Representations of pointed Hopf algebras and their Drinfel'd quantum doubles

TL;DR

This paper investigates the representation theory of certain pointed Hopf algebras and their Drinfel'd doubles, constructing modules, classifying simples, and analyzing their structure.

Contribution

It introduces a family of Verma-type modules, provides a parametrization of simple modules, and computes their Loewy and socle series.

Findings

Constructed Verma-type modules for the algebras and their doubles.

Parametrized simple modules and described their bases and dimensions.

Determined Loewy and socle series under mild restrictions.

Abstract

We study representations of nilpotent type nontrivial liftings of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma- type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.