) (a, b). Note that the sum of the coordnates s constant (as always, modulo n) n a coray. Lemma 2.2. Let X and Y be two ndecomposable objects n C T. A D-map n Hom CT (X, Y ) factors through the ray R X startng n X and the coray C Y endng n Y. Proof. Let X and Y be objects n T whch correspond to X and Y n C T. Let Ỹ = τ 1 Y [1] n D b (T ). We wll prove that a map f Hom Db (T )(X, Ỹ ) factors through the ray startng n X n the AR-quver of D b (T ). Ths wll mply that the mage of f n C T factors through the ray startng n X n the AR-quver of C T. Snce f : X Ỹ s not an somorphsm n Db (T ) for any choce of X and Y, t s enough to show that f factors through the rreducble map g whch forms the start of the ray, and we can do ths by nducton on the quas-length of X. If ql X = 1, then g s a left almost splt map n D b (T ), and the clam holds.

10 10 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE Suppose now that qlx 2. There s an almost splt trangle n D b (T ) g Z h X τ 1 X g Z h Snce f s not an somorphsm, f factors through the left almost splt map g g. We have that qlz = ql X 1, so by nducton we can assume that any map n Hom Db (T )(Z, Ỹ ) factors through h. By the mesh relaton, h g factors through g. Thus f factors through g. The other asserton s proved dually. From now on, suppose T s a maxmal rgd object n C Tn. We wll gve a complete descrpton of the possble confguratons of ndecomposable summands of T. As a consequence, we wll fnd that there are no cluster-tltng objects n C T. Lemma 2.3. All summands T of T must have qlt n 1. Proof. If ql T n, then there s a T -map T τt = T [1], so n partcular Ext 1 C Tn (T, T ) = Hom CT (T, T [1]) 0, and T s not a summand of a rgd object. In what follows, for any ndecomposable X = (a, b) n C T, the wng determned by X wll mean the set of ndecomposables whose poston n the AR-quver s n the trangle whch has X on top, that s, (a, b ) such that a a and a + b a + b. We wll use the notaton W X for ths. Lemma 2.4. If T 0 s an ndecomposable summand of T such that no other summand of T has hgher quas-length, then all the summands of T are n the wng W T0 determned by T 0. Proof. We assume, wthout loss of generalty, that T 0 has coordnates T 0 = (1, q). Let T 1 = (s, m) be a summand of T whch maxmses the sum x+y wth (x, y) the coordnates of summands of T. Recall that such sums are constant modulo n on corays, so ths means that all summands of T are n the regon bounded by the ray R (1,1) and the coray C (s+m 1,1) whch passes through T 1 and ends n the quassmple (s + m 1, 1). We note that for n = 2, one sees mmedately that maxmal rgd objects have only one ndecomposable summand, and then T 1 = T 0 and the clam s trval. So for the rest of the proof, let n 3. Assume that T 1 s not n W T0. See Fgure 4. Snce Ext 1 C T (T 0, T 1 ) = 0, we have q + 2 s and s + m n. Therefore all summands of T st nsde the wng W X where X = (1, s + m 1). Furthermore, snce s + m 1 n 1, X has no self-extensons and X has no extensons wth T, so Ext 1 C T (T X, T X) = 0. Hence X s a summand of T. Snce the quas-length of X s at least the quas-length of T 0, and X and T 0 are on the same ray, we must have X = T 0, but ths contradcts the fact that T 1 s not n W T0. We conclude that all summands are n W T0. For our maxmal rgd object T we wll n the rest of ths secton denote by T 0 the unque summand of maxmal quas-length, and we wll sometmes call t the top summand. Lemma 2.5. The quas-length of T 0 s n 1. Proof. Suppose qlt 0 = l 0 < n 1. Then T 0 = (m, l 0 ) for some m {1,..., n}. The object Y = (m, n 1) wll satsfy Ext 1 C Tn (T Y, T Y ) = 0. But ths contradcts the fact that T s maxmal rgd. Wth the preceedng seres of lemmas at our dsposal, we get the followng. Proposton 2.6. There s a natural bjecton between the set of maxmal rgd objects n C Tn and the set { tltng modules over A } {1,..., n} where A s a lnearly orented quver of Dynkn type A n 1.

11 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 11 X T 0 T 0 T 1 1 q s s+m 1 n Fgure 4. The shaded regon s the Ext-hammock of T 0. In the proof of Lemma 2.4, f T 1 s outsde the wng determned by T 0, then T X s rgd. Note that qlx = s + m 1 n 1, snce Ext 1 C Tn (T 0, T 1 ) = 0. Proof. We have already establshed that all the summands of T must be n the wng determned by the summand T 0 of maxmal quas-length. Ths wng has exactly the same shape as the AR-quver of ka. One can see easly that for an ndecomposable object n one such wng, the restrcton of the Ext-hammock to the wng exactly matches the (forwards and backwards) Ext-hammocks of the correspondng ndecomposable module n the AR-quver of ka. Thus the possble arrangements of parwse Ext-orthogonal ndecomposable objects nsde the wng match the possble arrangements of parwse Ext-orthogonal ndecomposable modules n the AR-quver of ka. Snce we have n choces for the top summand, we get the bjecton by mappng a maxmal rgd object n the cluster tube to the par consstng of the correspondng tltng module over ka and the frst coordnate of ts top summand. A rgd object C n a trangulated 2-CY category C s called cluster-tltng f Ext 1 C(C, X) = 0 mples that X add C. In partcular, all cluster tltng objects are maxmal rgd. For cluster categores arsng from module categores of fnte dmensonal heredtary algebras, the opposte mplcaton s also true, namely that all maxmal rgd objects are cluster-tltng. The cluster tubes provde examples n whch ths s not the case. Corollary 2.7. The category C T has no cluster-tltng objects. Frst a techncal lemma: Lemma 2.8. Let T be a maxmal rgd object wth top summand T 0, and X an ndecomposable whch s not n W T0 and not n W τt0. Then Hom CT (T, X) = 0 f and only f Hom CT (T 0, X) = 0. Proof. For any ndecomposable object A and an ndecomposable B n the wng W A determned by A, the restrcton of the Hom-hammock of B to nd C T \ (W A W τa ) s contaned n the Hom-hammock of A. (See Fgure 3.) Proof of Corollary 2.7. Let T be maxmal rgd, and assume wthout loss of generalty that T 0 = (1, n 1) s the top summand. Then from the shape of the Hom-hammock of T 0 we see that there are no T -maps from T 0 to any ndecomposable object on the ray R (n,1). Smlarly, there are no D-maps from T 0 to any of the ndecomposables on the coray C (n 2,1). Consder the object X = (n, 2n 1). Then X sts on R (n,1) and the coray C (n 2,1), as we see snce the sum of the coordnates s congruent to n 1 mod n. So there are no maps n C T from T 0 to X. Therefore, by Lemma 2.8, we have Hom CT (T, X) = 0. Consequently, f Y = τ 1 X, then Ext 1 C T (T, Y ) = Hom CT (T, X) = 0. But Y W T0, so Y addt, so T s not cluster-tltng.

12 12 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE Y 1 Y 2 T T T j 1 a a+2 n 1 a+1 a+b+1=n 1 Fgure 5. A stuaton from the proof of Proposton 2.9. When T = (1, a), the exstence of a T -map to τ 2 T j mples that T j = (a + 2, b). By the same reasonng, f there s a D-map also n the other drecton, then a+b+1 = n 1. The only possble postons for T 0 are Y 1 = (1, n 1) and Y 2 = (a + 2, n 1). As remarked after the defnton of cluster structures n Secton 1, an nterpretaton of condton (c) n the defnton s that the quver of the endomorphsm rng of the maxmal rgd object does not have 2-cycles. Ths s the case here: Proposton 2.9. The set of maxmal rgd objects n C T has a cluster structure. Proof. Snce n 2 there are rgd ndecomposables, and snce there are only a fnte number of rgd ndecomposables by Lemma 2.3, the set of maxmal rgd objects s non-empty. By the fact that C T s a Hom-fnte 2-Calab-Yau trangulated category, and Theorem 1.1, t s only necessary to show that for any maxmal rgd T n C T, there are no 2-cycles n the quver of End CT (T). Recall that the vertces of the quver are n correspondence wth the ndecomposable summands of T, and the arrows correspond to maps that are rreducble n addt. We frst recall from Lemma 2.4 that all the summands of T le n the wng W T0, where T 0 s the top summand of T, and from Lemma 2.5 that T 0 has quaslength n 1. Let T and T j be two non-somorphc ndecomposable summands of T. Assume frst that one of these s somorphc to T 0, say T T 0. If X s an ndecomposable n W T0 such that Hom CT (X, T 0 ) 0, then T 0 must be on the ray R X. But then Hom CT (T 0, X) = 0 unless X T 0. So n partcular t s mpossble that Hom CT (T 0, T j ) and Hom CT (T j, T 0 ) are both non-zero, and therefore there s no 2-cycle n the quver of End CT (T) traversng the vertex correspondng to T 0. We now consder the case when nether T nor T j s somorphc to T 0. Recall that the exstence of a non-zero D-map X Y s equvalent to the exstence of a non-zero T -map Y τ 2 X. Suppose frst that there s a non-zero T -map T T j. If there was also a non-zero T -map T τ 2 T j, then there would be a non-zero T -map T τt j. But ths s mpossble, snce Ext 1 T (T j, T ) = 0. So f there s a non-zero T -map T T j, there cannot be a non-zero D-map T j T. If there s a non-zero T -map T T j, then, snce T and T j both st n a wng of heght n 1, there cannot be a non-zero T -map T j T. We therefore conclude that f there s a 2-cycle traversng the vertces correspondng to T and T j, then one of the arrows corresponds to a D-map T T j, and the other corresponds to a D-map T j T, and both these maps are rreducble n addt. So t remans to show that f there are non-zero T -maps both T τ 2 T j and T j τ 2 T, then at least one of the correspondng D-maps must be reducble n addt. Assume therefore that two such maps exst. Wthout loss of generalty we can also assume that T = (1, a) for some a n 2, by f necessary redefnng the coordnates. Then, snce there s a T -map T τ 2 T j, and there are no maps T τt j, we fnd that T j must necessarly have coordnates (a + 2, b) for some b n 1. So T j sts on the coray C (a+b+1,1). See Fgure 5. Smlarly, snce there s also a T -map T j τ 2 T, the summand T must be on the neghbourng ray beneath the Ext-hammock of T j, so snce T = (1, a), we must have that a + b + 1 = n 1, and n partcular that T j sts on the coray C (n 1,1). See agan Fgure 5.

13 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 13 Note now that W (1,n 1) contans both T and T j, and T sts on the left hand edge and T j sts on the rght hand edge. By symmetry, W (a+2,n 1) also contans both T and T j, wth T j on the left hand edge and T on the rght hand edge. It follows that no other wng of heght n 1 contans both T and T j, and thus T 0 = (1, n 1) or T 0 = (a + 2, n 1). If we assume that T 0 = (1, n 1), then by Lemma 2.2, the D-map T T j factors through both the ray R T and the coray C Tj. In partcular, t factors through the T -map from T to T 0, and does not correspond to an arrow n the quver of End CT (T). Smlarly, f T 0 = (a + 2, n 1), then the D-map T j T does not correspond to an arrow n the quver. In any case, one of the D-maps s reducble. We conclude that there are no 2-cycles n the quver. Remark One should notce that ths set does not satsfy the defnton of cluster structures n [BIRS]. The reason for ths s that the quver of End CT (T) wll have a loop for all maxmal rgd T. Ths s because there s a non-zero D-map from T 0 to tself. In addton, we have that the only ndecomposable objects n the wng of T 0 whch T 0 has non-zero maps to n C T are those on the rght hand edge of the wng, and the only ndecomposable objects n the wng of T 0 whch have non-zero maps n C T to T 0 are those on the left hand edge of the wng. It follows that the D-map from T 0 to tself does not factor through any other ndecomposable object n the wng, and therefore t does not factor through any other ndecomposable drect summand of T. 3. Relatonshp to type B cluster algebras Cluster algebras were ntroduced n [FZ1]; see, for example, [FR] or [FZ4] for an ntroducton. A smplcal complex was assocated n [FZ3] to any fnte root system, and t was conjectured there and later proved n [CFZ] that these smplcal complexes are the face complexes of certan polytopes whch were called generalsed assocahedra. In the fnte type classfcaton of cluster algebras [FZ2], t was shown that a cluster algebra s of fnte type f and only f ts cluster complex s one of these smplcal complexes. The generalsed assocahedron assocated to a type B root system turned out to be the cyclohedron, also known as the Bott-Taubes polytope [BT]. Ths polytope was ndependently dscovered by Smon [S]. We wll now recall the descrpton of the exchange graph from [FZ3] (whch corresponds to the geometrc descrpton of the correspondng polytope n [S]). Let G n denote a regular 2n-gon. The set of cluster varables n a type B n 1 cluster algebra s n bjecton wth the set D n of centrally symmetrc pars of dagonals of G n, where the dameters are ncluded as degenerate pars. Under ths bjecton, the clusters correspond to the centrally symmetrc trangulatons of G n, and exchange of a cluster varable corresponds to flppng ether a par of centrally symmetrc dagonals or a dameter. In ths secton we defne a bjecton from the set of ndecomposable rgd objects n the cluster tube C Tn to the set D n. Ths map nduces a correspondence between the maxmal rgd objects and centrally symmetrc trangulatons whch s compatble wth exchange. Thus rgd objects n C Tn model the cluster combnatorcs of type B n 1 cluster algebras. We label the corners of G n clockwse, say, from 1 to 2n: 2n 1 2 For two corners labelled a and b (whch are nether equal nor neghbours), we denote by [a, b] the correspondng dagonal. Thus [a, b] = [b, a], and the centrally symmetrc pars of dagonals are gven as ([a, b], [a + n, b + n]). (Here and n what follows, we reduce modulo 2n f necessary.) Snce ths par 3

14 14 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE Fgure 6. The AR-quver of C T4, wth the ndecomposable rgd objects replaced by ther mages under the map δ. n+b+2 1 c+n 1 n+b+2 n+b+2 1 c+n c+d+1 n c+n c c+d+1 c+d+1 n c+d+1 c n+1 c+d+1 b+2 n+1 c b+2 n+1 b+2 Fgure 7. Three cases n the proof of Proposton 3.2. Left, condton () only holds; mddle, condton () only holds; rght, condtons () and () both hold. s unquely determned by ether of the two dagonals, we wll sometmes denote the par by one of ts representatves. Now, for each ndecomposable rgd object n C Tn we assgn a centrally symmetrc par of dagonals as follows: δ (a, b) ([a, a + b + 1], [a + n, a + b n]) Note that the pars of dagonals assgned to ndecomposable objects n the same ray or coray of C Tn all share a centrally symmetrc par of corners. Objects of quas-length 1 correspond to the shortest dagonals, whle the objects of quas-length n 1 correspond to the dameters. See Fgure 6. The followng fact s readly verfed. Lemma 3.1. The map δ defned above s a bjecton from the set of ndecomposable rgd objects n C Tn to the set D n of centrally symmetrc pars of dagonals of G n. Proposton 3.2. Let T 1 = (a, b) and T 2 = (c, d) be ndecomposable rgd objects n C Tn. Then the number of crossng ponts of δ(a, b) and δ(c, d) s equal to 2 dmext 1 C T (T 1, T 2 ). Proof. Wthout loss of generalty, we may assume that a = 1. We have δ(1, b) = ([1, b + 2], [n + 1, b + n + 2]) δ(c, d) = ([c, c + d + 1], [c + n, c + d + n + 2]). It s easy to check that δ(1, b) and δ(c, d) cross f and only f one of the followng two condtons holds: () 1 < c < b + 2 and c + d > b + 1; () 1 < c + d + 1 n < b + 2 and 1 < c < n + 1 Furthermore, the number of crossng ponts s 4 when both condtons hold, and s 2 f only one holds. See Fgure 7. From the structure of the tube, t can be checked that condton () holds f and only f Hom T (T 1, τt 2 ) 0, and n ths case dmhom T (T 1, τt 2 ) = 1. Smlarly, condton () holds f and only f Hom T (T 2, τt 1 )

15 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 15 (2,n 1) () and () hold (b+1,n 1) (1,b) (2,b) () only holds () only holds (n,b) (1,b) (b+1,n b) (2,1) (b+1,1) (n,1) (b 1,1) Fgure 8. Three stuatons for extensons wth the object (1, b), as n the proof of Proposton 3.2 0, and n ths case dmhom T (T 2, τt 1 ) = 1. By Lemma 2.1 and the Auslander-Reten formula, and the result follows. Ext 1 C T (T 1, T 2 ) = D Hom T (T 2, τt 1 ) Hom T (T 1, τt 2 ) Corollary 3.3. The map δ nduces a bjecton between the ndecomposable rgd objects n C Tn and the cluster varables n a type B n 1 cluster algebra, and under ths bjecton, the maxmal rgd objects correspond to the clusters. Proof. By the work of [FZ2, FZ3], we need to show that the mage under δ of the summands of a maxmal rgd object concdes wth a set of pars of dagonals whch form a centrally symmetrc trangulaton of G n. Ths s clear from Lemma 3.1 and Proposton 3.2. Now let T nt be the zg-zag maxmal rgd object t T nt = (, n 2 + 1) (, n 2) =1 where t = n n 1 2 for n even and t = 2 for n odd, and any expresson wth zero n the last coordnate s to be dsregarded. Proposton 3.4. The Cartan counterpart of the matrx B Tnt s the Cartan matrx for the root system of type B n 1. Proof. We recall that the Cartan counterpart of a matrx B = (b j ) s the matrx A(B) = (a j ) gven by a = 2 and a j = b j when j. For ths proof, we set T to be the summand of T nt wth quas-length n, for = 1,..., n 1. In partcular the top summand s T 1. For convenence, we defne T n = 0. For each T wth even and 2 n 1, the exchange trangles are T 0 T T T 1 T +1 T whle for odd and 1 < n 1, the exchange trangles are T T 1 T +1 T T 0 T In the quver of End CT (T nt ), there s a loop on the vertex correspondng to the summand T 1. (See Remark 2.10.) However, twce around ths loop s a zero relaton, so the exchange trangles for ths summand are T1 T 2 T 2 T 1

16 16 ASLAK BAKKE BUAN, ROBERT J. MARSH, AND DAGFINN F. VATNE T 1 0 T 1 So the matrx of the exchange trangles s B Tnt = ( 1) n ( 1) n 1 0 and the Cartan counterpart s A(B Tnt ) = Notng that for a cluster algebra of type B, a cluster determnes ts seed [FZ2], we have thus proved the followng theorem: Theorem 3.5. There s a bjecton between the ndecomposable rgd objects of a cluster tube of rank n and the cluster varables of a cluster algebra of type B n 1, nducng a bjecton between the maxmal rgd objects of the cluster tube and the clusters of the cluster algebra. Furthermore, the exchange matrx of a seed concdes wth the matrx assocated to the correspondng maxmal rgd object n Secton 1. Proof. The frst part of the theorem has been shown above (Corollary 3.3). The second part follows from Theorem 1.1 and Proposton 3.4, notng that the exchange matrx correspondng to the ntal root cluster { α 1, α 2,..., α n 1 } of type B n 1 n [FZ2] s the matrx B Tnt arsng from the exchange trangles correspondng to the maxmal rgd object T nt n Proposton 3.4. (See also Fgure 5 n [FZ3]). Acknowledgements The frst and thrd named author wsh to thank Robert Marsh and the School of Mathematcs at the Unversty of Leeds for ther knd hosptalty n the sprng of The frst author s supported by an NFR Storforsk-grant. The second author acknowledges support from the EPSRC, grant number EP/C01040X/2, and s also currently an EPSRC Leadershp Fellow, grant number EP/G007497/1. We would also lke to thank the referee for valuable comments. References [ABS] I. Assem, T. Brüstle, R. Schffler Cluster-tlted algebras as trval extensons, Bull. London Math. Soc. 40 (1), (2008) [AS] M. Auslander, S. Smalø Preprojectve modules over Artn algebras, J. Algebra 66 (1), (1980) [BKL1] M. Barot, D. Kussn, H. Lenzng The Grothendeck group of a cluster category, J. Pure Appl. Algebra 212 (1), (2008) [BKL2] M. Barot, D. Kussn, H. Lenzng The cluster category of a canoncal algebra, to appear n Trans. Amer. Math. Soc, preprnt v. 3 arxv:math.rt/ (2008) [BT] R. Bott, C. Taubes On the self-lnkng of knots. Topology and physcs, J. Math. Phys. 35 (10), (1994) [BIRS] A. B. Buan, O. Iyama, I. Reten, J. Scott Cluster structures for 2-Calab-Yau categores and unpotent groups, to appear n Composto Math., preprnt v.3 arxv:math/ (2007) [BMR] A. B. Buan, R. Marsh, I. Reten Cluster mutaton va quver representatons, Comment. Math. Helv. 83 (1), (2008)

17 CLUSTER STRUCTURES FROM 2-CALABI-YAU CATEGORIES WITH LOOPS 17 [BMRRT] A. B. Buan, R. Marsh, M. Reneke, I. Reten, G. Todorov Tltng theory and cluster combnatorcs, Adv. Math. 204 (2), (2006) [BCKMRT] A. B. Buan, P. Caldero, B. Keller, R. Marsh, I. Reten, G. Todorov Appendx to Clusters and seeds for acyclc cluster algebras, Proc. Amer. Math. Soc. 135 (10), (2007) [BIKR] I. Burban, O. Iyama, B. Keller, I. Reten Cluster tltng for one-dmensonal hypersurface sngulartes, Adv. Math. 217 (6), (2008) [CC] P. Caldero, F. Chapoton Cluster algebras as Hall algebras of quver representatons, Comment. Math. Helv. 81 (3), (2006) [CCS] P. Caldero, F. Chapoton, R. Schffler Quvers wth relatons arsng from clusters (A n case), Trans. Amer. Math. Soc. 358 (3), (2006) [CK] P. Caldero, B. Keller From trangulated categores to cluster algebras II, Ann. Sc. École Norm. Sup. (4) 39 (6), (2006) [CFZ] F. Chapoton, S. Fomn, A. Zelevnsky Polytopal realzatons of generalzed assocahedra, Canad. Math. Bull. 45 (4), (2002) [FR] S. Fomn, N. Readng Root systems and generalzed assocahedra, IAS/Park Cty Math. Ser. 13, (2004) [FZ1] S. Fomn, A. Zelevnsky Cluster algebras I: Foundatons, J. Amer. Math. Soc. 15 (2), (2002) [FZ2] S. Fomn, A. Zelevnsky Cluster algebras II: Fnte type classfcaton, Invent. Math. 154 (1), (2003) [FZ3] S. Fomn, A. Zelevnsky Y-systems and generalzed assocahedra, Ann. Math. (2) 158 (3), (2003) [FZ4] S. Fomn, A. Zelevnsky Cluster algebras: Notes for the CDM-03 conference, CDM 2003: Current Developments n Mathematcs, Internatonal Press (2004) [GLS] C. Geß, B. Leclerc, J. Schröer Rgd modules over preprojectve algebras, Invent. Math. 165 (3), (2006) [IY] O. Iyama, Y. Yoshno Mutaton n trangulated categores and rgd Cohen-Macaulay modules, Invent. Math. 172 (1), (2008) [K] B. Keller On trangulated orbt categores, Doc. Math. 10, (2005) [KR] B. Keller, I. Reten Cluster-tlted algebras are Gorensten and stably Calab-Yau, Adv. Math. 211 (1), (2007) [MRZ] R. Marsh, M. Reneke, A. Zelevnsky Generalzed assocahedra va quver representatons, Trans. Amer. Math. Soc. 355 (10), (2003) [S] R. Smon A type-b assocahedron, Adv. n Appl. Math. 30, 2-25 (2003) Insttutt for matematske fag, Norges teknsk-naturvtenskapelge unverstet, N-7491 Trondhem, Norway E-mal address: aslakb@math.ntnu.no Department of Pure Mathematcs, Unversty of Leeds, Leeds LS2 9JT, England E-mal address: marsh@maths.leeds.ac.uk Insttutt for matematske fag, Norges teknsk-naturvtenskapelge unverstet, N-7491 Trondhem, Norway E-mal address: dvatne@math.ntnu.no