On projective modules over finite quantum groups

CRISTIAN VAY

TRANSFORMATION GROUPS

BIRKHAUSER BOSTON INC

Año: 2019 vol. 24 p. 279 - 299

1083-4362

Let D be the Drinfeld double of the bosonization B(V)#kG of a finite-dimensional Nichols algebra B(V) over a finite group G. It is known that the simple D-modules are parametrized by the simple modules over D(G), the Drinfeld double of G. This parametrization can be obtained by considering the head L(lambda) of the Verma module M(lambda) for every simple D(G)-module lambda. In the present work, we show that the projective D-modules are filtered by Verma modules and the BGG Reciprocity [P(mu):M(lambda)]=[M(lambda):L(mu)]$ holds for the projective cover P(mu) of L(mu). We use graded characters to prove the BGG Reciprocity and obtain a graded version of it. As a by-product we show that a Verma module is simple if and only if it is projective. We also describe the tensor product between projective modules.