On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon

Abstract

Let k be a field and let D be a k-linear algebraic triangulated category with split idempotents. Let be the suspension functor of D and let s be a 2-spherical object of D, that is, the morphism space D(s, i s) is k for i = 0 and i = 2 and vanishes otherwise. Assume that s classically generates D, that is, each object of D can be built from s using (de)suspensions, direct sums, direct summands, and distinguished triangles. It was proved in [15, thm. 2.1] that D is uniquely determined by these properties. As we will explain, D is a good candidate for a cluster category of Dynkin type A∞. For instance, we show that there is a bijection between the cluster tilting subcategories of D and certain triangulations of the∞-gon. We use this to give an example of a subcategory A which is weak cluster tilting, that is, satisfies A = ( −1A )⊥ = ⊥( A ), but fails to be functorially finite. Perpendicular