The cluster category of Dynkin type $A_\infty$

The cluster category of Dynkin type $A_\infty$

Tue, 01/06/2010
17:00 		Peter Jorgensen (Newcastle) Algebra Seminar Add to calendar L2
The cluster category of Dynkin type $ A_\infty $
   The cluster category of Dynkin type $ A_\infty $ is a ubiquitous object with interesting properties, some of which will be explained in this talk.
   Let us denote the category by $ \mathcal{D} $. Then $ \mathcal{D} $ is a 2-Calabi-Yau triangulated category which can be defined in a standard way as an orbit category, but it is also the compact derived category $ D^c(C^{∗}(S^2;k)) $ of the singular cochain algebra $ C^*(S^2;k) $ of the 2-sphere $ S^{2} $. There is also a “universal” definition: $ \mathcal{D} $ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.
  Just like cluster categories of finite quivers, $ \mathcal{D} $ has many cluster tilting subcategories, with the crucial difference that in $ \mathcal{D} $, the cluster tilting subcategories have infinitely many indecomposable objects, so do not correspond to cluster tilting objects.
   The talk will show how the cluster tilting subcategories have a rich combinatorial structure: They can be parametrised by “triangulations of the $ \infty $-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.
  This will be used to show how to obtain a subcategory of $ \mathcal{D} $ which has all the properties of a cluster tilting subcategory, except that it is not functorially finite. There will also be remarks on how $ \mathcal{D} $ generalises the situation from Dynkin type $ A_n $ , and how triangulations of the $ \infty $-gon are new and interesting combinatorial objects.