Subalgebras of Quasi-hereditary Algebras Arising from Algebraic and Quantum Groups

Quasi-hereditary algebras arise in several natural contexts. For example, their module categories often appear in connection with the representation theory of algebraic groups over elds of positive characteristic or of quantum groups at a root of unity. This point of view has been exploited by various authors, e. g., [CPS3{8], [G2], etc. Classical Schur algebras attached to the general linear groups GLn (see [G1]) as well as their quantized versions associated to the quantum general linear groups (see [DD, PW], etc.) provide typical examples of quasi-hereditary algebras arising in this way. A second importance source of quasi-hereditary algebras comes from geometry, in the study of perverse sheaves. For example, it is proved in [PS1, PS2] that the perverse sheaf category on a suitably nice strati ed topological space X is the module category for a quasi-hereditary algebra. The rst example of a Borel subalgebra of a quasi-hereditary algebra was given by Green [G2], though he did not give a general de nition of the concept. Later, Konig [K1] introduced the notion of an exact Borel subalgebra, which did not t Green's example, but did apply for the category O associated to a simple complex Lie algebra. The second author of this paper introduced in [K1; appendix] a further notion, which synthesized these two cases, but still does not obviously capture other interesting examples, such as given in [Dy] and [DR]. The notion of a Borel subalgebra, introduced in Section 2, does seem to capture all the known examples, and it is su ciently rich to have an interesting theory. The original notion of [K1; appendix] is recast here as a homological Borel subalgebra,