On the radical of Cluster tilted algebras

TL;DR

This paper determines the minimal power at which the radical of the module category of a representation-finite cluster tilted algebra becomes zero, linking it to the quiver's vertices, and explores morphism compositions within this context.

Contribution

It provides a formula for the nilpotency index of the radical in representation-finite cluster tilted algebras based on quiver vertices and analyzes morphism compositions.

Findings

The minimal power of the radical's nilpotency index is determined by the quiver's vertices.

Explicit bounds for the nilpotency index in self-injective cases are given.

The structure of irreducible morphism compositions in the radical is characterized.

Abstract

We determine the minimal lower bound $n$, with $n \geq 1$, where the $n$-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of $m$, $m \ge 2$, irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the $(m+1)$-th power of the radical of their module category.