Wei Xing: Cluster tilting for higher Nakayama algebras

Abstract

Auslander-Reiten theory is a fundamental tool to study  representation theory from a homological point of view. A higher dimensional analogue, developed by Iyama, is naturally framed in terms of dZ-cluster tilting subcategories in both abelian and triangulated settings. In this thesis, we develop methods to  construct and identify such subcategories and show that they witness derived equivalences, with applications to higher Auslander algebras of type A, as well as singular equivalences, with applications to higher Nakayama algebras.

In Paper I, we construct a singular equivalence between d-homological pairs (A, M) and (B, N). An as application, we show that every d-Nakayama algebra is singular equivalent to a self-injective d-Nakayama algebra. This equivalence can be recovered from the combinatorial invariant given by the resolution quiver. Our result generalizes the classical singular equivalence for Nakayama algebras and provides an alternative proof of a similar equivalence of higher Nakayama algebras due to McMahon.

In Paper II, we introduce 2-subhomogeneous d-representation finite algebras and show how they can be   constructed using certain tilting complexes over a fractionally Calabi-Yau algebra. As an application, for each higher Auslander algebra of type A satisfying certain coprimality condition, we obtain a new derived equivalence induced by an explicit tilting complex. This yields certain replicated algebras that are 2-subhomogeneous d-representation finite.

In Paper III, we study which d-Nakayama algebras admit an ndZ-cluster tilting subcategory for an integer n>1. The radical square zero case is already covered by results on classical Nakayama algebras due to Herschend-Kvamme-Vaso.For each remaining non-self-injective d-Nakayama algebra, we give a complete classification of its ndZ-cluster tilting subcategories,showing that at most one exists for a suitable integer n. For self-injective d-Nakayama algebras satisfying an additional condition,we show that such a subcategory does exist by constructing an explicit example, applying the methods of Darpö-Iyama to the algebras obtained in Paper II.