Tilting mutation for $m$-replicated algebras

TL;DR

This paper explores the properties of tilting modules over $m$-replicated algebras and demonstrates how $m$-cluster mutations can be realized within module categories, extending previous duplicated algebra results.

Contribution

It generalizes the realization of $m$-cluster mutations from duplicated algebras to $m$-replicated algebras, providing new insights into their module categories.

Findings

Properties of complements to faithful almost complete tilting modules are characterized.

$m$-cluster mutation can be realized in the module category of $A^{(m)}$.

Generalization of previous results on duplicated algebras to $m$-replicated algebras.

Abstract

Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $A^{(m)}$ be the $m$-replicated algebra of $A$ and $\mathscr{C}_{m}(A)$ be the $m$-cluster category of $ A$. We investigate properties of complements to a faithful almost complete tilting $A^{(m)}$-module and prove that the $m$-cluster mutation in $\mathscr{C}_{m}(A)$ can be realized in ${\rm mod} A^{(m)}$, which generalizes corresponding results on duplicated algebras established in [Z1].