I have a family of finite dimensional algebras that are directed quasihereditary. I think they might be Koszul algebras and I am wondering what approaches there are to check Koszulness or even quadraticity. I know the quivers of these algebras and can compute Ext^n between simple modules for all n, but I do not have a quiver presentation. I know that there are paths of length 2 and of higher lengths between all vertices of the quiver with nonvanishing Ext^2 so I cannot prove or eliminate quadraticity for trivial reasons. Any thoughts?
I should add that I do not have explicit minimal projective resolutions of the simple modules.