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

TL;DR

This paper demonstrates that quasi-hereditary algebras in Lie theory have an analogue of the PBW theorem, showing each has an exact Borel subalgebra and relating module categories to directed boxes via $A_{}$-structures.

Contribution

It establishes an analogue of the PBW theorem for quasi-hereditary algebras and links module categories with directed boxes constructed from $A_{}$-structures.

Findings

Every quasi-hereditary algebra has an exact Borel subalgebra up to Morita equivalence.

The category of modules with standard filtration is equivalent to representations of a directed box.

The constructed box's underlying algebra is an exact Borel subalgebra.

Abstract

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.