  Given a quasi-hereditary algebra $B$, we present conditions which guarantee
that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is
Koszul and at the same time inherits the quasi-hereditary property and other
good Lie-theoretic properties that $B$ might possess. The method involves
working with a pair $(A,{\mathfrak a})$ consisting of a quasi-hereditary
algebra $A$ and a (positively) graded subalgebra $\mathfrak a$. The algebra $B$
arises as a quotient $B=A/J$ of $A$ by a defining ideal $J$ of $A$. Along the
way, we also show that the standard (Weyl) modules for $B$ have a structure as
graded modules for $\mathfrak a$. These results are applied to obtain new
information about the finite dimensional algebras (e.g., the $q$-Schur
algebras) which arise as quotients of quantum enveloping algebras. Further
applications, perhaps the most penetrating, yield results for the finite
dimensional algebras associated to semisimple algebraic groups in positive
characteristic $p$. These results require, at least presently, considerable
restrictions on the size of $p$.
