On projective modules over finite quantum groups

Abstract

Let (Formula presented.) be the Drinfeld double of the bosonization (Formula presented.)(V)(Formula presented.)G of a finite-dimensional Nichols algebra (Formula presented.)(V ) over a finite group G. It is known that the simple (Formula presented.)-modules are parametrized by the simple modules over (Formula presented.)(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 (Formula presented.)(G)-module λ. In the present work, we show that the projective (Formula presented.)-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.