Quasi-hereditary covers of higher zigzag-algebras

The aim of this paper is to define and study some quasi-hereditary covers for higher zigzag algebras. We will show how these algebras satisfy three different Koszul properties: they are Koszul in the classical sense, standard Koszul and Koszul with respect to the standard module $\mathrm{\Delta }$, according with the definition given in \cite{Madsen1}. This last property gives rise to a well defined duality and the $\mathrm{\Delta }$-Koszul dual will be computed as the path algebra of a quiver with relations.