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

Published in Advances in Mathematics, 2014

Recommended citation: Steffen Koenig, Julian Külshammer, and Sergiy Ovsienko (2014). "Quasi-hereditary algebras, exact Borel subalgebras, A-algebras and boxes." Advances in Mathematics. 262. https://doi.org/10.1016/j.aim.2014.05.016

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 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 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-structure on Ext(Δ,Δ). Its underlying algebra is an exact Borel subalgebra.