$\tau$-Tilting Finite Algebras With Nonempty Left Or Right Parts Are Representation-Finite

TL;DR

This paper establishes that a finite dimensional algebra with a nonempty left or right part is $ au$-tilting finite if and only if it is representation-finite, linking $ au$-tilting finiteness to classical finiteness.

Contribution

It proves a characterization of $ au$-tilting finite algebras with nonempty parts, connecting $ au$-tilting finiteness directly to representation finiteness.

Findings

$ au$-tilting finite iff representation-finite for algebras with nonempty parts

Provides a criterion to determine $ au$-tilting finiteness based on classical finiteness

Establishes a new link between $ au$-tilting theory and representation theory

Abstract

Let $\Lambda$ be a finite dimensional algebra such that $\mathcal{L}_{\Lambda}$ or $\mathcal{R}_{\Lambda}\neq\emptyset$. Then $\Lambda$ is $\tau$-tilting finite if and only if $\Lambda$ is representation-finite.