Strong Exact Borel Subalgebras of Quasi-hereditary Algebras and Abstract Kazhdan–Lusztig Theory

Strong exact Borel subalgebras and strong Δ-subalgebras are shown to exist for quasi-hereditary algebras which possess exact Borel subalgebras and Δ-subalgebras. This implies that the algebras associated with blocks of category O have strong exact Borel subalgebras and strong Δ-subalgebras. The structure of these subalgebras is shown to be closely related to abstract Kazhdan–Lusztig theory. The main technical tool in this paper is a construction which has an exact Borel subalgebras (of a given quasi-hereditary algebra) as input and a strong exact Borel subalgebra as output. From this result, Morita invariance of the existence of exact Borel subalgebras is derived.