FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS (AFTER PALU) Contents

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1 FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS (AFTER PALU) DONG YANG Abstract. This is a report on Yann Palu s PhD thesis. It is based on my talk on Bielefeld Representation Seminar in July 2009 (which is in turn based on Palu s thesis defence). Contents 1. Cluster algebras (without coefficients) 1 2. Categorifications 2 3. The framework 2 4. The cluster character 4 5. The multiplication formula 6 6. An example 7 References 10 The main reference is Palu s thesis [20], which can be downloaded from palu. The example in Section 6 is new. 1. Cluster algebras (without coefficients) Cluster algebras were invented by Fomin Zelevinsky [9]. They are commutative algebras generated by the so-called cluster variables. Here we give a brief review of the definition of a cluster algebra without coeffiecients. Let Q be a finite quiver witout loops or 2-cycles. We label the vertices of Qby1,...,n. LetQ(x 1,...,x n )bethefieldofrational functionsinvariables x 1,...,x n. We call the pair (x = (x 1,...,x n ),Q) the initial seed. A seed is a pair (x,q ) obtained from (x,q) by applying a finite sequence of (seed) mutations. Precisely, for a seed (x,q ) and each vertex i of Q, we define a new pair (x,q ), where Q is obtained from Q by quiver mutation at i, and x = (x 1,...,x i,...,x n) with x i is defined via the exchange relation, x i x i = x j + x j. j i in Q i j in Q 1

2 2 DONG YANG The first entry of a seed is called a cluster, and an element of a cluster is called a cluster variable. The cluster algebra A Q is the Z-subalgebra of Q(x 1,...,x n ) generated by all cluster variables. 2. Categorifications Nowadays people are fond of categorifications, because usually the rich structure of a category tells us much information about whatever it categorifies. For cluster algebras, there are categorifications using tensor categories ([13]), Frobenius categories ([10], etc.) and triangulated categories ([2] [3], etc.). Here we focus our attention on the third approach. The first example is that for an acyclic quiver Q the cluster algebra A Q is categorified by the cluster category C Q [2] [3] defined by Buan Marsh Reineke Reiten Todorov [2] (also Caldero Chapoton Schiffler [5] in type A). The relation between A Q and C Q is summarized in the following table ([16, Section 6.1]) cluster algebra multiplication cluster variables clusters seeds mutation exchange relation xx = m+m cluster category direct sum rigid indecomposables cluster-tilting objects cluster-tilting objects with the quiver of their endomorphism algebras mutation exchange triangles T i E Ti ΣT i Ti E T i ΣTi Here the translator from the right hand side to the left hand side is the Caldero Chapoton map. In this relation the most relevant properties of the cluster category C Q are that C Q is 2-Calabi Yau and has a cluster-tilting object (precisely, theimage of the path algebra of Q). These nice properties are shared by many other triangulated categories. We shall see that they categorify some algebras, which may or may not be cluster algebras. 3. The framework Let k be an algebraically closed field, and let C be a k-linear triangulated category with suspension functor Σ. Assume that C is Krull Schmidt, i.e. each object is a finite direct sum of indecomposable object, and each indecomposable object has local endomorphism algebra;

3 FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS (AFTER PALU) 3 C is Hom-finite, i.e. for any two objects X and Y of C, the morphism space Hom C (X,Y) is finite dimensional over k; C is 2-Calabi Yau, i.e. we have bifunctorial isomorphisms DHom C (X,Y) = Hom C (Y,Σ 2 X), where X and Y are objects of C and D is the k-duality Hom k (?,k); C has acluster-titling object T, i.e. T satisfies the following property: Hom C (T,ΣX) = 0 iff X add(t), where add(t) is the smallest full subcategory of C containing T and stable under taking direct summands and finite direct sums. Such categories include: the cluster category C Q associated to an acyclic quiver Q [2]; the triangulated category C M studied in [11]; the generalized cluster category C (Q,W) of Amiot associated to a Jacobi-finite quiver with potential (Q, W) [1]; some stable Cohen Macaulay module category studied in [4]. Let B = End C (T) be the endomorphism algebra of T. Then we have a functor F = Hom C (T,?) : C mod(b),x Hom C (T,X), where mod(b) is the category of finite dimensional (right) modules over B. According to [18], the functor F induces an equivalence C/(ΣT) mod(b), where (ΣT) is the ideal of C which consists of morphisms factoring through add(σt). Assume that T = T 1 T n, where T 1,...,T n are pairwise non-isomorphic indecomposable objects. Then F(T i ) = P i, F(ΣT i ) = 0, F(Σ 2 T i ) = I i, where P i respectively I i is the indecomposable projective respectively injective B-module corresponding to i. For two B-modules M and N we define an anti-symmetric form M,N a = dim k Hom B (M,N) dim k Ext 1 B(M,N) dim k Hom B (N,M)+dim k Ext 1 B(N,M). Proposition 3.1. (Palu) The anti-symmetric form?,? a descends to the Grothendieck group of mod(b). Proof. Here we give a proof for B being the Jacobian algebra of some quiver with potential (Q,W). The idea is borrowed from [20, Section 4.3.3]. To the quiver with potential (Q,W) we associate a differential graded algebra Γ (called the Ginzburg dg algebra, cf. [12], also [19]) which satisfies

4 4 DONG YANG H 0 Γ = B; the finite dimensional derived category D = D fd (Γ) is 3-Calabi Yau [17], i.e. we have bifunctorial isomorphisms DHom D (X,Y) = Hom D (Y,Σ 3 X), for objects X and Y of D; D has a t-structure whose heart is equivalent to mod(b) [1]. Consider the Euler form on D: X,Y := i Z( 1) i dim k Hom D (X,Σ i Y). Now let M and N be two B-modules, considered as objects of D. Since they are in the heart of a t-structure, it follows that Hom D (M,Σ i N) = 0 for i < 0. The 3-Calabi Yau property implies that Thus we have Hom D (M,Σ i N) = DHom D (N,Σ 3 i M) = 0 for i > 3. M,N = i Z( 1) i dim k Hom D (M,Σ i N) = dim k Hom D (M,N) dim k Hom D (M,ΣN) +dim k Hom D (M,Σ 2 N) dim k Hom D (M,Σ 3 N) = dim k Hom D (M,N) dim k Hom D (M,ΣN) +dim k Hom D (N,ΣM) dim k Hom D (N,M) = dim k Hom B (M,N) dim k Ext 1 B(M,N) +dim k Ext 1 B(N,M) dim k Hom B (N,M) = M,N a, where the third equality follows from the 3-Calabi Yau property. Therefore?,? a descends to the Grothendieck group since so does the Euler form?,?. 4. The cluster character Let k be an algebraically closed field. Let C be a 2-Calabi Yau triangulated category and A a commutative ring. Definition 4.1. (Palu) A cluster character on C with values in A is a map satisfying X? : objc A if M is isomorphic to N then X M equals X N ;

5 FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS (AFTER PALU) 5 X M N = X M X N ; if Hom C (M,ΣN) = k, then X M X N = X E +X E, where E and E are the middle terms of the nonsplit triangles N E M ΣN M E N ΣM. From now on we assume C and T as in Section 3. Recall that T = T 1 T n, where T 1,...,T n are pairwise nonisomorphic indecomposable objects. The functor F = Hom C (T,?) : C mod(b) takes T i to P i, the indecomposable projective B-module corresponding to i. Let M be an object of C. According to [18], there is a triangle M Σ 2 T 0 M Σ2 T 1 M ΣM, where TM 0 and T1 M belong to add(t). Define coind(m), the coindex of M, to be the difference of the images of FTM 0 and FT1 M in the Grothendieck group K 0 (projb)ofprojb, thecategory offinitelygeneratedprojectiveb-modules, i.e. coind(m) := [FT 0 M ] [FT1 M ] K 0(projB). The Grothendieck group K 0 (projb) is a free abelian group of rank n and it has a standard basis [P 1 ],...,[P n ]. By abuse of notation, we denote the coordinate vector of coind(m) in the basis [P 1 ],...,[P n ] by the same symbol coind(m). Theorem 4.2. (Palu) The map X? T : objc Q(x 1,...,x n ) n n M χ(gr e (FM)) i=1 x coind(m) i i e i=1 x S i,e a i is a cluster character on C. Here χ is the Euler characteristic, the sum is over all elements of the Grothendieck group K 0 (modb), and for e K 0 (modb), Gr e (FM) is the set of submodules of FM whose image in K 0 (modb) is e. Remark 4.3. In the case where C = C Q and T = kq, the map X T? is the Caldero Chapoton map, which was proved to be a cluster character by Caldero Keller [6]. Let T = T 1 T n be a cluster-tilting object of C. According to [15], for each direct summand T i of T, there is a unique T i (up to isomorphism) which is indecomposable and which is not isomorphic to T i such that T =

6 6 DONG YANG T 1 T i T n is again a cluster-tilting object. We say that T is obtainedfromt bymutation in direction i. WesaythatT isreachable from T if T can be obtained from T by applying a finite sequence of mutations. We say that a rigid object is reachable from T if it is a direct summand of a cluster-tilting object reachable from T. Corollary Assume that for all cluster-tilting object T reachable from T the quiver of End C (T ) has no loops or 2-cycles. Then we have surjective maps and {cluster-tilting objects reachable from T} {clusters in A Q }, {indecomposable rigid objects reachable from T} {cluster variables in A Q }, where Q is the quiver of End C (T). In particular, each cluster variable of A Q is a Laurent polynomial in the variables x 1,...,x n. Proof. In this case, there are one-dimensional extensions between the two complements T i and T i, cf. [16, Lemma 7.5]. Thus the two maps are surjective. 5. The multiplication formula Let k = C be the field of complex numbers. Let V be a variety over k and A an abelian group. A function f : V A is said to be constructible if it takes a finite number of values and its fibers are constructible subsets of V. For a constructible function f : V A, the integral V f means f = a Aχ(f 1 (a))a. V Assume C and T as in the previous section. We assume further that C has constructible cones, cf. [20, Section 5.1.3]. Theorem 5.1. (Palu) For two objects M and N of C, the map PHom C (M,ΣN) Q(x 1,...,x n ),ǫ X T mt(ǫ) is constructible, where mt(ǫ) is define by the triangle N mt(ǫ) M ǫ ΣN. Moreover, we have the following multiplication formula χ(phom C (M,ΣN))XM T XT N = Xmt(ǫ) T + ǫ PHom C (M,ΣN) ǫ PHom C (N,ΣM) X T mt(ǫ). 1 Thanks due to Bernhard Keller, who pointed an error in the statement of an earlier version.

7 FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS (AFTER PALU) 7 Theorem 5.2. (Palu) C has constructible cones if or C is the stable category of a Hom-finite Frobenius Krull Schmidt category, C is the generalized cluster category C (Q,W) of Amiot associated to a Jacobi-finite quiver with potential (Q, W) [1]. Remark 5.3. a) In the case where C = C Q and T = kq, Theorem 5.1 was proved by Caldero Keller [7] for Q a Dynkin quiver, by Hubery [14] for Q an extended Dynkin quiver, and by Xiao Xu [21] and also Xu [22] for Q an arbitrary acyclic quiver b) The multiplication formula was used by Caldero Keller [7] in the Dynkin case and by Ding Xiao Xu [8] in the extended Dynkin case to obtain a basis of the corresponding cluster algebra. 6. An example In this section we discuss a concrete example of a generalized cluster category in the sense of Amiot. Fixing a cluster-tilting object we will compute the images of all indecomposable objects under the cluster character X T? and test the multiplication formula (Theorem 5.1). Let k = C be the field of complex numbers. Let Q be the Jordan quiver, which has a unique loop ϕ. We take the potential W to be ϕ 3. Then the associated generalized cluster category C = C (Q,W) is a k-linear triangulated category. (If you do not know the precise definition of C, do not worry. I will give more details on the structure of this category, which will be enough for our purpose.) The Jacobian algebra of (Q,W) is 2 dimensional. Thus according to Amiot [1], the category C is Hom-finite, Krull Schmidt, 2-Calabi Yau and has cluster-tilting objects. The category C has exactly 3 indecomposable objects: T +, T, M. We have T + = ΣT, T = ΣT +, M = ΣM. Inparticular, Σ 2 = id. Themorphismspacesbetweenthe3indecomposables are as follows a) Hom C (T +,T + ) = k[u]/u 2, Hom C (T,T ) = k[v]/v 2, b) Hom C (T +,T ) = 0 = Hom C (T,T + ), c) Hom C (T +,M) = k = Hom C (T,M), Hom C (M,T + ) = k = Hom C (M,T ), d) Hom C (M,M) = k[w]/w 2. The Auslander Reiten triangles are given as follows T + M T v T,

8 8 DONG YANG So the quiver of C is T M T + u T +, M T + T M w M. T M By b), the objects T + and T are rigid. By b) and c) they are actually cluster-tilting objects. There are no other basic cluster-tilting objects. Take T = T + and put B = End C (T) = k[u]/u 2. The cluster character X T? will be denoted by X? for short. Recall from Section 3 that we have the following funtor F = Hom C (T,?): C F T + T T mod(b) B 0 S. Here S is the unique simple B-module (up to isomorphism). TheGrothendieck groupk 0 (projb)isfreeofrank1, andit hasastandard basis [B]. We have the following triangles T + id T + 0 T, T 0 T + id T +, M T + u T + M, from which we obtain the coindices of T +, T and M: coind(t + ) = 1 0 = 1, coind(t ) = 0 1 = 1, coind(m) = 1 1 = 0. TheGrothendieck group K 0 (modb)is freeof rank1andit hasastandard basis [S]. We identify an element in K 0 (modb) with its coordinate in [S]. In this way, the image of a B-module in K 0 (modb) is identifited with its dimension over k. It is easy to check that the anti-symmetric form?,? a vanishes. So the image of an object N of C under the cluster map X? = X? T is X N = x coind(n) χ(gr e (FN)). e N {0} Notice that FT = 0, FT + = B has three submodules which are respectively of dimension 0, 1, and 2, and FM = S has two submodules which are M

9 FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS (AFTER PALU) 9 respectively of dimension 0 and 1. Therefore by substituting N by T, T + and M in the above formula we obtain X T = x 1 1 = x, X T + = x 1 (1+1+1) = 3 x, X M = x 0 (1+1) = 2. We see from c) that Hom C (T +,ΣM) = k = Hom C (M,ΣT + ). Let ǫ be a basis of Hom C (T +,ΣM) and ǫ a basis of Hom C (M,ΣT + ). We have two nonsplit trangles M T + u T + ǫ ΣM = M, T + u T + M ǫ ΣT + = T. In particular, mt(ǫ) = T + = mt(ǫ ). Therefore we have X T +X M = 2X T + = X T + +X T + = X mt(ǫ) +X mt(ǫ ), as claimed by Theorem 4.2. Next let us check the multiplication formula χ(phom C (T +,ΣT ))X T +X T = ǫ PHom C (T +,ΣT ) X mt(ǫ) + X mt(ǫ), ǫ PHom C (T,ΣT + ) as claimed by Theorem 5.1 (+ Theorem 5.2). First of all Hom C (T +,ΣT ) = Hom C (T +,T + ) = k[u]/u 2 is 2-dimensional, and hence LHS = 2 3 x = 6. x Now let us compute the first term of the right hand side. Let ǫ be an element in Hom C (T +,ΣT ) = Hom C (T +,T + ) = k[u]/u 2. It is easy to see thatǫbelongstok uiff mt(ǫ) = M. Ifthisisnot thecase, thenǫisinvertible and hence mt(ǫ) = 0. Therefore we have X mt(ǫ) = χ(p(k u))x M +χ(p(k[u]/u 2 ku))x 0 ǫ PHom C (T +,ΣT ) = = 3. Similarly we have Thus ǫ PHom C (T,ΣT + ) X mt(ǫ) = 3. RHS = 3+3 = 6 = LHS.

10 10 DONG YANG References. 1. Claire Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, arxiv: Aslak Bakke Buan, Robert J. Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Advances in Mathematics 204 (2) (2006), Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster mutation via quiver representations, Comm. Math. Helv. 83 (2008), Igor Burban, Osamu Iyama, Bernhard Keller, and Idun Reiten, Cluster tilting for onedimensional hypersurface singularities, Adv. Math., 217 (2008), no. 6, Philippe Caldero, Frédéric Chapoton and Ralf Schiffler, Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc. 358, no. 5 (2006), Philippe Caldero and Bernhard Keller, From triangulated categories to cluster algebras. II, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, , From triangulated categories to cluster algebras, Inv. Math. 172 (2008), Ming Ding, Jie Xiao and Fan Xu, Integral bases of cluster algebras and representations of tame quivers, arxiv: Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15, 2002, no. 2, (electronic). 10. Christof Geiß, Bernard Leclerc and Jan Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, , Cluster algebra structures and semicanonical bases for unipotent groups, arxiv:math/ Victor Ginzburg, Calabi-Yau algebras, arxiv:math/ v3 [math.ag]. 13. David Hernandez and Bernard Leclerc, Cluster algebras and quantum affine algebras, arxiv: Andrew Hubery, Acyclic cluster algebras via Ringel-Hall algebras. Preprint available at the author s home page. 15. Osamu Iyama and Yuji Yoshino, Mutations in triangulated categories and rigid Cohen- Macaulay modules, Inv. Math. 172 (2008), Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, arxiv: , Pseudocompact dg algebras and derived categores, appendix to Derived equivalence from mutations of quivers with potential, arxiv: v Bernhard Keller and Idun Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Advances in Mathematics 211 (2007), Bernhard Keller and Dong Yang, Derived equivalence from mutations of quivers with potential, arxiv: v Yann Palu, Des Catégories triangulées aux algèbres amassées, Ph. D. thesis, Université Paris Diderot - Paris 7, Juin Jie Xiao and Fan Xu, Green s formula with C -action and Caldero-Keller s formula for cluster algebras, to appear in Progress in Mathematics. Also arxiv: Fan Xu, On the cluster multiplication formula for acyclic cluster algebras, to appear in Trans. Amer. Math. Soc. Also arxiv: address: yang@mpim-bonn.mpg.de