On the relations for the cluster tilted algebra resulting from a monomial tilted algebra

Abstract

First constructed by Fomin and Zelevinski [13], cluster algebras have been studied from many different perspectives. One such perspective is the study of cluster tilted algebras. We focus on when C is a monomial tilted algebras and $\tilde{C}$ its associated cluster tilted algebra. We show the set of partial derivatives of the Keller potential form a minimal set of relations for $\tilde{C}$ and we show that if C is also Koszul, then there are overlap relations that can be used to determine if $\tilde{C}$ is Koszul. We use the tools of noncommutative Gröbner basis theory to prove these results.