We prove a structure theorem for triangulated Calabi-Yau categories: An algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable categor...

This thesis is concerned with higher cluster tilting objects in generalized higher cluster categories and tropical friezes associated with Dynkin diagrams. The generalized cluster category arising from a suitable 3-Calabi-Yau differential graded algebra was introduced by C. Amiot. It is Hom-finite, 2-Calabi-Yau and admits a canonical cluster-tilting object. In this thesis, we extend these resul...

We give a complete classification of torsion pairs in the cluster categories associated to tubes of finite rank. The classification is in terms of combinatorial objects called Ptolemy diagrams which already appeared in our earlier work on torsion pairs in cluster categories of Dynkin type A. As a consequence of our classification we establish closed formulae enumerating the torsion pairs in clu...

We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An, Dn, E6, E7 or E8 are triangulated equivalent to ucluster categories of the corresponding Dynkin type. The proof relies on the “Morita” theorem for u-cluster categories by Keller and Reiten, along with the recent computation of Calabi-Yau dimensions of stable module cat...

Clustering geographic data using traditional methods often result in clusters that look dispersed over the geographic space and poorly reflect any underlying spatial structure. We propose a polygon-based spatial clustering approach, which models a spatial object as a polygon with three groups of attributes: general attributes, boundary attributes, and spatial events. We have developed a general...

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to the matrix A (see section 4 for precise meaning). When A is of finite type, these isomorphisms behave nicely, they...

In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradin...

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to the matrix A (see section 4 for precise meaning). When A is of finite type, these isomorphisms behave nicely, they...