Preprojective algebras and c-sortable words

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1 Preprojective algebras and c-sortable words Claire Amiot, Osamu Iyama, Idun Reiten, Gordana Todorov To cite this version: Claire Amiot, Osamu Iyama, Idun Reiten, Gordana Todorov. Preprojective algebras and c-sortable words. 5 pages <hal v> HAL Id: hal Submitted on 22 Feb 200 (v), last revised 5 Jun 20 (v) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Abstract. Let Q be an acyclic quiver and Λ be the completion of the preprojective algebra of Q over an algebraically closed field k. To any element w in the Coxeter group of Q, Buan, Iyama, Reiten and Scott have introduced and studied in [BIRS09a] a finite dimensional algebra Λ w = Λ/I w. In this paper we look at filtrations of Λ w associated to any reduced expression w of w. We are specially interested in the case where the word w is c-sortable where c is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting kq-modules with finite torsionfree class. This nice description allows us to construct a triangle equivalence between the 2-Calabi-Yau triangulated category SubΛ w and the generalized cluster category associated with an Auslander algebra. Contents Introduction 2 Notation Acknowledgements. Background Calabi-Yau categories associated with reduced words 4.2. Mutation of tilting modules 5.. Reflections and reflection functors 6 2. Generalities on the layers Layers as images of simples The dimension vectors of the layers Reflection functors and ideals I i 2. Tilting modules and c-sortable words.. Three series of kq-modules 5.2. Tilting modules with finite torsionfree class 8.. Example Categories as cluster categories associated with Auslander algebras Canonical cluster-tilting object of SubΛ w Generalized cluster categories Computing endomorphism algebras Triangle equivalence Example 0 5. Problems and examples 2 References 4 all authors were supported by the Storforsk-grant 670 from the Norwegian Research Council. the fourth author was also supported by the NSA-grant MSPF-08G-228.

3 2 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Introduction Attempts to categorify the cluster algebras of Fomin and Zelevinsky [FZ02] have led to the investigation of categories with the 2-Calabi-Yau property (2-CY for short) and their cluster-tilting objects. Main early classes of examples were the cluster categories associated with finite dimensional path algebras [BMR + 06] and the preprojective algebras of Dynkin type [GLS06]. This paper is centered around the more general class of stably 2-CY and triangulated 2-CY categories associated with elements in Coxeter groups [BIRS09a] (the adaptable case was done independently in [GLS08]), and their relationship to the generalized cluster categories from [Ami09a] (see Section 4 for definition). Let Q be a finite connected quiver with vertices,...,n, and Λ the completion of the preprojective algebra of the quiver Q over a field k. Denote by s,...,s n the distinguished generators in the corresponding Coxeter group W Q. To an element w in W Q, there is associated a stably 2-CY category SubΛ w and a triangulated 2-CY category SubΛ w. The definitions are based on first associating an ideal I i in Λ to each s i, hence to any reduced word by taking products. This way we also get a finite dimensional algebra Λ w := Λ/I w. ObjectsofthecategorySubΛ w aresubmodulesoffinitedimensional freeλ w -modules. The cluster category is then equivalent to SubΛ w with w = c 2, where c is a Coxeter element such that c 2 is a reduced expression [BIRS09a, GLS08]. When Λ is a preprojective algebra of Dynkin type, then the category modλ as investigated in [GLS06] is also obtained as SubΛ w where w is the longest word [BIRS09a, III.5]. Using the construction of ideals we get for each reduced expression w = s u s u2...s ul a chain of ideals Λ I u I u u 2... I w, which gives rise to an interesting set of Λ-modules: L w := Λ I u, L 2 w := I u I u u 2,...,L l w := I u...u l I w which all turn out to be indecomposable and to lie in SubΛ w. The investigation of this set of modules, which we call layers, from different points of view, including connections with tilting theory, is one of the main themes of this paper, especially for a class of words called c-sortable. The modules L w,...,l l w provide a natural filtration for the cluster-tilting object M w associated with the reduced expression w = s u...s ul (see Section ). These modules can be used to show that the endomorphism algebras End Λ (M w ) are quasi-hereditary [IR0]. Here we show that these modules are rigid (Theorem 2.2), that is Ext Λ (Lj w,lj w ) = 0 and that their dimension vectors are real roots (Theorem 2.6), so that there are unique associated indecomposable kq-modules (L j w ) Q (which are not necessarily rigid). The situation is especially nice when all layers are indecomposable kq-modules, so that L j w = (L j w) Q. This is the case for c-sortable words. An element w of W Q is c- sortable when there exists a reduced expression of w of the form w = c (0) c ()...c (m) with c (m)... c () c (0) c where c is a Coxeter element, that is, a word containing each generator s i exactly once, and in an order admissible with respect to the orientation of Q.

4 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS Starting with the tilting kq-module kq (when c (0) = c), there is a natural way of performing exchanges of complements of almost complete tilting modules, determined by the given reduced expression. We denote the final tilting module by T w, and the indecomposable kq-modules used in the sequence of constructions by Tw j for j =,...,l. We show that L j w Tw j for all j (Theorem.8) and that the indecomposable modules in the torsionfree class Sub(T w ) are exactly the Tw j (Theorem.0). In particular this gives a one-one correspondence between c-sortable words and torsionfree classes, as first shown in [Tho] using different methods. There is another sequence Uw,...Ul w of indecomposable kq-modules, defined using restricted reflection functors, which coincide with the above sequences. This is both interesting in itself, and provides a method for proving L j w Tj w for j =,...,l. In another paper [AIRT], we give a description of the layers from a functorial point of view. When the c-sortable word is c m, and c = s...s n, then the successive layers are given by P,...,P n,τ P,...,τ P n,τ 2 P,...,τ m P n for the indecomposable projective kq-modules P i, where τ denotes the AR-translation. In the general case we will give a description of the layers using specific factor modules of the above modules. The generalized cluster categories C A for algebras A of global dimension at most two were introduced in [Ami09a]. It was shown that for a special class of words w, properly contained in the dual of the c-sortable words, the 2-CY category SubΛ w is triangle equivalent to some C A. We show that the procedure for choosing A works more generally for any (dual of a) c-sortable word (Theorem 4.0), with a simpler proof due to developments in the meantime. The paper is organized as follows. We start with some background material on 2- CY categories associated with reduced words, on complements of almost complete tilting modules andonreflection functors. In Section 2 we show that for any reduced word w, the associated layers are indecomposable rigid modules, which also are real roots. Hence there are unique associated indecomposable kq-modules. In Section we show that our three series of indecomposable modules {L j w }, {Tj w } and {Uj w } coincide in the c-sortable case. The description of the layers as specific factor modules of the τ i P for P indecomposable projective is given in Section 4. In Section 5 we show the relationship with generalized cluster categories in the c-sortable case. Section 6 is devoted to examples and questions beyond the c-sortable case. Some of this work was presented at a conference in Trondheim in August Notation. Throughout k is an algebraically closed field. The tensor product, when not specified, will be over the field k. For a k-algebra A, we denote by moda the category of finitely presented right A-modules, and by f.l. A the category of finite length right A-modules. For a quiver Q we denote by Q 0 the set of vertices and by Q the set of arrows, and for a Q we denote by s(a) its source and by t(a) its target. Acknowledgements. This work was done when the first author was a post doc in NTNU Trondheim. She would like to thank the Research Council of Norway for financial support.

5 4 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV. Background.. 2-Calabi-Yau categories associated with reduced words. Let Q be a finite quiver without oriented cycles and with vertices Q 0 = {,...,n}. For i,j Q 0 we denote by m ij the positive integer m ij := {a Q s(a) = i,t(a) = j}+ {a Q s(a) = j,t(a) = i}. The Coxeter group associated to Q is defined by the generators s,...,s n and relations s 2 i =, s i s j = s j s i if m ij = 0, s i s j s i = s j s i s j if m ij =. In this paper w will denote a word (i.e. an expression in the free abelian group generated by s i,i Q 0 ), and w will be its equivalence class in the Coxeter group W Q. An expression w = s u...s ul is reduced if l is smallest possible. An element c = s u...s ul is called Coxeter element if l = n and {u,...,u l } = {,...,n}. We say that a Coxeter element c = s u...s un is admissible with respect to the orientation of Q if i < j when there is an arrow u i u j. The preprojective algebra associated to Q is the algebra kq/ a Q aa a a where Q is the double quiver of Q, which is obtained from Q by adding for each arrow a : i j in Q an arrow a : i j pointing in the opposite direction. We denote by Λ the completion of the preprojective algebra associated to Q and by f.l.λ the category of right Λ-modules of finite length. The algebra Λ is selfinjective finite-dimensional if Q is a Dynkin quiver. Then the stable category mod Λ satisfies the 2-Calabi-Yau property (2-CY for short), that is, there is a functorial isomorphism DHom Λ (X,Y) Hom Λ (Y,X[2]), where D := Hom k (,k) and [] := Ω is the suspension functor. When Q is not Dynkin, then Λ is infinite dimensional and of global dimension 2. In this case the triangulated category D b (f.l.λ) is 2-CY. We now recall some work from [IR08, BIRS09a]. For each i =,...,n we have an ideal I i := Λ( e i )Λ, where e i is the idempotent of Λ associated with the vertex i. We write I w := I ul...i u2 I u when w = s u s u2...s ul is a reduced expression of w W Q. We collect the following information which is useful for Section 2: Proposition.. [BIRS09a] Let Λ be a preprojective algebra. (a) If w = s u...s ul and w = s v...s vl are two reduced expression of the same element in the Coxeter group, then I w = I w. (b) If w = w s i with w reduced, then I w I w. Moreover w is reduced if and only if I w I w. And for j i we have e j I w = e j I w. If Λ is not of Dynkin type we have moreover: (c) Any finite product I of the ideals I j is a tilting module of projective dimension at most one, and End Λ (I) Λ.

6 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 5 (d) If S is a simple Λ-module and I is a tilting module of projective dimension at most one, then S Λ I = 0 or Tor Λ (S,I) = 0. (e) If S i := Λ/I i and Tor Λ (S i,i) = 0, then I i L Λ I = I i Λ I = I i I for a tilting ideal I of projective dimension at most one. By (a) the ideal I w does not depend on the choice of the reduced expression w of w. Thefore we write I w for the ideal I w and denote Λ w := Λ/I w. This is a finite dimensional algebra. We denote by SubΛ w the category of submodules of free Λ w -modules. This is a Frobenius category, that is an exact category with enough projectives and injectives, and the projectives and injectives coincide. Its stable category SubΛ w is a triangulated category which satisfies the 2-Calabi-Yau property [BIRS09a]. The category SubΛ w is then said to be stably 2-Calabi-Yau. Recall that a cluster-tilting object in a Frobenius stably 2-CY category C with finite dimensional morphisms spaces is an object T C such that Ext C(T,T) = 0 Ext C(T,X) = 0 implies that X addt. For any reduced word w = s u...s ul, we write M j w := e u j Λ I uj...i u. Theorem.2. [BIRS09a, Thm III.2.8] For any reduced expression w = s u...s ul of w W Q, the object M w := l j= Mj w is a cluster-tilting object in the stably 2-CY category SubΛ w. For any reduced word w = s u...s ul, we have the chain of ideals Λ I u I u u 2... I w, which is strict by Proposition. (b). For j =,...,l we define the layer L j w := I u j...i u I uj...i u. Using Proposition. (b) it is immediate to see the following Proposition.. We have isomorphisms in f.l. Λ: L j w e u j L j w e u j I ui...i u I uj...i u Ker( M j w where i is the greatest integer < j satisfying u i = u j. M i w ), Therefore the layers L w,...,ll w give a filtration of the cluster-tilting object M w..2. Mutation of tilting modules. Let Q be finite quiver with vertices {,...,n} and without oriented cycles. Definition.4. A tilting kq-module T is a basic module with n indecomposable summands such that Ext kq (T,T) = 0. For each indecomposable summand T i of T, it is known that there is at most one indecomposable Ti T i such that T/T i Ti is a tilting module [RS90, Ung90], and that there is exactly one if and only if T/T i is a sincere kq-module [HU89]. We then say that

7 6 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV T i (and possibly T i ) is a complement for the almost complete tilting module T/T i. The (possibly) other complement of T/T i can be obtained using the following result: f Proposition.5. (a) If the minimal left add(t/t i )-approximation T i monomorphism, then Cokerf is a complement for T/T i. (b) If the minimal right add(t/t i )-approximation B then Kerg is a complement for T/T i. g B is a T i is an epimorphism, There is a one-one correspondence between tilting modules T and contravariantly finite torsionfree classes F = Sub T containing the projective modules... Reflectionsand reflectionfunctors. LetQbefinitequiverwithvertices{,...,n} and without oriented cycles. Let i Q 0 be a source. Then the quiver Q := µ i (Q) is obtained by replacing all arrows starting at the vertex i by arrows in the opposite direction. Write kq = P P n where P j is the indecomposable projective kq-module associated with the vertex j. Then using results of [BGP7] and [APR79] we have functors: modkq R i R i modkq where R i := Hom kq (M, ), R i := kq M and M := τ P i kq/p i which induce inverse equivalences R i (modkq)/[e i kq] (modkq )/[e i DkQ ], R i where modkq/[e i kq] (resp. modkq /[e i DkQ ]) is obtained from the module category modkq (resp. modkq ) by annihilating morphisms factoring through P i = e i kq (resp. e i DkQ ). Since i is a source (resp. a sink) of Q (resp. Q ) the category modkq/[e i kq] (resp. modkq /[e i DkQ ]) is also a full subcategory of modkq (resp. modkq ). When the vertex i is not a sink or source, there is still defined a reflection on the level of the Grothendieck group K 0 (modkq). It is constructed using the semigroup with generators [X] for X modkq and relations [X] + [Z] = [Y] if there is a short exact sequence X Y Z. This is a free abelian group with basis {[S ],...[S n ]}, where S,...,S n are the simple kq-modules. With respect to this basis we define R i ([S j ]) = [S j ]+(m ij 2δ ij )[S i ], where m ij is the number of edges of the underlying graph of Q as before. This definition is coherent with the previous one. Indeed if i is a source and M is an indecomposable in modkq which is not isomorphic to P i, then we have R i ([M]) = [R i (M)]. 2. Generalities on the layers Let w be an element in the Coxeter group of an acyclic quiver Q, and fix w = s u...s ul a reduced expression of w. For j =,...,l we have defined in Section the layer L j w as

8 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 7 the quotient L j w := I u j...i u I uj...i u. In this section, we investigate some main properties of these layers. We show that each layer can be seen as the image of a simple Λ-module under an auto-equivalence of D b (f.l.λ). Hence they are rigid indecomposable Λ-modules of finite length, and we compute explicitly their dimension vectors and show that they are real roots. Hence to each layer we can associate a unique indecomposable kq-module with the same dimension vector, but which is not necessarily rigid. Note that some of the results of this section have been proven independently in [GLS0] but with different proofs. 2.. Layers as images of simples. Proposition 2.. Let Q a non Dynkin quiver and Λ the completion of the preprojective algebra. For j =,...,l we have isomorphisms in D(ModΛ): L j w S uj L Λ (I uj...i u ) S uj L Λ I uj L Λ L Λ I u where S uj is the simple Λ-module associated to the vertex u j. Proof. Weset w := s u...s uj andw := s u...s uj. Sincew isreduced, byproposition.(e) we have I w I uj Λ... Λ I u I uj L Λ... L Λ I u, and hence we get the second isomorphism. Sincew = w s uj isreduced, wehavei w = I uj I w I w,andthereforetor Λ (S uj,i w ) = 0 by Proposition. (d). Thus we have S uj L Λ I w S uj Λ I w Λ I uj Λ I w I w I uj I w = L j w. Immediately we have the following result, which implies that L j w is an indecomposable rigid Λ-module of finite length. Theorem 2.2. For j =,...,l we have L j w if Λ is of non Dynkin type: if Λ is of Dynkin type: dimext i Λ (Lj w,lj w ) = { i = 0,2, 0 otherwise. dimext i Λ (Lj w,lj w ) = { i = 0,2, 0 i =. Note that there can be higher extensions in the Dynkin case. In the non Dynkin case, is then said to be 2-spherical in the sense of Seidel-Thomas [ST0].

9 8 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Proof. We separate the proof when Λ is of non Dynkin type and when Λ is of Dynkin type. Non Dynkin case: By Proposition. (c), I w is a tilting Λ-module with End Λ (I w ) Λ. Hence the functor L Λ I w is an autoequivalence of D(ModΛ). We have End Λ (S j ) k and hence Ext 2 Λ(S j,s j ) k since D b (f.l.λ) is 2-CY. Moreover since Q has no loops, Ext Λ(S j,s j ) vanishes and since Λ is known to have global dimension 2, Ext n Λ (S j,s j ) vanishes for n. Hence S j is 2-spherical. Since by Proposition 2. the layer L j w is the image of the simple S j by an autoequivalence of D b (f.l.λ), it follows that L j w is also 2-spherical. Dynkin case: Let Q be a Dynkin quiver and Q be an acyclic extended Dynkin quiver containing Q as a subquiver. Let Λ := Λ Q and Λ := Λ Q be the corresponding (completion of) their preprojective algebras. Then we have Λ Λ/ Λe Λ where e is the idempotent associated to the additional vertex of Q. The restriction functor R : modλ mod Λ is fully faithful and modλ can be seen as an extension closed subcategory of mod Λ. It is immediate to check that for a reduced expression w of w W Q we have L j w,λ. Using the first part of the proof, we get L j w, Λ End Λ (L j w ) End Λ(L j w ) k and Ext Λ (Lj w,lj w ) Ext Λ(L j w,lj w ) = 0. Finally using the fact that modλ is stably 2-CY we get Ext 2 Λ(L j w,l j w) k. Here we state a property about two consecutive layers of the same type, which gives rise to special non split short exact sequences in f.l.λ. Proposition 2.. Let i < j < k l be integers such that u i = u j = u k and such that j is the only integer satisfying i < j < k and u i = u j = u k. Then we have dim k Ext Λ(L j w,l k w) =. In order to prove this proposition, we first need a lemma. For h l, we denote as before by Mw h the Λ-module Mh w := e Λ u h I uh...i u. Lemma 2.4. Let i < j < k be as in Proposition 2.. (a) The map Hom Λ (Mw k,mj w ) Hom Λ(Mw k,mi w ) induced by the irreducible map Mw j Mi w is an epimorphism. (b) The image of the map Hom Λ (Mw,M i w) j Hom Λ (Mw,M j w) j induced by the irreducible map Mw j Mw i is in Rad Λ (Mw,M j w) j. Proof. (a) Since i < j < k, then by Lemma III..4 of [BIRS09a], we have isomorphisms Hom Λ (Mw,M k w) j Λ e e and Hom Λ (M I uj...i w,m k i Λ w) e e, u I ui...i u

10 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 9 where e is the idempotent e := e ui = e uj = e uk. Then the map Hom Λ (Mw,M k w) j Hom Λ (Mw k,mi w ) is the epimorphism e Λ Λ I uj...i u e uk ei ui...i u e uk induced by the inclusion I uj...i u I ui...i u. (b) It is clear that the image is contained in the radical. By Lemma III..4 of [BIRS09a], we have isomorphisms Hom Λ (Mw,M i w) j e I u j...i ui+ e and Rad Λ (M I uj...i w,m j w) j e e. u I uj...i u The map Hom Λ (Mw i,mj w ) Rad Λ(Mw j,mj w ) is induced by the inclusion of ideals I uj...i ui+ I uj. But since j is the only integer satisfying i < j < k and u i = u i = u k, we have ei uj...i ui+ e ei uj e and hence the map Hom Λ (Mw i,mj w ) Rad Λ (Mw j,mj w ) is an isomorphism. Proof of Proposition 2.. By definition of the layers, we have the following short exact sequences I uj (j) L j w M j w M i w and (k) L k w M k w M j w Let K be the kernel of the composition map Mw k Mj w Mi w. Then we have a short exact sequence (l) K M k w M i w which gives rise to the following long exact sequence in modend Λ (M w ), where M w = l h= Mh w DExt Λ (Mi w,m w) DHom Λ (K,M w ) DHom Λ (M k w,m w ) DHom Λ (M i w,m w )... The space DExt Λ (Mi w,m w) is zero by Lemma III.2. of [BIRS09a], and the End Λ (M w )- moduledhom Λ (Mw k,m w)isindecomposableinjective. ThereforethemoduleDHom Λ (K,M w ) has simple socle, and hence K is indecomposable. Moreover from the sequences (j), (k) and (l), we deduce that we have a short exact sequence L k w K L j w which is non split since K is indecomposable. Hence we get dim k Ext Λ(L j w,l k w) From (j) we deduce the following long exact sequence Hom Λ (M k w,m j w) Hom Λ (M k w,m i w) Ext Λ (Mk w,lj w ) Ext Λ (Mk w,mj w ) = 0. Hence by Lemma 2.4 (a) we get Ext Λ (Mk w,lj w ) = 0. From (j) we also deduce the following long exact sequence 0 Hom Λ (M i w,m j w) Hom Λ (M j w,m j w) Hom Λ (L j w,m j w) Ext Λ (Mi w,mj w ) = 0. Hence by Lemma 2.4 (b) we get Hom Λ (L j w,mj w ) Hom Λ(Mw j,mj w )/Rad Λ(Mw j,mj w ) which is one dimensional since Mw j is indecomposable.

11 0 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Finally using (k) we get the long exact sequence Ext Λ (Mk w,lj w ) Ext Λ (Lk w,lj w ) Ext 2 Λ (Mj w,lj w ) By the 2-CY property and the previous remarks we have Ext Λ (Mk w,lj w ) = 0 and Ext2 Λ (Mj w,lj w ) DHom Λ(L j w,mj w ) k and therefore dim k Ext Λ (Lj w,lk w ) The dimension vectors of the layers. In this section we investigate the action of the functor L Λ I w at the level of the Grothendieck group of D b (f.l.λ) when Λ is not of Dynkin type. We show that this action has interesting connections with known actions. We denote by [ L Λ I w ] the induced automorphism of K 0 (D b (f.l.λ)). Lemma 2.5. Let Q be a non Dynkin quiver. For all i,j in Q 0 we have [S j L Λ I i ] = [S j ]+(m ij 2δ ij )[S i ] in K 0 (D b (f.l.λ)), where m ij is the number of arrows between i and j in Q. Proof. Since S j is a simple Λ-bimodule, we have DS i S i as Λ-bimodules. Hence we have the following isomorphisms in Mod(Λ op Λ): Therefore we have S j L Λ S i DHom k (S j L Λ S i,k) DRHom Λ (S j,hom k (S i,k)) DRHom Λ (S j,ds i ) DRHom Λ (S j,s i ). [S j L Λ S i ] = ( t ( ) t dimext t Λ (S j,s i ))[S i ] = (2δ ij m ij )[S i ]. From the triangle S i [ ] I i Λ S i we get a triangle S j L Λ S i [ ] S j L Λ I i S j S j L Λ S i. Hence we have [S j L Λ I i ] = [S j ] [S j L Λ S i ] = [S j ] (2δ ij m ij )[S i ]. From Lemma 2.5, we deduce the following results. Theorem 2.6. Let Λ be the completion of a preprojective algebra of any type. () For j =,...,l we have [L j w ] = R u...r uj ([S uj ]), where the R t are the reflection functors defined in Section. (2) For j =,...,l, there exists a unique indecomposable kq-module (L j w ) Q such that [L j w] = [(L j w) Q ].

12 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS Proof. () As in the previous subsection we treat separately the Dynkin and the non Dynkin case. The non Dynkin case is a direct consequence of Lemma 2.5 and Proposition 2.. For the Dynkin case, we can follow the strategy of the proof of Proposition 2.. We introduce an extended Dynkin quiver containing Q as subquiver. Then applying reflection functors associated to the vertices of Q to modules whose support do not contain the additional vertex is the same as applying the reflection functors of Q. Thus the equality coming from the non Dynkin quiver gives us the equality for Q. (2) From () it follows that the dimension vector of the layer L j w is a positive real root, and we get the result applying Kac s Theorem. The layer L j w is always rigid as Λ-module, but the associated indecomposable kqmodule (L j w ) Q is not always rigid as shown in the following. Example 2.. Let Q be the quiver 2, and w := s s 2 s s 2 s s. Then we have L w =, L2 w = 2, L w = 2, L 4 w =, L5 w = 2 2, and L6 w = 2. Thus the associated indecomposable kq-modules are the follwing: (L j w ) Q = L j w for j =,...4, (L5 w ) Q = 2 2, and (L 6 w ) Q = 2 The module (L 6 w ) Q lies in the tube of rank 2, with indecomposable objects and 2 on the border of the tube. Since (L 6 w) Q is not on the border of the tube, it is not rigid. Definition 2.7. [BB05]LetQbeanacyclicquiver withnvertices, andw Q bethecoxeter group of Q. Let V be the vector space with basis v,...,v n. The geometric representation W GL(W) of W is defined by s i v j := v j +(m ij 2δ ij )v i. The contragradient of the geometric representation W GL(V) is then { s i vj = vj i j vj + t j m tjvt i = j The Grothendieck group K 0 (D b (f.l.λ)) has a basis consisting of the simple Λ-modules, and K 0 (K b (projλ)) has a basis consisting of the indecomposable projective Λ-modules. Proposition 2.8. (a) The Coxeter group W acts on K 0 (D b (f.l.λ)) by w [ L Λ I w ] as the geometric representation. (b) The Coxeter group W acts on K 0 (K b (projλ)) by w [ L Λ I w ] as the contragradient of the geometric representation. Proof. (a) This follows directly from Lemma 2.5. (b) This is shown in [IR08, Theorem 6.6]. It is assumed in [IR08] that Q is extended Dynkin, but this assumption is not used in the proof for this statement..

13 2 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV 2.. Reflection functors and ideals I i. In this subsection, we state some basic properties of the first layers. In particular we show that the equivalence L Λ I i, when Q is not Dynkin, can be interpreted as a reflection functor of the category D b (f.l.λ). Lemma 2.9. Let Q be an acyclic quiver, and Λ = Λ Q. Let c W Q be a Coxeter element admissible with respect to the orientation of Q. Let i Q 0 be a source of Q. Then we have the following isomorphisms in mod Λ: () Λ/I c kq, (2) I i /I csi τ P i kq/p i = R i (kq) where P i = e i kq is the indecomposable projective kq-module associated to i and τ is the AR-translation of modkq. () I c n/i c n+ τ n (kq). Proof. () This is Propositions II..2 and II.. of [BIRS09a]. (2) We separate the case whether Q is of Dynkin type and of non Dynkin type. Note that by Proposition. (b) we have e j I i = e j Λ and e j I csi = e j I i I c = e j I c if j i. Therefore by () it is enough to prove that e i I i /I csi τ (e i kq). Assume first that Q is of non Dynkin type. The projective resolution of e i I i in modλ has the form: ( ) 0 e i Λ a Q,s(a)=i e t(a)λ e i I i 0 Applying thefunctor Λ I c to theexact sequence ( ), we getanexact sequence ( ) 0 e i I c a Q,s(a)=i e t(a)i c e i I i Λ I c 0. By Proposition. (e), we have I i Λ I c = I i I c = I csi. Hence we deduce from ( ) and ( ) the short exact sequence 0 e i Λ I c a Q,s(a)=i e t(a) Λ I c e i I i I csi 0. Since i is a source in Q, we have the set equality {a Q, with s(a) = i} = {a Q, with s(a) = i}. Therefore by () this short exact sequence is 0 e i kq a Q,s(a)=i e t(a)kq e i I i I csi 0. I Hence we have e i i I csi τ (e i kq). Let Q be of Dynkin type. Denote by Q an acyclic extended Dynkin quiver containing Q as a subquiver and such that the additional vertex is a sink. Let Λ := Λ Q and Λ := Λ Q be the corresponding (completion of) their preprojective algebras. Denote by c Q the Coxeter element of W Q admissible with respect to the orientation of Q. Using the above argument for the quiver Q and for Ĩc Q we get a short exact sequence 0 e i k Q a Q,s(a)=i e t(a)k Q e i Ĩ i Ĩ cq s i 0.

14 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS Since the additional vertex i 0 is a sink, we get that e j k Q e j kq for j i 0 and Λ/Ĩc Q Λ/I cq kq. Hence we have e i I i I i I cq e i I i Λ Λ I cq e i Ĩ i Λ () This is a direct consequence of () and (2). Λ Ĩ cq e i Ĩ i Ĩ cq s i τ Q (e ikq) τ Q (e ikq). From Lemma 2.9 we deduce the following result which gives another interpretation of the tilting ideals I i. Corollary 2.0. Let Q be an acyclic quiver which is not Dynkin, and Λ = Λ Q. Let i Q 0 be a sink of Q. Denote Q := µ i (Q). Then the following diagram commute modkq/[e i DkQ] R i modkq /[e i kq ], D b (f.l.λ) L Λ I i where the vertical functors are the natural inclusions. D b (f.l.λ) Proof. Denote by c the Coxeter element admissible with respect to the orientation of Q, and by c = s i cs i the Coxeter element admissible with respect to the orientation of Q. We have the following isomorphisms in D b (f.l.λ). kq L L Λ I i Λ/I c Λ I i by Lemma 2.9 () Λ/I c Λ I i by Proposition. (e) I i /I c I i I i /I i I c kq by Lemma 2.9 (2). Tilting modules and c-sortable words In this section Q is a finite acyclic quiver, Λ is the completion of the preprojective algebra associated with Q and c a Coxeter element admissible with respect to the orientation of Q. The purpose of this section is to investigate the layers for words w satsifying a certain property called c-sortable. Definition.. [Rea07] Let c be a Coxeter element of the Coxeter group W Q. An element w of W Q is called c-sortable if there exists a reduced expression w of w of the form w = c (0) c ()...c (m) where all c (t) are subwords of c whose supports satisfy supp(c (m) ) supp(c (m ) )... supp(c () ) supp(c (0) ) Q 0. For i Q 0, if s i is in the support of c (t), by abuse of notation, we will write i c (t). Here is an immediate result [Rea07].

15 4 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Lemma.2. Letw beac-sortableelementofw Q. Thenthe expressionw = c (0) c ()...c (m) is unique. Let w be an element of W Q, and w = s u...s ul a reduced expression. Recall from Section that for j =,...,l the layer L j w is defined to be the Λ-module: L j w = e u j I uk...i u I uj...i u = I u j...i u I uj...i u where k < j satisfies u k = u j and is maximal with this property. Here is a Theorem giving a nice charasterization of c-sortable words. Theorem.. Let w be an element of W Q and w = s u s u2...s ul be a reduced expression of w. Then we have the following: () if there exists a Coxeter c such that w is c-sortable and w is the c-sortable expression of w, then for all j =,...,l L j w is in modkq, where Q is admissible for the Coxeter element c; (2) if for all j =,...,l the layer L j w is in modkq for a certain orientation of Q, then w is c-sortable, where c is the Coxeter element admissible for the orientation of Q. Proof. () Assume that w = s u...s ul is a c-sortable word. Let j, and k be the (possibly) last index < j such that u j = u k. Since w is c-sortable, the word s u...s uj is a subsequence of cs u...s uk. Therefore we have an inclusion Hence there is a surjection e uj I u...u k I c = e uj I cu...u k e uj I u...u j e uj I u...u k I u...u k I c e uj I u...u k I u...u j = L j w. The left term is a kq-module, indeed it is isomorphic to e uj I u...u k Λ Λ I c = e uj I u...u k Λ kq by Lemma 2.9 (). Thus the right term L j w is also a kq-module. (2)Forthisstatement we againhave totreat separately thedynkinandthenondynkin case. Assume first that Q is not Dynkin. We prove this assertion by induction on the length of the word w. For l(w) = the result is immediate. Assume that (2) is true for any word w of length l and let w := s u...s ul be a reduced expression such that L j w is a kq-module for all j =,...,l. Without loss of generality we can assume that the support of w contains all the vertices of Q. We first show that u is a source of Q. Assume it is not, then there exists k 2 such that there is an arrow u k u in Q. Take the smallest such number. It is then not hard to check that the top of L k w is the simple S u k and that the kernel of the map L k w S u k contains S u in its top. Thus L k w is not a kq-module, which is a contradiction. Therefore u is a source of the quiver Q and we have L j w = e uj I uk...i u I uj...i u (e uj I uk...i u2 I uj...i u2 ) L Λ I u by Proposition. (e).

16 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 5 Hence we have L j w = L Lj w Λ I u for j = 2,...,l, where w := s u2...s ul. By Theorem 2.6 () we have [L j w] = R u... R uj ([S uj ]) in the Grothendieck group K 0 (D(f.l.Λ). Since w is reduced we then have [L j w ] [S u ] for j 2. Thus L j w is not isomorphic to the simple projective e u kq = S u if j 2. Then by Corollary 2.0, we get L j w R u (L j w ) modkq /[e u DkQ ] where Q = µ u (Q). By induction hypothesis we get that w is c -sortable where c is the Coxeter element admissible for the orientation of Q, i.e. c = s u cs u. We get the conclusion using the following criterion which detects c-sortability: Lemma.4. [Rea07, Lemma 2.] Let c := s u...s un be a Coxeter element. If l(s u w) < l(w), then w is c-sortable if and only if s u w is s u cs u -sortable. If Q is Dynkin, we introduce an extended Dynkin quiver Q such that the additional vertex is a source. And then we conclude by the above argument for non Dynkin quivers... Three series of kq-modules. To the c-sortable word w = s u...s ul, we associate three different series of kq-modules, and show that they coincide. For j =,...,l, we define kq-modules Tw. j For j l(c (0) ), Tw j is the projective kq-module e uj kq. For j > l(c (0) ), let k be the maximal integer such that k < j and u k = u j. We define Tw j as the cokernel of the map where f is a minimal left (T k+ w f : T k w E Tj w )-approximation. Definition.5. An admissible triple is a triple (Q,c,w) consisting of an acyclic quiver Q, a Coxeter element c admissible with respect to the orientation of Q, and a c-sortable word w = c (0) c ()...c (m) such that c = c (0) v for some v as words. We denote by Q (j) the quiver Q restricted to the support of c (j). Definition.6. Let (Q,c,w) be an admissible triple, with w = s u w. The reduction of (Q,c,w) at s u is the triple (Q,c,w ) with Q = µ u (Q (0) ), where µ u is the reflection at u and c = s u c (0) s u. It is not hard to check the following property: Lemma.7. The triple (Q,c,w ) is admissible. Note that since u is a source on the restriction of Q to supp(c (0) ), it is always possible to apply the reflection functor µ u. Let (Q,c,w) be an admissible triple with w = s u s u2...s ul. For j =,...,l, we define kq-modules Uw j by induction on l. If l = then we define Uw = e u kq, the projective indecomposable kq-module associated to the vertex u.

17 6 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Assume l 2. Then we write w = s u w, and by Lemma.7 the triple (Q = µ u (Q (0) ),s u cs u,w ) is an admissible triple with l(w ) = l. Therefore by the induction hypothesis we have kq-modules U w,...ul w. For j = 2,...,l we define where R u is the composition U j w = R u (U j w ) modkq = modk(µ u Q (0) ) R u modkq (0) modkq. Theorem.8. Let w = s u...s ul be a c-sortable word where c is admissible for the orientation of Q. Then for j =,...,l, we have L j w Uj w Tj w, where the Lj w are the layers and the kq-modules Tw j and Uw j are defined as above. Proof. We first prove that L j w Uj w. By definition L w = e u Λ/I u = S u. Since by assumption c = c (0) v, we have e u kq (0) = e u kq = S u. Hence we get Uw = L w. Let w be the word s u2...s ul. We will prove that L j w = R u (L j w ) for j 2. By Lemma 2.9 (2) we have Ru I ( ) = u kq. Hence we can write I c (0) su L j w = e u j I uk...i u2 e uj I uj...i u2 =: Y X. We have the following exact commutative diagram: X Λ I u I c (0) X Λ I u X Λ I u I u I c (0) 0 Y Λ I u I c (0) f a Y Λ I u Y Λ I u I u I c (0) 0 g Y X Λ I u I c (0) d b Y X Λ I u e Y X Λ I u I u I c (0) By Proposition. (b) we have the inclusion YI c X since u 2 u j is a subword of c u 2...u k. By definition c (0) s u is s u c, so the map f factors through a. Therefore the composition dg vanishes and since g is epi, d vanishes and hence e is an isomorphism. Moreover since w = s u s u2...s uk is a subword of s u...s uj, then XI u is contained in YI u by Proposition. (b). Hence a is mono. Finally we get an isomorphism Y X Λ I u I u I c (0) Y Λ I u X Λ I u L j w. We will now prove that U j w Tj w. For j l(c(0) ) this is clear because of a basic property of reflection functors.

18 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 7 Assume j > l(c (0) ). Let k be the maximal integer < j such that u k = u j. It exists because j > l(c (0) ) and because w is c-sortable. We define the subwords w = s u...s uk and w = s uk...s uj of w. Let c be s uk...s uj, and Q be the quiver µ w (Q) = µ uk µ u (Q). Then (Q,c,w ) is an admissible triple. We have Uw = S u k and U j k+ w = R c (S uk ) = τ kq (S uk ), thus we have an almost split sequence: 0 Uw E Uj k+ w 0 Applying the reflection functor R w to this short exact sequence we still get a short exact sequence: 0 R w (U w ) R w (E) R w (U j k+ w ) 0 which is 0 Uw k R w (E) U j w 0 and the left map is a left add(r w (U 2 w ) R w (U j k w ))-approximation, thus a left add(uw k+ Uj w )-approximation. Note moreover that this approximation is always mono. Corollary.9. Let w be a c-sortable word, where c is admissible with respect to the orientation of Q. Then the kq-modules L j w satisfy the following properties: () They are non zero. (2) They are pairwise non-isomorphic. () The space Hom kq (L j w,l k w) vanishes if j > k. (4) The minimal left add{l k+ w,...,lj w }-approximation map f : Lk w E is a monomorphism, where k and j are consecutive of same type. Proof. () This is Proposition. (c). (2) This is clearly true for the Uw j because reflection functors preserve isoclasses. () Using reflection functors, we can assume that Uw k is simple projective, and then this is clear. (4) The fact that the approximation map is mono comes from the fact that reflection functors preserves short exact sequences. Theorem.0. Let w = s u...s ul be a c-sortable word, where c is admissible for the orientation of Q. For i Q (0) 0, denote by t w (i) the maximal integer such that u tw(i) = i. Then the kq-module T w := i Q (0) 0 L tw(i) w is a kq (0) -tilting module and we have Sub(T w ) = {L w,...,ll w }. Proof. The fact that T w is a kq (0) -tilting module can easily be seen using the fact that L j w = Tw, j and Corollary.9 ()-(4). We prove that Sub(T w ) = {L w,...,ll w } by induction on l = l(w). If l(w) =, then the assertion is clear. Assume that l 2 and write w = s u w.

19 8 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV Case : u is in the support of w : this means that t w (u ) 2. Thus we have T w = i Q (0) 0 = i Q (0) 0 = i Q (0) 0 = Ru (T w) Then using the induction hypothesis we get U tw(i) w Ru (U tw(i) w ) Ru (U t w (i) w ) {R u (U w ),...,R u (U l(w ) w )} SubT w {S u,r u (U w ),...,R u (U l(w ) w )} By definition of the T j w there exists a short exact sequence: S u = T w E T j w 0 where E is in add(tw 2... Tj w ) and where j is the minimal integer 2 such that u j = u. It exists since u is in the support of w. The approximation map is a monomorphism, thus S u is in Sub(E) Sub(Tw 2... Tw j ) SubT w. Case 2: u is not in the support of w. Then it is easy to see that T w = S u Ru (T w ). And we get SubT w = {S u,ru (Uw ),...,R u (U l(w ) w )} = {Uw,U w,...,u 2 l(w) w }. Remark.. (a) The short exact sequence L k w E L j w in modkq is an almost split sequence of the category Sub(T w ). (b) This almost split sequence is an element of Ext Λ (Lj w,lk w ), which is the 2-Calabi- Yau complement of the short exact sequence L j w K L k w of Proposition Tilting modules with finite torsionfree class. In this section we establish a converse of Theorem.0. Hence we get a natural bijection between tilting kq-module with finite torsionfree class and c-sortable elements in W Q. Proposition.. Let w = s u...s ul = c (0)...c (m) be a c-sortable word where c is a Coxeter word admissible for the orientation of Q. We define T w := j c T tw(j) (0) w as in Theorem.0. Let i Q 0 such that c (m) s i is a subword of c (m ) or i c (m). We define L I := e w i I i I w and T as the cokernel of Tw tw(i) E where f is a minimal leftadd(t tw(i)+ w... Tw l )- approximation. Then we have the following () T L ; (2) the kq (0) -module T j i Ttw(j) w ws i is c-sortable. f f is a tilting module if and only if the expression

20 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 9 Proof. If c (m) s i is a subword of c (m ) then we can write ws i = c (0)...c (m ) c with c := c (m) s i and we have supp(c ) supp(c (m ) )... supp(c () ) supp(c (0) ). If i c (m) then we write ws i = c (0)...c (m ) c (m+) with c (m+) := s i and then supp(c (m+) ) supp(c (m) )... supp(c () ) supp(c (0) ). To prove () it is then enough to observe that the proof of Theorem.8 does not use the fact that the expression w is reduced. By Theorem.0 it is enough to check that if T j j Ttw(j) w is tilting then ws i is reduced. If ws i is not reduced we have L = 0 by Proposition. (b), and therefore T = 0 by (). Since w is c-sortable all Tw j = Lj w are non zero indecomposable modules by Theorem 2.2. Therefore the module T j j Ttw(j) w has l(c (0) ) indecomposable summands, so it can not be a tilting module over kq (0). From Proposition. we deduce a nice consequence. Theorem.2. Let Q be an acyclic quiver. Let c be a Coxeter element admissible with respect to the orientation of Q. Let T be a tilting module over kq. Assume that SubT has finitely many indecomposable modules. Then there exists a unique c-sortable word w such that T w T. Proof. Assume that the orientation of Q is admissible for the Coxeter element s s 2...s n. The category Sub T has almost split sequences. Denote by τ the AR-translation of this category. Since SubT is finite, then for any i Q 0 there exists m i such that τ m i (e i kq) = 0. And for each indecomposable X in SubT, there exist unique t 0 and i Q 0 such that X τ t (e i kq). Indeed since SubT is finite, the AR quiver of SubT is connected and since the algebra kq is hereditary it is not hard to see that there are no periodic modules. Then for t 0 we look at the set {i Q 0 τ t (e i kq) 0} = {i (t) < i (t) 2 < < i (t) p t } and set c (t) := s (t) i s (t) i...s (t) 2 i. It then clear that the word w := c (0) c ()...c (m) where p t m := max{m i i Q 0 } satisfies supp(c (m) )... supp(c () ) supp(c (0) ). We have to check that w is reduced. Assume it is not and write w = w s i v where w is reduced and w s i is not reduced. The word w is again c-sortable so can be written as w := c (0)...c (m ). For j Q 0 denote by m j the integer such that j c(m j ) and j / c (m j +). Then by hypothesis m i < m i. Using the almost split sequences of SubT, it is immediate that T w i Q 0 τ m i (ei kq). T t w (i)+2 w w which is zero by Proposition. Then by Proposition. the cokernel T of the minimal left add{t t w (i)+ w T l(w ) w }-approximation map T t w (i) w E is L l(w )+

21 20 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV (b). Therefore we have τ (T t w (i) w ) = 0 which is a contradiction since τ (T t w (i) w ) = τ m i (e i kq) and m i + m i. As a consequence we get the following: Corollary.. If T is a tilting kq-module such that SubT is of finite type, then all indecomposables Sub T are rigid as kq-modules. Combining Theorem.2 with Theorem.0 we get the following result which was first proved using other methods in [Tho]. Corollary.4. There is a - correspondence {finite torsionfree class of modkq} : {c-sortable words with c (0) = c}.. Example. Let Q be the following graph 2, and let w be the word s s 2 s s s 2 s in the Coxeter group W Q. An admissible orientation for Q is the following 2. The canonical cluster-tilting object M w in SubΛ w has the following direct summands Mw =, Mw 2 = 2, M w = 2, M 4 w = 2 2, Mw 5 = 2 2 2, M 6 w = 2 2. Then we can easily compute the layers L w,...,l 6 w. They are the indecomposable summands of the Mw i as kq-modules: L w =, L 2 w = 2, L w = 2, L 4 w = 2 2, L 5 w = 2 2, and L 6 w =. Let us compute the Tw j. For j the Tj w are the projective kq-modules, thus we have T w =, T 2 w = 2, and T w = 2. Then we have to compute approximations. We have a short exact sequence , where the map 2 2 is the minimal left add(t 2 w T w )-approximation of T w. Hence we have T 4 w = 2 2. We have an exact sequence ,

22 PREPROJECTIVE ALGEBRAS AND C-SORTABLE WORDS 2 where is the minimal left add(t w Tw)-approximation 4 of Tw. 2 Hence we have T 5 w = 2 2. There is an exact sequence , hence Tw 6 =. So we have Tj w = Lj w as in Theorem.8. The module T w is by definition Tw T5 w T6 w. It is easy to check Theorem.0. The module T w is a tilting module over kq, and we have SubT w = {, 2, 2, 2 2, 2 2, }. Let us now compute the U j w s. By definition U w =. Then we have Uw 2 = R ( 2) = 2, U w = R R 2 ( ) = R ( 2 ) = 2 2 and Uw 4 = RR 2R ( ) = RR 2( ) = R ( 2 ) = AndfinallywehaveUw 6 = R R 2 R R R 2 ( )where R i isthereflectionfunctorassociated to the quiver 2. Therefore we have U 6 w = R R 2R R ( 2 ) = R R 2R ( 2) = R R 2( 2 ) = R ( ) =. 4. Categories as cluster categories associated with Auslander algebras In this section Q is an acyclic quiver, c is the Coxeter element admissible with respect to the orientation of Q and w = c (0) c ()...c (m) is a c-sortable word with c (0) = c. We denote by M w the canonical cluster-tilting object of SubΛ w associated with the c-sortable expression w of w. This section is devoted to proving that the triangulated category SubΛ w is triangle equivalent to a generalized cluster category associated to an algebra of global dimension at most two. Note that the result also holds in the case of general words [ART09], but with a very different construction. A link between the construction given in this paper and the construction of [ART09] is given in [Ami09b]. The first subsection is devoted to recalling results on Jacobian algebras defined in [DWZ08],andontheendomorphismalgebraofthecluster-tiltingobjectM w from[birs09a] and [BIRS09b]. In the second subsection we recall some definitions and basic properties for generalized cluster categories. In the third subsection we construct an algebra A of global dimension at most two such that the endomorphism algebra of the canonical cluster-tilting object in the generalized cluster category C A is isomorphic to the endomorphism algebra of M w in the category SubΛ w (Proposition 4.9). In the fourth subsection we construct a triangle functor from C A to the category SubΛ w using a consequence of the universal property of the generalized cluster category (see Proposition 4.4). Using a criterion of [KR08] (Proposition 4.4), we show that this functor is an equivalence. In the last subsection we describe an example.

23 22 CLAIRE AMIOT, OSAMU IYAMA, IDUN REITEN, AND GORDANA TODOROV 4.. Canonical cluster-tilting object of SubΛ w. Quivers with potentials and their associated Jacobian algebras have been investigated in [DWZ08]. Let Q be a finite quiver. For each arrow a in Q, the cyclic derivative a with respect to a is the unique linear map a : kq/[kq,kq] kq which takes the class of a path p to the sum p=uavvu taken over all decompositions of the path p (where u and v are possibly idempotent elements e i associated to the vertex i). An element W in kq/[kq,kq] is a potential on Q, and is given by a linear combination of cycles in Q. The associated Jacobian algebra Jac(Q,W) is by definition the algebra kq/ a W;a Q. There is a generalization of quivers with potentials (Q,W) to frozen quivers with potentials (Q,W,F) in [BIRS09b] (see also [ART09]), where F = (F 0,F ) is a pair of a subset F 0 of vertices of Q (called frozen vertices) and a subset F of arrows contained in the set {a Q,s(a) F 0 and t(a) F 0 } (called frozen arrows). The associated frozen Jacobian algebra is by definition the algebra Jac(Q,W,F) = kq/ a W,a / F. Let w = c (0) c ()...c (m) be a c-sortable word. Assume that the orientation of Q is admissible with respect to c and that c (0) = c. For t 0, we define Q (t) to be the full subquiver of Q with vertices in the support of c (t). For each i in Q 0 we denote by m i the integer such that i c (m i) and i / c (m i+). Let Q w be the following quiver: the vertices are {(i,r),r = 0,...,m, i c (r) }. for each r, for each i in Q (r+) 0, one arrow p i r : (i,r +) (i,r) for each a : i j Q, if r < m i and r m j, one arrow a r : (i,r) (j,r), for each a : i j Q, if m i m j, one arrow a mi : (i,m i ) (j,m j ), foreacha : i j Q, ifr < m i andr < m j then onearrowa r : (j,r) (i,r+), for each a : i j Q, if m j < m i, one arrow a m j : (j,m j ) (i,m i ). We define the potential W w to be the sum W w = p i r a r a r r<m i,r<m j a:i j a:i j,m i m j p j m i...pj m j a m i a m i + p j r a r+a r r m i,r<m j a:i j,m i >m j p mj...p mi a m j a mj Let us denote by Q w the full subquiver of Q w with vertices (i,r) where r m i. And let W w be the potential W w = p i ra ra r p j ra r+ a r a:i j r<m i,r<m j r m i,r<m j Then we have the following result: