Christoffer Söderberg: Higher Auslander-Reiten Theory for Species with Relations

Abstract

One of the main topics in higher dimensional Auslander-Reiten theory is the study of so-called d-representation finite algebras. These are algebras of global dimension d that admit a d-cluster tilting module. In particular, 1-representation finite algebras are representation finite and hereditary. Over an algebraically closed field 1-representation finite algebras are therefore classified as path algebras of Dynkin quivers by Gabriel's Theorem. Over a general field (not necessarily algebraically closed) there is a similar classification result of 1-representation finite algebras due to Dlab and Ringel, in which Dynkin quivers are replaced by species of Dynkin type. In this thesis we study d-representation finite algebras over general fields and (in the spirit of the Dlab-Ringel Theorem) produce d-representation finite algebras given by species with relations.

The thesis consists of three papers. The first paper describes preprojective algebras of the representation finite species that appear in the classification of Dlab and Ringel. It is shown that they are almost Koszul. Moreover, taking tensor products of l-homogeneous representation finite species one obtains 2-representation finite algebras. Their 3-preprojective algebras are described using Segre products. The second paper uses species with potentials, which generalise quivers with potentials to describe the 3-preprojective algebras from the previous paper. Buan, Iyama, Reiten and Smith showed that there is a strong connection with mutation of quivers with potential and mutation in 2-Calabi-Yau categories. We generalise some of their results to species with potentials. This leads to new examples of self-injective species with potentials, which fit in the derived Auslander-Iyama correspondence due to Jasso and Muro. The third paper considers 2-APR tilting of 2-representation finite algebras whose 3-preprojective algebras are given by species with potentials. It is shown that under certain conditions 2-APR tilting is given by cut mutation (generalising a result by Herschend and Iyama for quivers with potentials). Moreover, a sufficient condition is given for transitivity of cut mutation showing that in these cases all 2-representation finite algebras that appear for the same species with potentials are derived equivalent.