Finite mutation classes of coloured quivers
aa r X i v : . [ m a t h . R T ] J a n FINITE MUTATION CLASSES OF COLOURED QUIVERS
HERMUND ANDR´E TORKILDSEN
Abstract.
We consider the general notion of coloured quiver mutation andshow that the mutation class of a coloured quiver Q , arising from an m -clustertilting object associated with H , is finite if and only if H is of finite or tamerepresentation type, or it has at most 2 simples. This generalizes a resultknown for 1-cluster categories. Introduction
Mutation of skew-symmetric matrices, or equivalently quiver mutation, is verycentral in the topic of cluster algebras [FZ]. Quiver mutation induces an equivalencerelation on the set of quivers. The mutation class of a quiver Q consists of all quiversmutation equivalent to Q . In [BR] it was shown that the mutation class of an acyclicquiver Q is finite if and only if the underlying graph of Q is either Dynkin, extendedDynkin or has at most two vertices.Cluster categories were defined in [BMRRT] in the general case and in [CCS] inthe A n -case as a categorical model of the combinatorics of cluster algebras. Somecluster categories have a nice geometric description in terms of triangulations ofcertain polygons, see [CCS, S]. This was used in [To, BTo] to count the number ofquivers in the mutation classes of quivers of Dynkin type A and D . In [BRS] theyused different methods to count the number of quivers in the mutation classes ofquivers of type e A .A generalization of cluster categories, the m -cluster categories, have been investi-gated by several authors. See for example [BM1, BM2, BT, IY, K, T, W, Z, ZZ]. In[BT] mutation on coloured quivers was defined, and we can define mutation classesof coloured quivers. It is a natural question to ask when the mutation classes ofcoloured quivers are finite. In this paper we want to show the following theorem,analogous to the main theorem in [BR]. Theorem.
Let k be an algebraically closed field and Q a connected finite quiverwithout oriented cycles. The following are equivalent for H = kQ . (1) There are only a finite number of basic m -cluster tilted algebras associatedwith H , up to isomorphism. (2) There are only a finite number of Gabriel quivers occurring for m -clustertilted algebras associated with H , up to isomorphism. (3) H is of finite or tame representation type, or has at most two non-isomorphicsimple modules. (4) There are only a finite number of τ -orbits of cluster tilting objects associatedwith H . (5) There are only a finite number of coloured quivers occurring for m -clustertilting objects associated with H , up to isomorphism. (6) The mutation class of a coloured quiver Q , arising from an m -cluster tiltingobject associated with H , is finite. Background
Let H = kQ be a finite dimensional hereditary algebra over an algebraicallyclosed field k , with Q a quiver with n vertices. The cluster category was definedin [BMRRT] and independently in [CCS] in the A n case. Consider the boundedderived category D b ( H ) of mod H . Then the cluster category is defined as the orbitcategory C H = D b ( H ) /τ − [1], where τ is the Auslander-Reiten translation and [1]is the shift functor.As a generalization of cluster categories, we can consider the m -cluster categoriesdefined as C mH = D b ( H ) /τ − [ m ]. The m -cluster category was shown in [K] to betriangulated. The m -cluster category is a Krull-Schmidt category, an ( m + 1)-Calabi-Yau category, and it has an AR-translate τ = [ m ]. The indecomposableobjects in C mH are of the form X [ i ], with 0 ≤ i < m , where X is an indecomposable H -module, and of the form P [ m ], where P is a projective H -module.An m -cluster tilting object is an object T in C mH with the property that X isin add T if and only if Ext i C mH ( T, X ) = 0 for all i ∈ { , , ..., m } . It was shown in[W, ZZ] that an object which is maximal m -rigid, i.e. it has the property that X ∈ add T if and only if Ext i C mH ( T ⊕ X, T ⊕ X ) = 0 for all i ∈ { , , ..., m } , is alsoan m -cluster tilting object. They also showed that an m -cluster tilting object T always has n non-isomorphic indecomposable summands.An almost complete m -cluster tilting object ¯ T is an object with n − i C mH ( ¯ T , ¯ T ) = 0 for i ∈{ , , ..., m } . It is known from [W, ZZ] that any almost complete m -cluster tiltingobject has exactly m + 1 complements, i.e. there exist m + 1 non-isomorphicindecomposable objects T ′ such that ¯ T ⊕ T ′ is an m -cluster tilting object.Let ¯ T be an almost complete m -cluster tilting object and denote by T ( c ) k , where c ∈ { , , , ..., m } , the complements of ¯ T . In [IY] it is shown that the complementsare connected by m + 1 exchange triangles T ( c ) k → B ( c ) k → T ( c +1) k → , where B ( c ) k are in add ¯ T .An m -cluster tilted algebra is an algebra of the form End C mH ( T ), where T is an m -cluster tilting object in C mH .2. Coloured quiver mutation
In the case when m = 1 there is a well-known procedure for the exchange of inde-composable direct summands of a cluster-tilting object. Given an almost completecluster-tilting object, there exist exactly two complements, and the correspondingquivers are given by quiver mutation. For an arbitrary m ≥
1, the procedure isa little more complicated. Since an almost complete m -cluster tilting object has,up to isomorphism, exactly m + 1 complements, the Gabriel quiver does not giveenough information to keep track of the exchange procedure. Buan and Thomastherefore defined a class of coloured quivers in [BT], and they define a mutationprocedure on such quivers to model the exchange on m -cluster tilting objects. Inthis section we recall some results from this paper.To an m -cluster tilting object T , Buan and Thomas associate a coloured quiver Q T , with arrows of colours chosen from the set { , , , ..., m } . For each indecom-posable summand of T there is a vertex in Q T . If T i and T j are two indecomposablesummands of T corresponding to vertex i and j in Q T , there are r arrows from i to j of colour c , where r is the multiplicity of T j in B ( c ) i .They show that such quivers have the following properties. INITE MUTATION CLASSES OF COLOURED QUIVERS 3 (1) The quiver has no loops.(2) If there is an arrow from i to j with colour c , then there exist no arrowfrom i to j with colour c ′ = c .(3) If there are r arrows from i to j of colour c , then there are r arrows from j to i of colour m − c .They also define coloured quiver mutation, and they give an algorithm for theprocedure. Let Q = Q T , for an m -cluster tilting object T , be a coloured quiver andlet j be a vertex in Q . The mutation of Q at vertex j is a quiver µ j ( Q ) obtainedas follows.(1) For each pair of arrows i ( c ) / / j (0) / / k where i = k and c ∈ { , , ..., m } , add an arrow from i to k of colour c andan arrow from k to i of colour m − c .(2) If there exist arrows of different colours from a vertex i to a vertex k , cancelthe same number of arrows of each colour until there are only arrows of thesame colour from i to k .(3) Add one to the colour of all arrows that goes into j , and subtract one fromthe colour of all arrows going out of j .See Figure 1 for an example.1 (0) ( ( (2) h h (0) ( ( (2) h h µ −→ (2) ( ( (0) h h (0) ( ( (2) h h µ −→ (1) ( ( (1) h h (0) ( ( (2) h h Figure 1.
Examples of mutation of coloured quivers for Dynkintype A and m = 2.In [BT] the following theorem is proved. Theorem 2.1.
Let T = ⊕ ni =1 T i be an m -cluster tilting object in C mH . Let T ′ = T /T j ⊕ T (1) j be an m -cluster tilting object where there is an exchange triangle T j → B (0) j → T (1) j → . Then Q T ′ = µ j ( Q T ) . The quiver obtained from Q T by removing all arrows of colour different from 0 isthe Gabriel quiver of the m -cluster tilted algebra End C mH ( T ). Quivers of m -clustertilted algebras can be reached by repeated coloured quiver mutation [ZZ] (see also[BT]). Proposition 2.2.
Any m -cluster tilting object can be reached from any other m -cluster tilting object via iterated mutation. They obtain the following corollary.
Corollary 2.3.
For an m -cluster category C mH of the acyclic quiver Q , all quiversof m -cluster tilted algebras are given by repeated mutation of Q . Let us always denote by Q G the Gabriel quiver of the coloured quiver Q . In thispaper we are only interested in coloured quivers which arises from an m -clustertilting object. Let Q G be an acyclic quiver and Q the coloured quiver obtainedfrom Q G by adding the necessary arrows of colour m , i.e. if there exist r arrowsfrom i to j of colour 0, then add r arrows from j to i of colour m . Then the HERMUND ANDR´E TORKILDSEN quivers which arises from m -cluster tilting objects are exactly the quivers mutationequivalent to Q .Let Q be a coloured quiver with arrows only of colour 0 and m , as above, andwhere the underlying graph of the Gabriel quiver Q G is of Dynkin type ∆. Thencertainly Q G is a quiver of an m -cluster tilted algebra. Let us call the set of quiversmutation equivalent to Q the mutation class of type ∆. Certainly, all orientationsof ∆ (as a Gabriel quiver) is in the mutation class of type ∆.Figure 2 shows all non-isomorphic coloured quivers in the mutation class of type A for m = 2. 1 (0) ( ( (2) h h (0) ( ( (2) h h (2) ( ( (0) h h (0) ( ( (2) h h (1) ( ( (1) h h (0) ( ( (2) h h (0) ( ( (2) h h (1) ( ( (1) h h (0) ( ( (2) h h (2) ( ( (0) h h (1) ( ( (1) h h (1) ( ( (1) h h (1) ( ( (0) D D (1) h h (1) ( ( (2) (cid:4) (cid:4) (1) h h Figure 2.
All non-isomorphic coloured quivers in the mutationclass of A for m = 2.We note that in a mutation class, there can be several non-isomorphic colouredquivers with the same underlying Gabriel quiver, and that the Gabriel quiver of an m -cluster tilted algebra might be disconnected.To any m -cluster tilting object T there exist a coloured quiver Q T , but we alsohave the following. Lemma 2.4.
Suppose Q is a coloured quiver in some mutation class of a quiver ofan m -cluster tilted algebra. Then there exist an m -cluster tilting object T such that Q = Q T .Proof. This follows directly from the corollary, since mutation of m -cluster tiltingobjects corresponds to mutation of coloured quivers. (cid:3) We know that [ i ] is an equivalence on the m -cluster category for all integers i .In particular, τ = [ m ] is an equivalence. Proposition 2.5. If T is an m -cluster tilting object, then Q T is isomorphic to Q T [ i ] for all i Proof.
It is enough to prove that Q T is isomorphic to Q T [ ± . Suppose there are r arrows in Q T from i to j with colour c . Let T i and T j be the indecomposable directsummands of T corresponding to vertex i and j in Q T respectively. Let ¯ T = T /T i INITE MUTATION CLASSES OF COLOURED QUIVERS 5 be the almost complete m -cluster tilting object obtained from T by removing T i .Then there exist an exchange triangle T ( c ) i → B ( c ) i → T ( c +1) i → with B ( c ) i in add( ¯ T ). There are r arrows from i to j , with colour c , so hence T j hasmultiplicity r in B ( c ) i . Clearly T i [1] and T j [1] are indecomposable direct summandsof T [1] and we have the exchange triangle T ( c ) i [1] → B ( c ) i [1] → T ( c +1) i [1] → . Since T j has multiplicity r in B ( c ) i , T j [1] has multiplicity r in B ( c ) i [1]. It followsthat there are r arrows in Q T [1] from i to j with colour c . The same proof holdsfor [ − (cid:3) Finiteness of the number of non-isomorphic m -cluster tiltedalgebras In [BR] the authors showed that if Q is a finite quiver with no oriented cycles,then there is only a finite number of quivers in the mutation class of Q if andonly if the underlying graph of Q is Dynkin, extended Dynkin or has at most twovertices. In these cases there are only a finite number of non-isomorphic cluster-tilted algebras of some fixed type. In this section we want to prove an analogousresult for coloured quivers by generalizing the results and proofs in [BR].Let H = kQ be a finite dimensional hereditary algebra. We know that H is offinite representation type if and only if the underlying graph of Q is Dynkin. Fur-thermore, H is tame if and only if the underlying graph of Q is extended Dynkin.Objects in the module category of H , when H is of infinite type, are either prepro-jective, preinjective or regular. In the case when H is tame, the regular componentsof the AR-quiver are disjoint tubes of the form Z A ∞ / (cid:10) τ i (cid:11) for some i , and in thewild case they are of the form Z A ∞ .If X is a preprojective or preinjective H -module, it is known that X is rigid, i.e.Ext H ( X, X ) = 0. The following is a well-known result, see for example [R].
Lemma 3.1.
Let H = kQ be a finite dimensional hereditary algebra of infiniterepresentation type, then if H has exactly two simples, no indecomposable regularobject is rigid. In [W] it was shown that if T is an m -cluster tilting object in C mH , then it isinduced from a tilting object in mod H ∨ mod H [1] ∨ ... ∨ mod H [ m − H is derived equivalent to H . If H is of finite or tame representation type, it wasshown in [BR] that for each indecomposable projective H -module P , there are onlya finite number of indecomposable objects X such that Ext C H ( X, P ). Lemma 3.2.
Let P [ i ] be a shift of an indecomposable projective H -module, where H is of finite or tame representation type. Then there is only a finite set of objects X in C mH with Ext k C mH ( X, P [ i ]) = 0 for all k ∈ { , , ..., m } .Proof. We can assume that an m -cluster tilting object is induced from a tiltingobject in mod H ∨ mod H [1] ∨ ... ∨ mod H [ m − X such that Ext C mH ( X, P ) = 0, where P is a projective H -module, since the shiftfunctor is an equivalence on the m -cluster category. It follows from [BR] that thereare only a finite number of indecomposable objects X lying inside mod H [ i ], withExt C mH ( X, P [ i ]) = 0 for all i . HERMUND ANDR´E TORKILDSEN
We have Ext j +1 C mH ( X, P ) = Ext C mH ( X, P [ j ]), so there are only finitely many inde-composable objects X in mod H [ j ] such that Ext j +1 C mH ( X, P ) = 0. Consequently thereare only a finite number of indecomposable objects X such that Ext k C mH ( X, P ) = 0for all k ∈ { , , ..., m } , and we are finished. (cid:3) It is known from [BKL] that in the tame case, a collection of one or more tubesis triangulated. We give the proof of the following for the convenience of the reader.
Proposition 3.3.
Let H be a finite dimensional tame hereditary algebra over afield k , and C mH the corresponding m -cluster category. Let X → Y → Z → be a triangle in C mH , where two of the terms are shifts of regular modules. Then allterms are shifts of regular modules.Proof. It is enough to show that if X and Z are shifts of regular modules, then Y is a shift of a regular module. There exist a homogeneous tube T , i.e. τ M = M for all M ∈ T , such that no direct summands of X or Z are in T . Let W be aquasi-simple object in T . We have that W is sincere (see [DR]). We get the exactsequence Hom( Z, W ) → Hom(
Y, W ) → Hom(
X, W ) . We have that Hom(
Z, W ) = Hom(
X, W ) = 0, since there are no maps betweendisjoint tubes. It follows that Hom(
Y, W ) = 0. Since W is sincere, we have thatHom( U, W ) = 0 for any projective U , hence for any preprojective since τ W = W .We can do similarly for preinjectives. It follows that all direct summands of Y areshifts of regulars. (cid:3) Proposition 3.4.
Let C mH be an m -cluster category, where H is of tame represen-tation type. Let T be an m -cluster tilting object in C mH . Then T has, up to τ , atleast one direct summand which is a shift of a projective or injective.Proof. It is clearly enough to prove that there are no m -cluster tilting objects in C mH with only shifts of regular H -modules as direct summands. So suppose, for acontradiction, that such a T exists.We can decompose T into indecomposable summands, where T = T ⊕ T ⊕ ... ⊕ T n and n is the number of simple H -modules. If all direct summands are of the samedegree, we already have a contradiction, since a tilting module has at least onedirect summand which is preprojective or preinjective (see [R]).Assume that T n is a direct summand of degree k ≤ m . Let ¯ T = T ⊕ T ⊕ ... ⊕ T n − be the almost complete m -cluster tilting object obtained from T by removing thedirect summand T n . Then we know that the complements of ¯ T are connected by m + 1 AR-triangles, M i +1 → X i → M i → , where i ∈ { , , , ..., m } and X i ∈ add ¯ T .The direct summands of X i are by assumption shifts of regular modules. Wealso have that T n is a shift of a regular module and that it is equal to M j for some j , since it is a complement of ¯ T . It follows that M i is a shift of a regular modulefor all i by Proposition 3.3, since these are connected by the exhange triangles.So all m -cluster tilting objects that can be reached from T by a finite number ofmutations, have only regular direct summands.This leads to a contradiction, because we know from Proposition 2.2 that all m -cluster tilting objects can be reached from T by a finite number of mutations,and a tilting module in H induces an m -cluster tilting object in C mH with at leastone direct summand preprojective or preinjective. (cid:3) INITE MUTATION CLASSES OF COLOURED QUIVERS 7
From this it follows that we can assume that an m -cluster tilting object has atleast one direct summand which is a shift of a projective up to τ .We also need a lemma proven in [BR]. Lemma 3.5.
Let H be wild with at least non-isomorphic simples. Let t be apositive integer. Then there is a tilting module T in H with indecomposable directsummands T and T , such that dim Hom H ( T , T ) ≥ t . To prove the next lemma, which was observed in [BR] for 1-cluster tilted algebras,we use the following fact from [W]. Let F = τ − [ m ]. If X and Y are two objectsin some chosen fundamental domain in D b ( H ), then Hom D b ( H ) ( X, F i Y ) = 0 for all i = 0 , Lemma 3.6.
If a path in the quiver of an m -cluster tilted algebra goes through twooriented cycles, then it is zero.Proof. We have thatHom C H ( X, Y ) = ⊕ i ∈ Z Hom D b ( H ) ( X, F i Y ) . Let X and Y be two indecomposable m -rigid objects in a chosen fundamental do-main. It is well known that since Ext D b ( H ) ( X, X ) = 0, we have that End D b ( H ) ( X ) = k . It follows that in an oriented cycle, one of the maps lifts to a map of the form X → F Y in D b ( H ). If there is a path that goes through two oriented cycles, wehave a map of the form X → F Y → F Z , and this is 0 by the above. (cid:3) The following theorem generalizes the main theorem in [BR].
Theorem 3.7.
Let k be an algebraically closed field and Q a connected finite quiverwithout oriented cycles. The following are equivalent for H = kQ . (1) There are only a finite number of basic m -cluster tilted algebras associatedwith H , up to isomorphism. (2) There are only a finite number of Gabriel quivers occurring for m -clustertilted algebras associated with H , up to isomorphism. (3) H is of finite or tame representation type, or has at most two non-isomorphicsimple modules. (4) There are only a finite number of τ -orbits of cluster tilting objects associatedwith H . (5) There are only a finite number of coloured quivers occurring for m -clustertilting objects associated with H , up to isomorphism. (6) The mutation class of a coloured quiver Q , arising from an m -cluster tiltingobject associated with H , is finite.Proof. (1) implies (2) and (4) implies (5) is clear.(2) implies (3): Suppose there are only a finite number of quivers occurring for m -cluster tilted algebras associated with H , and let u be the maximal number of arrowsbetween to vertices in the quiver. Then by Lemma 3.6, for any two indecomposablesummands T and T of an m -cluster tilting object T , dim Hom C mH ( T , T ) < u n ,where n is the number of simple H -modules. Then it follows from Lemma 3.5 that H is not wild with more than 3 simples.(3) implies (4): If H is of finite representation type this is clear, since we onlyhave a finite number of indecomposables.Next, suppose H has at most two non-isomorphic simple modules. If there is onlyone simple module we have H ≃ k , so we can assume there are two simples. Suppose R is a regular indecomposable H -module. Then it follows from Lemma 3.1 that R is not rigid, i.e. Ext C mH ( R, R ) = 0. Then we also have that Ext C mH ( R [ i ] , R [ i ]) = 0for any i ∈ { , , ..., m − } . Up to τ in C mH we can assume that an m -cluster tilting HERMUND ANDR´E TORKILDSEN object has a direct summand which is a shift of a projective H -module, say P [ j ].Then P [ j ] has m + 1 indecomposable complements. It follows that there are onlya finite number of m -cluster tilting objects up to τ , since there are only a finitenumber of choices for P [ j ].Suppose H is tame. By Proposition 3.4, an m -cluster tilting object has at leastone direct summand which is a shift of a projective or injective, and hence up to τ we can assume it has an indecomposable direct summand which is a shift of aprojective. From Lemma 3.2 we have that there is only a finite number of m -clustertilting objects with a shift of an indecomposable projective H -module as a directsummand.(5) implies (6): This is clear, since mutation of m -cluster tilting objects corre-sponds to mutation of coloured quivers.We have that (4) implies (1) by using Lemma 2.5. (6) implies (2) is trivial, andso we are done. (cid:3) We get the following corollary.
Corollary 3.8.
A coloured quiver Q corresponding to an m -cluster tilting object,has finite mutation class if and only if Q is mutation equivalent to a quiver Q ′ ,where Q ′ G has underlying graph Dynkin or extended Dynkin, or it has at most twovertices, and there are only arrows of colour and m in Q ′ . Acknowledgements:
The author would like to thank Aslak Bakke Buan forvaluable discussions and comments.
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