Published online by Cambridge University Press: 16 May 2012
Given a quasi-hereditary algebra , we present conditions which guarantee that the algebra obtained by grading by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that might possess. The method involves working with a pair consisting of a quasi-hereditary algebra and a (positively) graded subalgebra . The algebra arises as a quotient of by a defining ideal of . Along the way, we also show that the standard (Weyl) modules for have a structure as graded modules for . These results are applied to obtain new information about the finite dimensional algebras (e.g., the -Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated with semisimple algebraic groups in positive characteristic . These results require, at least at present, considerable restrictions on the size of .
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