Cluster tilting vs. weak cluster tilting in Dynkin type A infinity

TL;DR

This paper explores a new phenomenon in higher cluster tilting theory within a specific algebraic triangulated category, revealing differences in mutation behaviour between cluster tilting and weak cluster tilting subcategories.

Contribution

It demonstrates that weakly d-cluster tilting subcategories can exhibit more complex mutation patterns than their cluster tilting counterparts in Dynkin type A infinity.

Findings

d-cluster tilting subcategories have exactly d mutations per indecomposable object

Weakly d-cluster tilting subcategories can have objects with fewer mutations, up to  mutations

The category studied is generated by a (d+1)-spherical object, representing a higher cluster category of Dynkin type A infinity

Abstract

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell mutations. The category C is the algebraic triangulated category generated by a (d+1)-spherical object and can be thought of as a higher cluster category of Dynkin type A infinity.