EXACT BOREL SUBALGEBRAS OF QUASI-HEREDITARY ALGEBRAS

By analogy with the theory of Lie groups and Lie algebras, we examine (exact) Borel subalgebras of a quasi-hereditary algebra A. We prove their existence if A is associated with a block of the category O of a finite-dimensional semisimple complex Lie algebra or if A is a generalized Schur algebra of a simply connected semisimple algebraic group over an algebraically closed field. In these cases, the structure of exact Borel subalgebras is closely related to the conjectures of Kazhdan-Lusztig and of Lusztig respectively.