Quasi-hereditary algebras, exact Borel subalgebras, A∞A∞-categories and boxes

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 OO. 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 F(Δ)F(Δ) of modules with standard (Verma, Weyl, …) 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∞A∞-structure on Ext⁎(Δ,Δ)Ext⁎(Δ,Δ). Its underlying algebra is an exact Borel subalgebra.