Linkage principle for small quantum groups

TL;DR

This paper establishes a linkage principle for small quantum groups of various types, characterizes their module categories, and introduces a notion of atypicality akin to Lie superalgebra representation theory.

Contribution

It adapts techniques from classical quantum group theory to small quantum groups, providing a linkage principle and character formulas for atypical modules.

Findings

Proved a linkage principle for small quantum groups.

Characterized blocks of the module category.

Derived character formulas for 1-atypical simple modules.

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.