Strong exact Borel subalgebras and strong 2-subalgebras are shown to exist for quasi-hereditary algebras which possess exact Borel subalgebras and 2-subalgebras. This implies that the algebras associated with blocks of category O have strong exact Borel subalgebras and strong 2-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.
1999 Academic Press
1. Introduction
The motivation for studying strong exact Borel subalgebras of quasihereditary algebras comes from Lie theory, thus we start by shortly listing some of the relevant features of Lie theory, in particular those which we want to transfer to associative algebras.
Let g be a finite-dimensional semisimple complex Lie algebras. Fix a Cartan decomposition g=n + Ä h Ä n & . The Lie algebra b + :=n + Ä h is by definition a Borel subalgebra of g. This Borel subalgebra b + is a solvable Lie algebra. The theorem of Poincare , Birkhoff, and Witt implies that the universal enveloping algebra U(g) is a free right module over U(b + ), hence tensor induction U(g) U(b+) & is an exact functor. For each weight * # h* there is a one-dimensional simple b + -module C * on which h acts via *. Inducing this module defines the Verma module 2(*) :=U(g) U(b+) C * of highest weight *. This Verma module 2(*) has a unique simple quotient module L(*) and each simple highest weight module is a quotient of a Verma module. Character theory of g is the problem to determine the dimensions of the weight spaces of simple highest weight modules or equivalently to determine the (finitely many) simple composition factors of a Verma module 2(*). This problem is solved by Kazhdan Lusztig conjecture (which is a theorem, see [4,7]).