Oberseminar zur Algebra und Algebraischen Kombinatorik
Montag, 8.6.2009, ab 16:30 Uhr in Raum A 410
A cluster structure of type A infinity and triangulations of the infinity-gon
PD Dr. Thorsten Holm (Hannover)
Cluster categories have been very successful in modelling certain aspects of Fomin-Zelevinsky's cluster algebras algebraically. Usually, cluster categories are defined from finite quivers, as orbit categories of the derived category of the path algebra.
The talk concerns a certain triangulated category which in many ways behaves like a cluster category, in particular it is 2-Calabi-Yau, but whose Auslander-Reiten quiver is of tree class A infinity.
We show that this category has lots of cluster tilting subcategories and as a main result we parametrise them combinatorially by 'triangulations of the infinity-gon'.
Both cluster tilting subcategories and triangulations have a mutation rule, and the rules correspond to each other under the parametrisation.
Many open questions arise from such cluster structures of infinite Dynkin type.
This is a report on joint work with Peter Jorgensen (Newcastle).