Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category $\mathcal O$. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category $\mathcal{ F}(\Delta)$ of modules with standard (Verma, Weyl, \dots) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the $A_{ \infty}$-structure on $\mathcal{ F}(\Delta)$. Its underlying algebra is an exact Borel subalgebra.
Categories: math.RT math.SG