On $\tau$-tilting finite simply connected algebras

TL;DR

This paper proves that $ au$-tilting finite simply connected algebras are representation-finite, explores related algebra classes, and classifies sincere algebras with this property, providing a comprehensive list.

Contribution

It establishes the equivalence between $ au$-tilting finiteness and representation-finiteness for simply connected algebras and classifies sincere cases.

Findings

$ au$-tilting finite simply connected algebras are representation-finite

Reduction of $ au$-tilting finiteness from non-sincere to sincere algebras

Complete classification of $ au$-tilting finite sincere simply connected algebras

Abstract

We show that $\tau$-tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the $\tau$-tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of $\tau$-tilting finite sincere simply connected algebras.