Abstract. Building on work by Geiss-Leclerc-Schröer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in 'Cluster algebras IV: Coefficients'. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.
Inspired by the tropical dualities in cluster algebras, we introduce c-vectors for finite-dimensional algebras via τ -tilting theory. Let A be a finitedimensional algebra over a field k. Each c-vector of A can be realized as the (negative) dimension vector of certain indecomposable A-module and hence we establish the sign-coherence property for this kind of c-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras including quasitilted algebras and cluster-tilted algebras. In particular, we recover the equalities of c-vectors for acyclic cluster algebras and skew-symmetric cluster algebras of finite type respectively obtained by Nájera Chávez. To this end, a short proof for the sign-coherence of c-vectors for skew-symmetric cluster algebras has been given in the appendix. introductionThe c-vectors and g-vectors introduced by Fomin-Zelevinsky [15] are two kinds of integer vectors, which have played important roles in the theory of cluster algebras with coefficients. Both the vectors are conjectured to have a so-called signcoherence property [15], which has been recently proved by Gross-Hacking-Keel-Kontsevich [21]. For skew-symmetric cluster algebras, Nakanishi [34] found the socalled tropical dualities between c-vectors and g-vectors (cf. also [25,31,38] ). With the assumption of sign-coherence of c-vectors, the tropical dualities between c-vectors and g-vectors has been further generalized to skew-symmetrizable cluster algebras by Nakanishi-Zelevinsky [36]. Moreover, they showed that many properties or conjectures of cluster algebras follow from the tropical dualities and hence follow from the sign-coherence of c-vectors.On the other hand, c-vectors may be seen as a generalization of root systems. It follows from Nagao's work [31] that each c-vector of a given skew-symmetric cluster algebra can be realized as the (negative) dimension vector of certain exceptional module for the associated Jacobian algebra. In particular, the set of c-vectors of an acyclic cluster algebra is a subset of the real Schur roots for the corresponding Kac-Moody algebra. In [32], Nájera Chávez showed the inverse inclusion is also true for acyclic cluster algebras (cf. also [42]). Moreover, he also proved in [33] 1 2 CHANGJIAN FU that the set of positive c-vectors of a skew-symmetric cluster algebra of finite type coincides with the set of dimension vectors of all the exceptional modules over the corresponding representation-finite cluster-tilted algebra. Nakanishi-Stella [35] gave a diagrammatic description of c-vectors for cluster algebras of finite type. They proposed the root conjecture for any cluster algebras: for any skew-symmetrizable matrix B any c-vector of the cluster algebra A(B) is a root of the associated Kac-Moody algebra g(A(B)) , where A(B) is the Cartan counterpart of B. We refer to [35] for more details on c-vectors of cluster algebras of finite type.In this paper, we pursue the representation-theoretic approach to study c-vectors.We introduce the notion of c-vector for any fini...
For any given symmetrizable Cartan matrix C with a symmetrizer D, Geiß et al. (2016) introduced a generalized preprojective algebra Π(C, D). We study tilting modules and support τtilting modules for the generalized preprojective algebra Π(C, D) and show that there is a bijection between the set of all cofinite tilting ideals of Π(C, D) and the corresponding Weyl group W (C) provided that C has no component of Dynkin type. When C is of Dynkin type, we also establish a bijection between the set of all basic support τ -tilting Π(C, D)-modules and the corresponding Weyl group W (C). These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035-1079, 2009) and Mizuno (Math. Zeit. 277(3), 665-690, 2014) over classical preprojective algebras.Recently, Geiß et al. [10] introduced a class of Iwanaga-Gorenstein algebras via quivers with relations for any symmetrizable Cartan matrices with symmetrizers, which generalizes the path algebras of quivers associated with symmetric Cartan matrices. They also introduced the corresponding generalized preprojective algebras. This new class of preprojective algebras reduces to the classical one provided that the Cartan matrix is symmetric and the symmetrizer is the identity matrix. Surprisingly, the generalized preprojective algebras still share many properties with the classical one. Since the classical preprojective algebras have many important applications in different fields of mathematics, it is an interesting question to find out which results or constructions for classical preprojective algebras can be generalized to the general setting. For example, if one can generalize the constructions of [4,9] to the new preprojective algebras, then one may obtain new categorifications for certain skew-symmetrizable cluster algebras. This note gives a first attempt to generalize certain classification results in tilting theory of preprojective algebras to this new setting. For a given algebra, a basic question in tilting theory is to classify all the tilting modules or support τ -tilting modules. For the classical preprojective algebras, the classification has been obtained by Buan et al. [4] for preprojective algebras of non-Dynkin type (cf.
Abstract. For an integer w, let S w be the algebraic triangulated category generated by a w-spherical object. We determine the Picard group of S w and show that each orbit category of S w is triangulated and is triangle equivalent to a certain orbit category of the bounded derived category of a standard tube. When n = 2, the orbit category2 is 2-periodic triangulated, and we characterize the associated Ringel-Hall Lie algebra in the sense of Peng and Xiao.MSC classification 2010: 17B99, 18E30, 16E35, 16E45.
We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a clustertilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.
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