TITLE: How to recognize a finite dimensional algebra is Koszul or quadratic?
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: One incredibly useful fact is that it suffices to find linear resolutions of standard modules, not simples.  These are usually much easier to find "by hand." Strictly speaking this is stronger than Koszul (the term is "standard Koszul") but in practice it seems rare for a quasi-hereditary algebra to be Koszul and not standard Koszul.