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... algebra. A finite-dimensional A-module T is a τ-rigid module if HomApT, τT q “ 0 and a τ-tilting module if in addition |T | “ |A| holds. Here τ denotes the Auslander– Reiten translation of modA (see =-=[8]-=-). An A-module T is a support τ-tilting module 16 Thomas Brüstle and Dong Yang if there is an idempotent e of A such that T is a τ -tilting module over A{AeA. Note that T is rigid over A and the idemp...

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...enerates C, i.e. C “ thickpMq, and a tilting object if further HomCpM,Σ iMq vanishes also for all i ă 0. Tilting objects play an essential role in the Morita theory of derived categories of algebras (=-=[44, 24, 91, 59]-=-) and the notion of silting objects, generalising that of tilting objects, was introduced by Keller and Vossieck in [72] to study t-structures on the bounded derived category of finitedimensional repr...

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... . , Xn generate C. This notion was first used by Rickard in [92] to help constructing derived equivalences of symmetric algebras from stable equivalences. Spherical collections in algebraic geometry =-=[97]-=- are examples of simple-minded collections. In representation theory, a typical example of a simple-minded collection is a complete collection of pairwise non-isomorphic simple modules over a finite-d...

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...is quickly expanding, and we do not aim to cover everything in full generality. The reader is encouraged to read the original references. More detailed surveys on some of the objects and maps include =-=[105, 63, 31, 78, 90, 66]-=-. In the appendix we provide some results on the derived category of a non-positive dg algebra which are used in Sections 3 and 4. Throughout this article, k denotes an uncountable algebraically close...

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...ver all decompositions of the cycle c (where u and v are possibly trivial paths). Writing W “ ř c:cycle λcc, we define BρpW q “ ř c:cycle λcBρpcq. Thanks to the work of Derksen, Weyman and Zelevinsky =-=[27]-=-, the quiver mutation (Section 3.9) extends to a mutation of quivers with potential. Given a quiver with potential pQ,W q such that Q has no loops or oriented 2-cycles and a vertex i of Q , the mutati...

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...] to use quiver representations to categorify cluster algebras (without coefficients) with defining quiver being of Dynkin type. This work was generalised to all acyclic quivers by Caldero and Keller =-=[21, 20]-=- (see also Hubery [50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller [36] and by Palu [85], and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with pote...

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... of their hearts. Lemma 3.9. Let pCď0, Cě0q and pC1ď0, C1ě0q be two bounded t-structures on C. Then pC1ď0, C1ě0q is intermediate with respect to pCď0, Cě0q if and only if C1ď0 X C1ě0 Ď Cď0 X ΣCě0. In =-=[45]-=-, Happel, Reiten and Smalø introduced an operation, called the Happel– Reiten–Smalø tilt, to produce new t-structures from given ones. Precisely, let pCď0, Cě0q be a bounded t-structure on C with coho...

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...ops or oriented 2-cycles. 2.5. The Ginzburg dg algebra and the Jacobian algebra. Let pQ,W q be a quiver with potential. The (complete) Ginzburg dg algebra pΓpQ,W q of pQ,W q is constructed as follows =-=[43]-=-: Let Q̃ be the graded quiver with the same vertices as Q and whose arrows are • the arrows of Q (they all have degree 0), • an arrow ρ˚ : j Ñ i of degree ´1 for each arrow ρ : iÑ j of Q, • a loop ti ...

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...ential W on Q such that the Jacobian algebra of pQ,W q is infinite-dimensional, then mutpQq does not have a sink. 3.10. Clusters, ClpQq. Introduced and further investigated by Fomin and Zelevinsky in =-=[32, 33, 35]-=-, cluster algebras are commutative rings equipped with a distinguished set of generators, the cluster variables which are grouped into overlapping sets of variables, the clusters. The cluster variable...

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...cluster algebras, which has proved powerful in understanding cluster algebras, for example in proving a number of Fomin and Zelevinsky’s conjectures. It was originally defined by Caldero and Chapoton =-=[19]-=- to use quiver representations to categorify cluster algebras (without coefficients) with defining quiver being of Dynkin type. This work was generalised to all acyclic quivers by Caldero and Keller [...

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...egory. A typical example of such a triangulated category is the cluster category CQ of an acyclic quiver Q (that is, a quiver without oriented cycles), defined as the orbit category Dbpmod kQq{τ´1˝Σ (=-=[17, 60]-=-), where τ is the Auslander–Reiten translation of Dbpmod kQq. An object M of C is called a rigid object if HompM,ΣMq “ 0 holds, and a cluster-tilting object if further the following equality holds add...

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...Leibniz rule holds for all homogeneous elements a of degree p and all elements b: dpabq “ dpaqb ` p´1qpadpbq. Ordered Exchange Graphs 9 Consider the derived category DpAq of (right) dg A-modules, see =-=[59, 61]-=-. This is a triangulated category. For a dg A-module M , we have HomDpAqpA,Σ mMq “ HmpMq. This formula will be used without further reference. We are interested in the following two triangulated subca...

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... ÝÑ CpQ,W q. (c) ([7, Proposition 2.12]) For X,Y P FΓ, there is a short exact sequence 0 // HomperpΓqpX,ΣY q // HomCpQ,W qppipXq,ΣpipY qq // DHomperpΓqpY,ΣXq // 0 . These results were proved by Amiot =-=[7]-=- for non-complete Ginzburg dg algebras and non-complete Jacobian algebras, but her proof works also in the complete setting. When Q is acyclic, the Amiot cluster category CpQ,0q is triangle equivalent...

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...he mutation of cluster-tilting objects was introduced by Buan, Marsh, Reineke, Reiten and Todorov [17] for the case when C “ CQ is the cluster category of an acyclic quiver Q and by Iyama and Yoshino =-=[55]-=- for general C. Let M “ M1‘ . . .‘Mn be a cluster-tilting object of C, whereM1, . . . ,Mn are pairwise nonisomorphic indecomposable objects. Fix i “ 1, . . . , n. Thanks to the 2-Calabi–Yau property o...

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...enerates C, i.e. C “ thickpMq, and a tilting object if further HomCpM,Σ iMq vanishes also for all i ă 0. Tilting objects play an essential role in the Morita theory of derived categories of algebras (=-=[44, 24, 91, 59]-=-) and the notion of silting objects, generalising that of tilting objects, was introduced by Keller and Vossieck in [72] to study t-structures on the bounded derived category of finitedimensional repr...

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... Note that T is rigid over A and the idempotent e is unique in the sense that if e1 is another such idempotent then addpeAq “ addpe1Aq holds. These notions were already studied by Auslander and Smalø =-=[9]-=- in the 1980’s, but we are using here the terminology adopted by Adachi, Iyama and Reiten in [2], which generalises the work of Ingalls and Thomas [52] on hereditary algebras. The notion of τ -tilting...

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...jects of perpΓq and related it to Donaldson–Thomas invariants. Geiss, Leclerc and Schröer took a different approach for stably 2-Calabi–Yau Frobenius categories arising from preprojective algebras in =-=[38, 39, 40]-=- and later they proved in [41] that the two approaches are closely related. Let ClpQq denote the set of clusters of the cluster algebra AQ with principal coefficients (Section 3.10). An important feat...

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...M // M2 // ΣM 1 (3.3) with M 1 P Cě0 and M 2 P ΣCď0. The above triangle (3.3) is not canonical. This notion was introduced by Pauksztello in [86] and independently by Bondarko as weight structures in =-=[12]-=-. The co-heart is defined as the intersection Cě0 X Cď0. Note that the co-heart is usually not an abelian category. As for t-structures, the subcategories Cě0 and Cď0 are called the aisle and co-aisle...

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...uiver being of Dynkin type. This work was generalised to all acyclic quivers by Caldero and Keller [21, 20] (see also Hubery [50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller =-=[36]-=- and by Palu [85], and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with potential by Plamondon [88, 87]. In parallel, instead 52 Thomas Brüstle and Dong Yang of objects in C...

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...ential role in the Morita theory of derived categories of algebras ([44, 24, 91, 59]) and the notion of silting objects, generalising that of tilting objects, was introduced by Keller and Vossieck in =-=[72]-=- to study t-structures on the bounded derived category of finitedimensional representations over a Dynkin quiver. A typical example of a silting object is the free module AA of rank 1 inH bpprojAq for...

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...nkin type. This work was generalised to all acyclic quivers by Caldero and Keller [21, 20] (see also Hubery [50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller [36] and by Palu =-=[85]-=-, and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with potential by Plamondon [88, 87]. In parallel, instead 52 Thomas Brüstle and Dong Yang of objects in CpQ,W q, Derksen, ...

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...50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller [36] and by Palu [85], and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with potential by Plamondon =-=[88, 87]-=-. In parallel, instead 52 Thomas Brüstle and Dong Yang of objects in CpQ,W q, Derksen, Weyman and Zelevinsky constructed in [28] the Caldero–Chapoton map for decorated representations over the Jacobia...

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...7 Ringel observed that the set of tilting modules over a finite-dimensional algebra A carries the structure of a simplicial complex. The study of this complex and its poset structure was initiated in =-=[95]-=- and further carried out by Happel and Unger [99, 100, 49, 47, 48]. See also the contributions of Ringel and of Unger in the Handbook of tilting theory [96, 101]. Let A be a finite-dimensional algebra...

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...s such that ¨ HomApM,Nq “ 0 for all M P T and N P F , ¨ for any M P A there is a short exact sequence 0 // M 1 // M // M2 // 0 (3.1) with M 1 P T and M2 P F . This notion was introduced by Dickson in =-=[29]-=-. The subcategories T and F are respectively called the torsion class and the torsion-free class. For example, let A be the category of fintely generated abelian groups; then the torsion groups form a...

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...ndofunctors σď0 and σě1 of C such that σď0M “ M 1 and σě1M “ M2. For i P Z, let σďi “ Σ´iσď0Σi and σěi “ Σ´i`1σě1Σi´1. The notion of t-structures was introduced by Beilinson, Bernstein and Deligne in =-=[10]-=- when studying perverse sheaves over stratified topological spaces. The two subcategories Cď0 and Cě0 are often called the aisle and the co-aisle of the t-structure. One observes that Cě0 “ ΣpCď0qK an...

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...“ bn`i,j . Columns of c-matrices are called c-vectors. For instance, we have (̧ pQq “ In and (̧ qQq “ ´In. For more details on c-vectors, we refer the reader to [35] where they were introduced and to =-=[84, 83, 98, 82, 66]-=- where they were studied. With this terminology, for a quiver R P Mutp pQq the vertex i P Q0 is green if and only if the i-th column cipRq of the c-matrix satisfies cipRq ě 0, and the vertex is red if...

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...ex and its poset structure was initiated in [95] and further carried out by Happel and Unger [99, 100, 49, 47, 48]. See also the contributions of Ringel and of Unger in the Handbook of tilting theory =-=[96, 101]-=-. Let A be a finite-dimensional algebra. An A-module T is a pretilting module if Ext1ApT, T q “ 0 and the projective dimension of T is at most 1. It is a tilting module if in addition there is a short...

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...enerates C, i.e. C “ thickpMq, and a tilting object if further HomCpM,Σ iMq vanishes also for all i ă 0. Tilting objects play an essential role in the Morita theory of derived categories of algebras (=-=[44, 24, 91, 59]-=-) and the notion of silting objects, generalising that of tilting objects, was introduced by Keller and Vossieck in [72] to study t-structures on the bounded derived category of finitedimensional repr...

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...Dong Yang ¨ for each M P C there is a triangle in C M 1 // M // M2 // ΣM 1 (3.3) with M 1 P Cě0 and M 2 P ΣCď0. The above triangle (3.3) is not canonical. This notion was introduced by Pauksztello in =-=[86]-=- and independently by Bondarko as weight structures in [12]. The co-heart is defined as the intersection Cě0 X Cď0. Note that the co-heart is usually not an abelian category. As for t-structures, the ...

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...tions were already studied by Auslander and Smalø [9] in the 1980’s, but we are using here the terminology adopted by Adachi, Iyama and Reiten in [2], which generalises the work of Ingalls and Thomas =-=[52]-=- on hereditary algebras. The notion of τ -tilting modules generalises that of tilting modules (Section 3.1). Theorem 3.5. ([2]) Let T be an A-module. (a) T is a tilting module if and only if T is a fa...

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...der that we will use the notation µ´ for left mutation with ´ meaning a decrease with respect to the partial order. This convention is the same as the one used in [66] and opposite to the one used in =-=[4, 75]-=-. We remark that for most of these sets the structure of an ordered exchange graph are defined only after having the bijections in Section 4. 3.1. Tilting modules, tiltpAq. In this subsection, we reca...

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...by EA. The underlying graph of EA is in general not an exchange graph as in Section 2.1.1, because it is not regular, see [46, Corollary 1.3], [95, Remark 1.3] and [48, Lemma 4.3]. Thanks to the work =-=[2]-=- of Adachi, Iyama and Reiten, it can be completed into an ordered exchange graph by considering support τ -tilting modules (see Section 3.3). For an A-module T , define TK1 :“ tX P modA | Ext1ApT,Xq “...

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...is quickly expanding, and we do not aim to cover everything in full generality. The reader is encouraged to read the original references. More detailed surveys on some of the objects and maps include =-=[105, 63, 31, 78, 90, 66]-=-. In the appendix we provide some results on the derived category of a non-positive dg algebra which are used in Sections 3 and 4. Throughout this article, k denotes an uncountable algebraically close...

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...Leibniz rule holds for all homogeneous elements a of degree p and all elements b: dpabq “ dpaqb ` p´1qpadpbq. Ordered Exchange Graphs 9 Consider the derived category DpAq of (right) dg A-modules, see =-=[59, 61]-=-. This is a triangulated category. For a dg A-module M , we have HomDpAqpA,Σ mMq “ HmpMq. This formula will be used without further reference. We are interested in the following two triangulated subca...

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...p : Γ Ñ J . Let p˚ : perpΓq Ñ perpJq (respectively, p˚ : DfdpJq Ñ DfdpΓq) be the induction (respectively, restriction) along the projection p. The functor p˚ is extensively studied by King and Qiu in =-=[74]-=- for the case of a Dynkin quiver with trivial potential. 2-siltpΓq ÝÑ 2-siltpJq: The assignment M ÞÑ p˚pMq defines a bijection from 2-siltpΓq to 2-siltpJq which commutes with mutations, by applying Pr...

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...is quickly expanding, and we do not aim to cover everything in full generality. The reader is encouraged to read the original references. More detailed surveys on some of the objects and maps include =-=[105, 63, 31, 78, 90, 66]-=-. In the appendix we provide some results on the derived category of a non-positive dg algebra which are used in Sections 3 and 4. Throughout this article, k denotes an uncountable algebraically close...

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...“ bn`i,j . Columns of c-matrices are called c-vectors. For instance, we have (̧ pQq “ In and (̧ qQq “ ´In. For more details on c-vectors, we refer the reader to [35] where they were introduced and to =-=[84, 83, 98, 82, 66]-=- where they were studied. With this terminology, for a quiver R P Mutp pQq the vertex i P Q0 is green if and only if the i-th column cipRq of the c-matrix satisfies cipRq ě 0, and the vertex is red if...

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...] to use quiver representations to categorify cluster algebras (without coefficients) with defining quiver being of Dynkin type. This work was generalised to all acyclic quivers by Caldero and Keller =-=[21, 20]-=- (see also Hubery [50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller [36] and by Palu [85], and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with pote...

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...i-th column cipRq of the c-matrix satisfies cipRq ě 0, and the vertex is red if and only if cipRq ď 0. The sign-coherence for c-vectors, conjectured by Fomin and Zelevinsky in [35] and established in =-=[28]-=-, ensures that each c-vector satisfies either cipRq ě 0 or cipRq ď 0 (this gives a proof of Proposition 3.18). In Ordered Exchange Graphs 35 this sense, the c-vectors behave like root systems from Lie...

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...jects of perpΓq and related it to Donaldson–Thomas invariants. Geiss, Leclerc and Schröer took a different approach for stably 2-Calabi–Yau Frobenius categories arising from preprojective algebras in =-=[38, 39, 40]-=- and later they proved in [41] that the two approaches are closely related. Let ClpQq denote the set of clusters of the cluster algebra AQ with principal coefficients (Section 3.10). An important feat...

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...e left mutation µ´S pD ď0,Dě0q of pDď0,Dě0q at S is the Happel–Reiten–Smalø tilt at the torsion pair pKAS,Sq. We point out that this mutation is different from the mutation defined by Zhou and Zhu in =-=[106]-=- for the more general notion of torsion pairs of triangulated categories. On the set of t-structures on C there is a natural partial order pCď0, Cě0q ď pC1ď0, C1ě0q :ô Cď0 Ď C1ď0, equivalently, Cě0 Ě ...

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...50]) and further to 2-Calabi–Yau triangulated categories by Fu and Keller [36] and by Palu [85], and to Amiot cluster categories of (not necessarily Jacobi-finite) quivers with potential by Plamondon =-=[88, 87]-=-. In parallel, instead 52 Thomas Brüstle and Dong Yang of objects in CpQ,W q, Derksen, Weyman and Zelevinsky constructed in [28] the Caldero–Chapoton map for decorated representations over the Jacobia...

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... identities and non-commutative Donaldson–Thomas invariants [65] and in Ordered Exchange Graphs 7 calculating the complete spectrum of a BPS (Bogomol’nyi–Prasad–Sommerfield) particle in string theory =-=[6, 22, 104]-=- (this also appears implicitly in [37]). The main feature of an ordered exchange graph is the property (i): the arrows in the Hasse quiver of the poset are given by mutations. This property has been e...

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...led the cohomology functors). The t-structure pCď0, Cě0q is said to be bounded ifď nPZ ΣnCď0 “ C “ ď nPZ ΣnCě0. A bounded t-structure is one of the two ingredients of a Bridgeland stability condition =-=[15]-=-. A typical example of a t-structure is the pair pDď0std,D ě0 stdq for the bounded derived category DbpmodAq of a finite-dimensional algebra A, where Dď0std consists of complexes with vanishing cohomo...

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... algebras. Due to the separation formulas [35, Theorem 3.7, Proposition 3.13 and Corollary 6.3], the cluster algebras with principal coefficients govern the combinatorics of all cluster algebras (see =-=[23]-=- for a stronger result in the skew-symmetric case). In this article we will only be concerned with skew-symmetric cluster algebras with principal coefficients, which are defined for skew-symmetric mat...

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...is quickly expanding, and we do not aim to cover everything in full generality. The reader is encouraged to read the original references. More detailed surveys on some of the objects and maps include =-=[105, 63, 31, 78, 90, 66]-=-. In the appendix we provide some results on the derived category of a non-positive dg algebra which are used in Sections 3 and 4. Throughout this article, k denotes an uncountable algebraically close...

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...y of perpAq which contains tΣ mM | m ě 0u and which is closed under taking extensions and direct summands. We point out that this map is well-defined and bijective in a much more general setting, see =-=[12, 4, 68, 80]-=-. 40 Thomas Brüstle and Dong Yang t-strpAq ÝÑ smcpAq: Let pDď0,Dě0q be a bounded t-structure on DfdpAq with length heart. The corresponding simple-minded collection is a complete collection of pairwis...

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..., Section 2] and [79, Example 4.4]. In general, Question 3.13 is open. It has a positive answer for C “ HbpprojAq, Ordered Exchange Graphs 25 where A is a representation-finite symmetric algebra, see =-=[1, 3]-=-, or A is a piecewise hereditary algebra1. Another direction is to consider presilting objects N which are 2-term with respect to the given silting object M , i.e. there is a triangle M´1 // M0 // N /...

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...ential W on Q such that the Jacobian algebra of pQ,W q is infinite-dimensional, then mutpQq does not have a sink. 3.10. Clusters, ClpQq. Introduced and further investigated by Fomin and Zelevinsky in =-=[32, 33, 35]-=-, cluster algebras are commutative rings equipped with a distinguished set of generators, the cluster variables which are grouped into overlapping sets of variables, the clusters. The cluster variable...

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...der that we will use the notation µ´ for left mutation with ´ meaning a decrease with respect to the partial order. This convention is the same as the one used in [66] and opposite to the one used in =-=[4, 75]-=-. We remark that for most of these sets the structure of an ordered exchange graph are defined only after having the bijections in Section 4. 3.1. Tilting modules, tiltpAq. In this subsection, we reca...

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...e M ˚ i is the cone of the minimal left addp À j‰iMjq-approximation of Mi Mi // E. The object µ´i pMq is again a silting object. Silting mutation was defined and studied by Buan, Reiten and Thomas in =-=[18]-=- for bounded derived categories of finite-dimensional hereditary algebras and independently by Aihara and Iyama in [4] for the general case. It was shown that the Brenner–Butler-tilting module is the ...

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...ift functor (respectively, the shift functor) and under taking extensions and direct summands. A bounded co-t-structure is one of the two ingredients of a Jørgensen–Pauksztello co-stability condition =-=[57]-=-. A typical example of a co-t-structure is the pair pPstdě0 ,P std ď0 q for the homotopy category HbpprojAq of a finite-dimensional algebra A, where Pstdě0 consists of complexes which are homotopy equ...

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...is quickly expanding, and we do not aim to cover everything in full generality. The reader is encouraged to read the original references. More detailed surveys on some of the objects and maps include =-=[105, 63, 31, 78, 90, 66]-=-. In the appendix we provide some results on the derived category of a non-positive dg algebra which are used in Sections 3 and 4. Throughout this article, k denotes an uncountable algebraically close...

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...on-frozen vertex i P Q0 is called green ifsj1 P F | D iÝÑ j1 P Q1 ( “ H. It is called red ifsj1 P F | D j1ÝÑ i P Q1 ( “ H. We warn the reader that we have adopted the conventions opposite to those in =-=[16, 65]-=- to keep coherent with mutations of the categorical objects. When we mutate at a green or red vertex, the above step (4) is redundant because in this case no arrows between frozen vertices are produce...

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...u reduction (Section 3.8), silting reduction and τ-tilting reduction, are respectively introduced by Aihara and Iyama in [4] and by Jasso in [56]. The compatibility of these reductions are studied in =-=[64, 54, 56]-=-. 4.4. The correspondences between silting objects, t-structures, torsion pairs and support τ -tilting modules. Let pQ,W q be a Jacobi-finite quiver with potential, Γ “ pΓpQ,W q and J “ pJpQ,W q. Theo...

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.... If sτ -tiltpAq has a finite connected component, then sτ -tiltpAq itself is connected. Examples of such algebras include representation-finite algebras and preprojective algebras of Dynkin quivers (=-=[81]-=-). Example. Let A be the hereditary algebra of type A2 as in Section 3.1. There are precisely five isomorphism classes of basic support τ -tilting A-modules: S1‘P2 “ A, P2 ‘ S2, S2, S1 and 0. The orde...

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..., Section 2] and [79, Example 4.4]. In general, Question 3.13 is open. It has a positive answer for C “ HbpprojAq, Ordered Exchange Graphs 25 where A is a representation-finite symmetric algebra, see =-=[1, 3]-=-, or A is a piecewise hereditary algebra1. Another direction is to consider presilting objects N which are 2-term with respect to the given silting object M , i.e. there is a triangle M´1 // M0 // N /...

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... 3.7. Simple-minded collections, 2-smcpAq. Let C be a triangulated category. A collection tX1, . . . , Xnu of objects of C is said to be simple-minded (cohomologically Schurian in =-=[5]-=-) if the following conditions hold for i, j “ 1, . . . , n ¨ HompXi,Σ pXjq “ 0, @ p ă 0, ¨ HompXi, Xjq “ $&%k if i “ j,0 otherwise, ¨ X1, . . . , Xn generate C. This notion was first used by Rickard i...

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... identities and non-commutative Donaldson–Thomas invariants [65] and in Ordered Exchange Graphs 7 calculating the complete spectrum of a BPS (Bogomol’nyi–Prasad–Sommerfield) particle in string theory =-=[6, 22, 104]-=- (this also appears implicitly in [37]). The main feature of an ordered exchange graph is the property (i): the arrows in the Hasse quiver of the poset are given by mutations. This property has been e...

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... was initiated by Keller and Vossieck [72]. They showed that the left vertical map is bijective for the case when A is the path algebra of a Dynkin quiver. Parts of this diagram have also appeared in =-=[69, 93]-=-. Remark 4.2. (a) The naturally defined mutations on the sets siltpAq, smcpAq and t-strpAq induce a mutation operation on co-t-strpAq. Let pPě0,Pď0q be a bounded co-t-structure on perpAq and let M “ M...

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...ential W on Q such that the Jacobian algebra of pQ,W q is infinite-dimensional, then mutpQq does not have a sink. 3.10. Clusters, ClpQq. Introduced and further investigated by Fomin and Zelevinsky in =-=[32, 33, 35]-=-, cluster algebras are commutative rings equipped with a distinguished set of generators, the cluster variables which are grouped into overlapping sets of variables, the clusters. The cluster variable...

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...as invariants [65] and in Ordered Exchange Graphs 7 calculating the complete spectrum of a BPS (Bogomol’nyi–Prasad–Sommerfield) particle in string theory [6, 22, 104] (this also appears implicitly in =-=[37]-=-). The main feature of an ordered exchange graph is the property (i): the arrows in the Hasse quiver of the poset are given by mutations. This property has been established by Happel and Unger [49] in...

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...jects of perpΓq and related it to Donaldson–Thomas invariants. Geiss, Leclerc and Schröer took a different approach for stably 2-Calabi–Yau Frobenius categories arising from preprojective algebras in =-=[38, 39, 40]-=- and later they proved in [41] that the two approaches are closely related. Let ClpQq denote the set of clusters of the cluster algebra AQ with principal coefficients (Section 3.10). An important feat...

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.... Remark 4.8. Reduction techniques analogous to 2-Calabi–Yau reduction (Section 3.8), silting reduction and τ-tilting reduction, are respectively introduced by Aihara and Iyama in [4] and by Jasso in =-=[56]-=-. The compatibility of these reductions are studied in [64, 54, 56]. 4.4. The correspondences between silting objects, t-structures, torsion pairs and support τ -tilting modules. Let pQ,W q be a Jacob...

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...“ bn`i,j . Columns of c-matrices are called c-vectors. For instance, we have (̧ pQq “ In and (̧ qQq “ ´In. For more details on c-vectors, we refer the reader to [35] where they were introduced and to =-=[84, 83, 98, 82, 66]-=- where they were studied. With this terminology, for a quiver R P Mutp pQq the vertex i P Q0 is green if and only if the i-th column cipRq of the c-matrix satisfies cipRq ě 0, and the vertex is red if...

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...s a triangle M´1 // M0 // N // ΣM´1 with M´1 and M0 in addpMq. The following result is due to Derksen–Fei [26, Section 5], Aihara [3, Proposition 2.16], Wei [102, Section 6] and Iyama–Jørgensen– Yang =-=[53]-=- in various generalities. The idea is to form an analogue of the Bongartz complement for tilting modules (Theorem 3.1). Proposition 3.14. Let C be a Hom-finite Krull–Schmidt triangulated category with...

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...l elements red. A maximal path in an ordered exchange graph C can thus be interpreted as a maximal sequence of mutations at green elements. These sequences, calledmaximal green sequences by B. Keller =-=[65]-=-, play an important role in finding quantum dilogarithm identities and non-commutative Donaldson–Thomas invariants [65] and in Ordered Exchange Graphs 7 calculating the complete spectrum of a BPS (Bog...

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...is quickly expanding, and we do not aim to cover everything in full generality. The reader is encouraged to read the original references. More detailed surveys on some of the objects and maps include =-=[105, 63, 31, 78, 90, 66]-=-. In the appendix we provide some results on the derived category of a non-positive dg algebra which are used in Sections 3 and 4. Throughout this article, k denotes an uncountable algebraically close...

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... was initiated by Keller and Vossieck [72]. They showed that the left vertical map is bijective for the case when A is the path algebra of a Dynkin quiver. Parts of this diagram have also appeared in =-=[69, 93]-=-. Remark 4.2. (a) The naturally defined mutations on the sets siltpAq, smcpAq and t-strpAq induce a mutation operation on co-t-strpAq. Let pPě0,Pď0q be a bounded co-t-structure on perpAq and let M “ M...

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...“ bn`i,j . Columns of c-matrices are called c-vectors. For instance, we have (̧ pQq “ In and (̧ qQq “ ´In. For more details on c-vectors, we refer the reader to [35] where they were introduced and to =-=[84, 83, 98, 82, 66]-=- where they were studied. With this terminology, for a quiver R P Mutp pQq the vertex i P Q0 is green if and only if the i-th column cipRq of the c-matrix satisfies cipRq ě 0, and the vertex is red if...

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...if the following conditions hold for i, j “ 1, . . . , n ¨ HompXi,Σ pXjq “ 0, @ p ă 0, ¨ HompXi, Xjq “ $&%k if i “ j,0 otherwise, ¨ X1, . . . , Xn generate C. This notion was first used by Rickard in =-=[92]-=- to help constructing derived equivalences of symmetric algebras from stable equivalences. Spherical collections in algebraic geometry [97] are examples of simple-minded collections. In representation...

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... over a finite-dimensional algebra A carries the structure of a simplicial complex. The study of this complex and its poset structure was initiated in [95] and further carried out by Happel and Unger =-=[99, 100, 49, 47, 48]-=-. See also the contributions of Ringel and of Unger in the Handbook of tilting theory [96, 101]. Let A be a finite-dimensional algebra. An A-module T is a pretilting module if Ext1ApT, T q “ 0 and the...

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...nd 2. We indicated in the diagram above also the colour-coding of green and red vertices. More examples can easily be computed using Bernhard Keller’s java applet [67] or the Quiver Mutation Explorer =-=[30]-=-. Proposition 3.19. ([16, Corollary 1.12]) Let Q be a cluster quiver. Then: (1) mutpQq has a unique source, which is r pQs. (2) mutpQq has a sink if and only if r qQs is a vertex in mutpQq and in this...

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...u reduction (Section 3.8), silting reduction and τ-tilting reduction, are respectively introduced by Aihara and Iyama in [4] and by Jasso in [56]. The compatibility of these reductions are studied in =-=[64, 54, 56]-=-. 4.4. The correspondences between silting objects, t-structures, torsion pairs and support τ -tilting modules. Let pQ,W q be a Jacobi-finite quiver with potential, Γ “ pΓpQ,W q and J “ pJpQ,W q. Theo...

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...ivers φ interchanges the vertices 1 and 2. We indicated in the diagram above also the colour-coding of green and red vertices. More examples can easily be computed using Bernhard Keller’s java applet =-=[67]-=- or the Quiver Mutation Explorer [30]. Proposition 3.19. ([16, Corollary 1.12]) Let Q be a cluster quiver. Then: (1) mutpQq has a unique source, which is r pQs. (2) mutpQq has a sink if and only if r ...

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...ity n. Below we say that a map commutes with mutations (respecitvely, preserves partial orders) if it commutes with existing mutations (respectively, preserves existing partial orders). Theorem 4.1. (=-=[75, 68]-=-) Let A “ Γ or J . Then there is a commutative diagram of bijections which commute with mutations and preserve partial orders siltpAq // ))❘❘❘ ❘❘❘ ❘❘❘ ❘❘❘ ❘❘❘ co-t-strpAqoo uu❧❧❧❧ ❧❧❧❧ ❧❧❧ ❧❧❧ t-st...

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...a triangle M´1 // M0 // X // ΣM´1 with M´1 and M0 in addpMq, see [70, Proposition 2.1 (b)]. Combining [107, Theorem 2.6] and [107, Corollary 3.7.2], we obtain the following result. Proposition 3.16. (=-=[107]-=-) Assume that C has a cluster-tilting object M . A rigid object N of C is a cluster-tilting object if and only if |N | “ |M |. The mutation of cluster-tilting objects was introduced by Buan, Marsh, Re...