On projective modules over finite quantum groups

TL;DR

This paper studies the structure of projective modules over finite quantum groups, showing they are filtered by Verma modules, establishing BGG reciprocity, and characterizing simple Verma modules, with implications for tensor products.

Contribution

It demonstrates that projective modules over the Drinfeld double of finite quantum groups are filtered by Verma modules and proves BGG reciprocity in this setting.

Findings

Projective modules are filtered by Verma modules.

BGG reciprocity holds for these modules.

A Verma module is simple if and only if it is projective.

Abstract

Let $\mathcal{D}$ be the Drinfeld double of the bosonization ${\mathfrak B}(V)\#\Bbbk G$ of a finite-dimensional Nichols algebra ${\mathfrak B}(V)$ 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.