$m$-cluster tilted algebras of euclidean type

TL;DR

This paper characterizes when $m$-cluster tilted algebras of Euclidean type are representation finite, using geometric and algebraic methods, and describes their structure in terms of angulations and relations.

Contribution

It provides necessary and sufficient conditions for representation finiteness of $m$-cluster tilted algebras of Euclidean type, including a geometric description for type $ ilde{A}$.

Findings

Characterization of representation finite $m$-cluster tilted algebras of Euclidean type.

Description of representation finite type via $(m+2)$-angulations for type $ ilde{A}$.

Identification of certain infinite type algebras as $m$-relation extensions of tilted algebras.

Abstract

We consider $m$-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the case $\widetilde{A}$, using the geometric realization, we get a description of representation finite type in terms of $(m+2)$-angulations. We establish which $m$-cluster tilted algebras arise at the same time from quivers of type $A$ and $\widetilde{A}$. Finally, we characterize representation infinite $m$-cluster tilted algebras arising from a quiver of type $\widetilde{A}$ as $m$-relations extensions of some iterated tilted algebra of type $\widetilde{A}$.