We investigate how to characterize subcategories of abelian categories in terms intrinsic axioms. In particular, we find axioms which generating cogenerating functorially finite subcategories, precluster tilting and cluster categories. As a consequence prove that any d-abelian category is equivalent d-cluster subcategory an category, without assumption on the being projectively generated.

Let X be a weighted projective line and CX the associated cluster category. It is known that can realized as generalized category of quiver with potential. In this note, under assumption has at most three weights or tubular type, we prove if C(Q,W) Jacobi-finite non-degenerate potential (Q,W) shares 2-CY tilted algebra CX, then triangle equivalent to CX. As byproduct, determined by its provided...

We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture is its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices. 0 Introduction Cluster categories were introduced in [BMRRT] and, independently, in [CCS1] for type An, as a means ...

We show that the m-cluster category of type Dn is equivalent to a certain geometrically-defined category of arcs in a punctured regular nm − m + 1-gon. This generalises a result of Schiffler for m = 1. We use the notion of the mth power of a translation quiver to realise the m-cluster category in terms of the cluster category. Introduction Let k be a field and Q a quiver of Dynkin type ∆. Let D...

Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.

The present study demonstrates the application of cluster analysis to examine the typology of student categories of novice Luxembourgish teachers. Student categories are mental representations of groups of students in which teachers classify their students. The investigation of student categories is a relevant topic in education, because subsequent assessments of students may be biased by prior...

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi-Yau triangulated categories, in particular stable categories of maximal Cohen-Macaul...

Subgroups within a population are often difficult to discover and describe except by subjective methods. In this study, cluster analysis (numerical taxonomy) methods were used on selected craniofacial measurements obtained from 308 North American White children of both sexes in the age range 6-18 to derive categories of skeletal facial types. Two different cluster analysis approaches were used ...

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (− ‖−) on any pair of “colored” almost positive real Schur roots which generalizes previous de...