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A new approach to the Koszul property in representation theory using graded subalgebras
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 12 / Issue 1 / January 2013
- Published online by Cambridge University Press:
- 16 May 2012, pp. 153-197
- Print publication:
- January 2013
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- Article
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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 
.
, 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 
.