# $\tau$-Tilting Finite Cluster-Tilted Algebras

## TL;DR

This paper establishes a precise equivalence between $ au$-tilting finiteness and representation-finiteness for cluster-tilted algebras, deepening understanding of their structural properties.

## Contribution

It proves that a cluster-tilted algebra is $ au$-tilting finite if and only if it is representation-finite, providing a complete characterization.

## Key findings

- $ au$-tilting finiteness coincides with representation-finiteness for cluster-tilted algebras
- The paper offers a new criterion for classifying cluster-tilted algebras
- Enhances understanding of the relationship between $ au$-tilting theory and representation theory

## Abstract

Let B be a cluster-tilted algebra. We prove that B is $\tau$-tilting finite if and only if B is representation-finite.