TILTING THEORY AND CLUSTER ALGEBRAS

IDUN REITEN

Introduction

The purpose of this paper to give an introduction to the theory of cluster categoriesand cluster-tilted algebras, with some background on the theory of cluster algebras,which motivated these topics. We will also discuss some of the interplay betweencluster algebras on one side and cluster categories/ cluster-tilted algebras on theother, as well as feedback from the latter theory to cluster algebras.

The theory of cluster algebras was initiated by Fomin and Zelevinsky [FZ1], andfurther developed by them in a series of papers, including [FZ2], some involving othercoauthors. This theory has in recent years had a large impact on the representationtheory of algebras. The first connection with quiver representations was given in[MRZ]. Then the cluster categories were introduced in [BMRRT] in order to modelsome of the ingredients in the definition of a cluster algebra. For this purpose atilting theory was developed in the cluster category. (See [CCS1] for the independentconstruction of a category in the An case which turned out to be equivalent to thecluster category [CCS2]). This further led to the theory of cluster-tilted algebrasinitiated in [BMR1] and further developed in many papers by various authors.

The theory of cluster-tilted algebras (and cluster categories) is closely connectedwith ordinary tilting theory. Much of the inspiration comes from usual tilting theory,and features missing in tilting theory when trying to model clusters from the theoryof cluster algebras made it necessary to replace the module category modH for afinite dimensional hereditary algebra H with a related category which is the clustercategory. On the other hand, the theory of cluster-tilted algebras provides a newpoint of view on the old tilting theory.

The Bernstein-Gelfand-Ponomarev (BGP) reflection functors were an importantsource of inspiration for the development of tilting theory, which provided a majorgeneralization of the work in [BGP]. The Fomin-Zelevinsky (FZ) mutation, which isan essential ingredient in the definition of cluster algebras, gives a generalization ofthese reflections in another direction.

We start with introducing cluster algebras in the first section. We illustrate theessential concepts with an example, which will be used throughout the paper. Wegive main results and conjectures about cluster algebras which are relevant for ourfurther discussion. In Section 2 we introduce and investigate cluster categories,followed by cluster-tilted algebras in Section 3 . In Section 4 we discuss the interplaybetween cluster algebras and cluster categories/cluster-tilted algebras, and we alsogive applications to cluster algebras. The cluster categories are a special case of themore general class of Hom-finite triangulated Calabi-Yau categories of dimension 2(2-CY categories), and much of the theory generalizes to this setting. An importantcase is the stable category mod, where is the preprojective algebra of a Dynkin

1

2 IDUN REITEN

quiver. The closely related category mod has been studied extensively by Geiss-Leclerc-Schroer, who extended results from cluster categories to this setting, andgave applications to cluster algebras [GLS1][GLS2]. We treat this in Section 5 .

We suppose that the reader is familiar with the basic theory of quiver represen-tations and almost split sequences (see [Rin1],[ARS],[ASS] and other papers in thisvolume). We also presuppose some background from ordinary tilting theory (see[Rin1],[ASS],[AHK]), but here we shall nevertheless recall relevant definitions andresults when they are needed. We generally do not give proofs, but sometimes weinclude some indication of proofs in order to stress some ideas. Instead we give ex-amples to illustrate the theory, and we try to give some motivation for the work.We should also emphasize that the selection of the material reflects our personalinterests.

For each section we add some historical notes with references at the end, ratherthan giving too many references as we go along. We also refer to the surveys[BM][Rin2]. We assume throughout that we work over a field k which is algebraicallyclosed.

These notes are based on the series of lectures I gave in Trieste in January 2006.I would like to thank I. Muchtadi Alumsayh and G. Bobinski for providing me witha copy of their notes, and Aslak Bakke Buan for helpful comments.

1. Cluster algebras

In this section we introduce a special class of cluster algebras and illustrate theunderlying concepts through a concrete example. We also give a selection of main re-sults and conjectures of Fomin-Zelevinsky which provide an appropriate backgroundfor our further discussion.

1.1. FOMIN-ZELEVINSKY MUTATION. Let Q be a finite connected quiverwith vertices 1, 2, , n. We say that Q is a cluster quiver if it has no loops ddand no 2-cycles

&&ff For each vertex i = 1, 2, , n, we define a new quiver

i(Q) obtained by mutating Q, and we call the process Fomin-Zelevinsky mutation,or FZ-mutation for short.

The quiver i(Q) is obtained from Q as follows.(i) Reverse all arrows starting or ending at i.(ii) If in Q we have n > 0 arrows from t to i and m > 0 arrows from i to s and rarrows from t to s (interpreted as r arrows from s to t if r < 0), then in the newquiver i(Q) we have nm r arrows from s to t (interpreted as r nm arrows froms to t if nm r < 0).

An important easily verified property of the mutation is the following.

Proposition 1.1. For a cluster quiver Q, we have i(i(Q)) = Q for each vertex iof Q.

We illustrate with some examples.(a) Let Q be the quiver

TILTING THEORY AND CLUSTER ALGEBRAS 3

1

===

====

2//// 4

//// 5 //

ssss ss

kk

6 // 7

3

@@

Then 4(Q) it the following quiver:

1

2

4oooo

__>>>>>>>

5oooo

ss

// 6 // 7

3

For example there are now 22=4 arrows from vertex 2 to vertex 5, and consideringthe paths between 3 and 5, we have 1 2 3 = 1, so that there is one arrow from5 to 3.(b) Let Q be the quiver 1 2 3. Then Q = 3(Q) is the quiver 1 2 3obtained by reversing the arrows involving 3. For in this case, since 3 is a sink inthe quiver, there is no path of length two with middle vertex 3. (The same thinghappens when we mutate at a vertex which is a source).

Hence we see that when we mutate at a sink or a source, the procedure coincideswith the BGP-reflections.

When we have a BGP reflection, like the above example, there is an equivalence be-tween the subcategories of the categories of finite dimensional representations repQand repQ obtained by removing in each case the simple representation at thevertex 3 [BGP].

1.2. DEFINITION OF CLUSTER ALGEBRAS. Let Q be a cluster quiverwith vertices 1, 2, , n and let F = Q(x1, , xn) be the function field in n inde-terminates over Q. Consider the pair (x,Q), where x = {x1, , xn}. The clusteralgebra C(x,Q) will be defined to be a subring F . The main ingredients involved inthe definition are the following concepts: cluster, cluster variable, seed, mutation ofseeds.

The pair (x,Q) consisting of a transcendence basis x for F over the rational num-bers Q, together with a quiver with n vertices, is called a seed. For i = 1, , n wedefine a mutation i taking the seed (x,Q) to a new seed (x

, Q), where Q = i(Q)as discussed in 1.1, and x is obtained from x by replacing xi by a new element x

i in

F . Here xi is defined by xixi = m1 + m2, where m1 is a monomial in the variables

x1, , xn, where the power of xj is the number of arrows from j to i, and m2 is themonomial where the power of xj is the number of arrows from i to j. (If there is noarrow from j to i, then m1 = 1, and if there is no arrow from i to j, then m2 = 1.)

4 IDUN REITEN

Note that while in the new seed the quiver Q only depends on the quiver Q, thenx depends on both x and Q. We have 2i (x,Q) = (x,Q).

We perform this operation for all i = 1, , n, then perform it on the new seedsetc. or we get back to one of the seeds already computed. The n-element subsetsx, x, x, occurring are by definition the clusters, the elements in the clusters arethe cluster variables, and the seeds are all pairs (x, Q) occurring. The correspondingcluster algebra C(x,Q) which as an algebra only depends on Q, is the subring of Fgenerated by the cluster variables.

When we are given the cluster algebra only, the information on the clusters, clustervariables and seeds may be lost, and also the rule for mutation of seeds. We wantto keep all this information in mind, in addition to the cluster algebra itself, whichis determined by this information.

We remark that the more general definition of cluster algebras includes the pos-sibility of having the so-called coefficients, and also allows valued quivers. In thelanguage of [FZ1] the last generalization means to consider skew symmetrizable ma-trixes rather than just skew symmetric ones. The correspondence between quiversand matrices is illustrated by the following example: The quiver Q : 1 2 3 is

sent to(

0 1 01 0 10 1 0

)

1.3. AN EXAMPLE. Let Q be the quiver 1 2 3 and x = {x1, x2, x3}, wherex1, x2, x3 are indeterminates, and F = Q(x1, x2, x3). We have 1(x,Q) = (x

, Q),where Q = 1(Q) is the quiver 1 2 3 and x

= {x1, x2, x3}, where x1x1 = 1+x2,

so that x1 =1+x2

x1. Further 2(x,Q) = (x

, Q), where Q = 2(Q) is the quiver

1 552oo 3oo , and x = {x1, x2, x3}, where x2x

2 = x1 + x3, so that x

2 =

x1+x3x2

.

Continuing , we have 3(x,Q) = (x, Q) where Q is the quiver 1 2 3 and

x = {x1, x2, x3 }, where x2x

3 = x2 + 1, so that x

3 =

x2+1x3

.

Hence we have the diagram shown in Figure 1. In this diagram, Q1 = // // ,Q2 = oo // , Q3 = 33oo oo , Q4 = // oo and y =(1+x2)x1+(1+x2)x3

x1x2x3. The clusters are: {x1, x2, x3}, {

1+x2x1

, x2, x3}, { x1,x1+x3

x2, x3}, {x1, x2,

1+x2x3},

{1+x2x1

, x1+(1+x2)x3x1x2

, x3}, {1+x2x1

, x2,1+x2

x3}, {x1+(1+x2)x3

x1x2, x1+x3

x2, x3}, {x1,

x1+x3x2

, (1+x2)x1+x3x2x3

},

{x1,(1+x2)x1+x3

x2x3, 1+x2

x3}, {1+x2

x1, x1+(1+x2)x3

x1x2, (1+x2)x1+(1+x2)x3

x1x2x3}, {1+x2

x1, (1+x2)x1+(1+x2)x3

x1x2x3, 1+x2

x3},

{x1+(1+x2)x3x1x2

, x1+x3x2

, (1+x2)x1+(1+x2)x3x1x2x3

},

{ (1+x2)x1+(1+x2)x3x1x2x3

, x1+x3x2

, (1+x2)x1+x3x2x3

}, { (1+x2)x1+(1+x2)x3x1x2x3

, (1+x2)x1+x3x2x3

, 1+x2x3}, and the clus-

ter variables are: x1, x2, x3,1+x2

x1, x1+x3

x2, 1+x2

x3, x1+(1+x2)x3

x1x2, (1+x2)x1+x3

x2x3, (1+x2)x1+(1+x2)x3

x1x2x3.

If Q is a quiver mutation equivalent to Q, then the cluster algebras C(Q) andC(Q) are isomorphic.

1.4. SOME MAIN RESULTS. There is a large number of interesting results inthe theory of cluster algebras. Here we give some of the main theorems and openproblems which are of special interest for these notes.

(a) Finiteness conditions. The cluster algebra C(Q) = C(x,Q) is said to be offinite type if there is only a finite number of cluster variables. This is equivalent tosaying that there is only a finite number of clusters, and also to the fact that there is

TILT

IN

GT

HE

ORY

AN

DC

LU

ST

ER

ALG

EB

RA

S5

({x1, x2, x3}, Q1)

1

tthhhhhhhhh

hhhhhhhh

h

2

3

**VVVVVVV

VVVVVVVV

VVV

({ 1+x2x1

, x2, x3},Q2)

2

yyssssss

ssss

3

({x1,x1+x3

x2, x3}, Q3)

1

yysssss

sssss 3

%%KKKK

KKKKK

K({x1, x2,

1+x2x3

}, Q4)

2

%%KKKK

KKKKKK

({ 1+x2x1

},x1+(1+x2)x3

x1x2, x3), Q4}

3

gg 77({

x1+(1+x2)x3x1x2

, x1+x3x2

, x3}, Q1)

3

000

0000

0000

0000

0000

0000

000

({x1,x1+x3

x2,(1+x2)x1+x3

x2x3}, Q1)

1

gg 77({x1,

(1+x2)x1+x3x2x3

, 1+x2x3

}, Q2)

1

({ 1+x2x1

}, x2,1+x2

x3}, Q1)

2

vv

22

({ 1+x2x1

,x1+(1+x2)x3

x1x2, y}, Q1)

ee

%%

mm

))

({y, x1+x3x2

,(1+x2)x1+x3

x2x3},Q2) ({y,

(1+x2)x1+x3x2x3

, 1+x2x3

},Q1)//oo

({ 1+x2x1

, y, 1+x2x3

}, Q3)qq

11

({x1+(1+x2)x3x1x2

, x1+x3x2

, y}, Q4)

yy

99

Figure 1. Example.

6 IDUN REITEN

only a finite number of seeds. But as we shall see later, it is not equivalent to havingonly a finite number of quivers. There is the following description of finite type.

Theorem 1.2. Let Q be a cluster quiver. Then the cluster algebra C(Q) is of finitetype if and only if Q is mutation equivalent to a Dynkin quiver.

Note that this result is similar to Gabriels classification theorem of the quivers offinite representation type.

In Section 4 we consider the following problem problem posed by Seven:

For which quivers Q is the mutation class of Q finite?

(b) Laurent phenomenon. Observe that in the example in 1.3 we see that alldenominators of the cluster variables (when written in reduced form) are monomials.Surprisingly enough, this is a special case of the following general result.

Theorem 1.3. Let C(Q) be a cluster algebra with initial seed (x,Q). Then for anycluster variable in reduced form, the denominator is a monomial in x1, , xn.

(c) The monomial in the denominators of cluster variables. Taking a closerlook at the monomials in the denominators in the example in 1.3, we see that in-terpreting the factors xi as the simple modules Si corresponding to vertex i, thedenominators correspond to indecomposable modules via the composition factors.This was already proved in [FZ2] for the case of a Dynkin quiver with no paths oflength greater than two. As we shall see later, there are more general results in thisdirection, obtained as application of the theory of cluster categories and cluster-tiltedalgebras.

(d) Positivity. Considering again our example, we see that in the numerator, allmonomials have positive coefficients. This has been conjectured to be true in general.

(e) Clusters and seeds. Another interesting problem is the following, proved forfinite type in [FZ2]Problem: Is a seed (x, Q) expressed in terms of the initial seed (x,Q) uniquelydetermined its cluster x?

We shall see that this in the case when the quiver Q in the initial seed has nooriented cycles.

(f) Clusters differing only at one cluster variable. When applying mutationof seeds, the new cluster x has exactly one cluster variable different from the oldcluster x. If we again consider the example in 1.3, we see that if removing a clustervariable from a cluster, there is a unique other cluster variable which can replace itto give a new cluster. More generally, the following was shown in [FZ2].

Theorem 1.4. Let C(Q) be the cluster algebra associated with a Dynkin quiver Q.Then there is a unique way to replace a cluster variable in a cluster by another clustervariable to give a new cluster.

In general, there is the following open problem.

TILTING THEORY AND CLUSTER ALGEBRAS 7

Question: For any cluster algebra, is there a unique way of replacing any clustervariable in a cluster by another cluster variable to give a new cluster?

We shall discuss this problem for acyclic cluster quivers, that is, quivers withoutoriented cycles. We remark that in the Dynkin case it is known that the clustervariables are in bijection with the almost positive roots, that is, the positive rootstogether with the negative simple roots.

1.5. POSSIBLE MODELLING. The theory of cluster algebras has many nicefeatures, and it is an interesting problem to see if one can find good analogs of themain ingredients involved in their definition, in some appropriate category.

We want the additive category C to have the following properties.(i) To have an analog of clusters we want a special class of objects, all having the

same number n of nonisomorphic indecomposable summands.(ii) To imitate the process of seed mutation, we would want that each indecompos-

able summand of an object in the class can be replaced by a (unique) nonisomorphicindecomposable object such that we still get an object in our class.

(iii) To get a categorical interpretation of the definition of the new cluster variablexi coming from x

i, we would want that when an indecomposable object M is re-

placed by an indecomposable object M , then there are exact sequences or trianglesconnecting M and M , in both directions.

(iv) We would want an interpretation of the FZ-mutation.

The hope would be that this point of view should lead to an interesting theory initself, and at the same time, or instead, give a better understanding of the clusteralgebras.

Notes: The material in 1.1,1.2,1.4 is taken from [FZ1][FZ2][FZ3][BFZ]; see [BIRS]for material related to 1.5.

2. Cluster categories

Associated with a given cluster algebra we want to find some category C having aset of objects which we can view as analogs of clusters and which satisfy some or allof the requirements listed in 1.5.

A cluster algebra is said to be acyclic if in the mutation class of the associatedcluster quivers there is some quiver Q with no oriented cycles. Then we have anassociated finite dimensional hereditary k-algebra kQ. So the category mod kQ offinite dimensional kQ-modules might be a natural choice of category for modellingacyclic cluster algebras.

2.1. TILTING MODULES OVER HEREDITARY ALGEBRAS. If we con-sider C = mod kQ as the category we are looking for, then a natural choice of objectswould be the tilting kQ-modules. On one hand the reason is that they have n non-isomorphic indecomposable summands, where n is the number of vertices in Q. Onthe other hand there is a special tilting module associated with a BGP-reflectionof a quiver, and as we have seen, BGP-reflection is a special case of FZ-mutation.

8 IDUN REITEN

It will be instructive to first discuss this connection with tilting. Recall that for ahereditary algebra H, an H-module T is tilting if Ext1H(T, T ) = 0 and T has exactlyn nonisomorphic indecomposable summands up to isomorphism.

Example: Let Q be the quiver 1 2 3.(a) We first do mutation at the vertex 3. Then 3(Q) = Q

: 1 2 3. IfH = kQ, the H-module H is clearly a tilting H-module. Write H = P1 P2 P3,where Pi is the indecomposable projective module associated with the vertex i, andlet Si denote the simple top of Pi. Let T = P1 P2

1S3, where denotes thetranslation associated with almost split sequences, so that 1S3 = S2. We thenhave the following AR-quiver.

P1

???

??

P2

??

???

??S1S2

???

??_ _ _ _ _

S3 = P3

::vvvvvvvS2

??_ _ _ _ _ S1_ _ _ _ _

Note that EndH(T )op ' kQ = H , and EndH(H)

op ' kQ. So we can pass fromkQ to kQ, and hence from Q to Q, by replacing the indecomposable summand P3of the tilting module H by 1S3 to get another tilting H-module, and then takingendomorphism algebras. Note that 1P3 is the only indecomposable H-modulewhich can replace P3 to give a new tilting module. Also HomH(T, ) : modH modH induces the equivalence between subcategories as discussed in 1.1.

This example illustrates the module theoretical interpretation of the BGP-reflectionfunctors. Hence we also get a close connection between the AR-quivers, and the AR-quiver for H is the following

;;;

;; S_ _ _ _

AA

===

==

??

AAA

AA_ _ _ _

@@_ _ _ _ _

(b) We now do FZ-mutation at vertex 2 in Q, and get 2(Q) = Q : 1

))2oo 3oo .

Then it is natural to try to replace P2 in H = P1 P2 P3 to see if we get anonisomorphic tilting module, and if there is a unique one. This is indeed thecase, and the new tilting module is T = P1 S1 P3. But here we have mapsP3 P1 S1 with zero composition, so that EndH(T )

op is given by the quiver withrelations 1

&&2oo 3 , where an arrow 3 2 is missing compared to Q.

So our procedure does not work from the point of view of getting a model for theFZ-mutation, but it is quite close to working. What we would need is to have moremaps in our category than what we have in modH, in particular we would like tohave nonzero maps from S1 to P3 = S3.

(c) We also consider 1(Q) = 1 2 3 from the same point of view. Nowwe would like to replace P1 in H = P1 P2 P3 with another indecomposable H-module to obtain a tilting module. But here we encounter a problem at an earlier

TILTING THEORY AND CLUSTER ALGEBRAS 9

stage. This is actually not possible. The general explanation is that a projectiveinjective module has to be a direct summand of any tilting module. So here we get aproblem which indicates that the category modH is not large enough for being ableto replace P1.

In conclusion, as illustrated by this example, there are the following problems withusing the tilting modules over hereditary algebras as a model for clusters.

(1) There are not enough objects in order to replace any indecomposable summandof a tilting module with a nonisomorphic indecomposable module to get a new tiltingmodule.

(2) The quiver of the endomorphism algebra of a tilting module is not the desiredone, the problem being that there are not enough maps.

Actually, the following is known for tilting H-modules.

Theorem 2.1. (a) If T is a tilting H-module, then each indecomposable summandM can be replaced by at most one indecomposable H-module to get a new tiltingmodule.

(b) There is exactly one if and only if T/M is sincere, that is, each simple H-module occurs as a composition factor.

Note that in our example P1 P2 and P1 P3 are sincere, whereas P2 P3 isnot. We call an H-module T with Ext1H(T , T ) = 0 and with n 1 nonisomorphicindecomposable summands an almost complete tilting H-module.

In the case when an almost complete tilting module T has two complements, thatis, there are two ways of completing it to a tilting module, they are connected asfollows:

Theorem 2.2. Let T be an almost complete tilting H-module and M and M

nonisomorphic indecomposable modules such that T M and T M are tiltingmodules. Then, by possibly exchanging M and M , there is an exact sequence0 M B M 0 where f : B M is a minimal right addT -approximationand g : M B is a minimal left addT -approximation.

There is an important class of algebras associated with tilting modules over hered-itary algebras. An algebra is said to be tilted if it is of the form EndH(T )

op, whereT is a tilting module over a finite dimensional hereditary algebra H. These algebrasare general enough to appear a lot in a natural way in representation theory, andthey are close enough to hereditary algebras to inherit nice properties.

For an H-module T , denote by FacT the subcategory of modH whose objectsare factors of finite direct sums of copies of T . Recall also that a subcategory ofmodH is a torsion class if it is closed under factors and extensions, and a torsionfreeclass if it is closed under submodules and extensions. Then we have the followingrelationship between hereditary algebras and tilted algebras.

Theorem 2.3. Let H be a hereditary finite dimensional algebra and T a tiltingH-module, and = EndH(T )

op.(a) T = FacT is a torsion class in modH, with associated torsionfree class F ={X; HomH(T,X) = 0}.(b)There exists a torsion pair (X ,Y) in mod, where X is a torsion class and Y atorsionfree class, such that

10 IDUN REITEN

(i) HomH(T, ) : modH mod induces an equivalence between T and Y(ii) Ext1H(T, ) : modH mod induces an equivalence between F and X(iii) each indecomposable object in mod is in X or Y.

An important homological property which can be proved for a tilted algebra isthat has global dimension at most 2.

2.2. DEFINITION AND EXAMPLES. The question is now how to modify thecategory modH to take care of the shortcomings discussed in 2.1. In addition weknow from Section 1 that for cluster algebras given by Dynkin quivers, the clustervariables are in one-one correspondence with the almost positive roots. Hence thereare n more cluster variables than the number of indecomposable modules for theDynkin quiver, where n is the number of vertices.

We now explain how to modify modH in view of of the above remarks.Let Db(H) be the bounded derived category of the finite dimensional hereditary

k-algebra H = kQ, where Q is a finite quiver without oriented cycles. Then the inde-composable objects are all isomorphic to stalk complexes. The translation , whichin this case gives an equivalence from the category modP H whose indecomposableH-modules are not projective to the category modI H whose indecomposable H-modules are not injective, induces an equivalence : Db(H) Db(H). Then (C)is the left hand term of the almost split triangle with right hand term C. Note thatunder the embedding modH D(H), almost split sequences go to almost splittriangles.

Let now F be the equivalence 1[1] from Db(H) to Db(H), where [1] is the shiftfunctor. Then we define the cluster category CH to be the orbit category D

b(H)/F .Here the objects are the F -orbits of objects in Db(H). In order to recall the defini-tion of maps, we consider the fundamental domain of indecomposable objects givenby indH {Pi[1]; i = 1, , n}, where P1, , Pn are the nonisomorphic indecom-posable projective H-modules. As is easy to see, any F -orbit of indecomposableobjects contains exactly one of the indecomposable objects listed. Let A, B be in-decomposable objects in this fundamental domain, and denote by A and B also theorbits in which they lie. Then by definition HomCH (A,B) = iZ HomCH (A, F

iB),which in our case equals HomDb(H)(A,B) HomDb(H)(A, FB). We illustrate withthe following.

Example: Let Q be the quiver 1 2 3, and let Si and Pi be the simple andindecomposable projective H-modules corresponding to the vertex i, where H = kQ.We then have the following AR-quiver for H, and for Db(H)