Quasi-hereditary algebras and generalized Koszul duality

We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to Δ. If a quasi-hereditary algebra Λ is Koszul with respect to Δ, then Λ and the Yoneda extension algebra of Δ are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of Δ is Koszul in the classical sense.