Linkage principle for small quantum groups

Abstract

We consider small quantum groups with root systems of Cartan, super and modular type, among others. These are constructed as Drinfeld doubles of finite-dimensional Nichols algebras of diagonal type. We prove a linkage principle for them by adapting techniques from the work of Andersen, Jantzen and Soergel in the context of small quantum groups at roots of unity. Consequently we characterize the blocks of the category of modules. We also find a notion of (a)typicality similar to the one in the representation theory of Lie superalgebras. The typical simple modules turn out to be the simple and projective Verma modules. Moreover, we deduce a character formula for 1-atypical simple modules.