Cluster-categories, cluster-tilted algebras and derived equivalences

• Applicant: Professor Dr. Thorsten Holm
• Subject Area: Mathematics
• Term: from 2009 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 123657482

Cluster algebras form an exciting new area of mathematics, linking representation theory, algebraic Lie theory, combinatorics and algebraic geometry. Cluster categories are a categorification of cluster algebras and allow to apply deep techniques from representation theory. In the proposed project we plan to study the structure of m-cluster categories and of the corresponding m-cluster-tilted algebras (endomorphism algebras of certain objects in the m-cluster categories). One of the main goals is to understand when cluster-tilted algebras, e.g. of Dynkin or extended Dynkin types, have equivalent derived categories. This will also require to obtain new results on derived invariants of these algebras. Moreover, we are looking for new and surprising occurrences of cluster behaviour in different locations. For instance, we recently studied an interesting triangulated category which shares cluster phenomena, but would be of type A1. We aim at finding more such surprising examples which should lead to completely new insights.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1388: Representation Theory