Quasi-hereditary algebras and generalized Koszul duality

We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to $\mathrm{\Delta }$. If a quasi-hereditary algebra $\text{L}$ is Koszul with respect to $\mathrm{\Delta }$, then $\text{L}$ and the Yoneda extension algebra of $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ is Koszul in the classical sense.