ON PROJECTIVE MODULES OVER FINITE QUANTUM GROUPS

Let D be the Drinfeld double of the bosonization V( V )k G of a finite-dimensional Nichols algebra V( 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(λ) of the Verma module M(λ) for every simple D ( G )-module λ. In the present work, we show that the projective D -modules are filtered by Verma modules and the BGG Reciprocity [P( μ ): M(λ)] = [M(λ): L( μ )] holds for the projective cover P( μ ) of L( μ ). 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.