On projective modules over finite quantum groups

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