Quantum doubles of certain rank two pointed Hopf algebras

TL;DR

This paper studies a specific class of rank two pointed Hopf algebras, describing their Drinfel'd double modules, computing tensor product socles, and identifying conditions for irreducibility, advancing understanding of their representation theory.

Contribution

It provides a detailed description of simple modules of the Drinfel'd double for these Hopf algebras and characterizes when tensor products are irreducible.

Findings

Description of simple modules of the Drinfel'd double

Formula for the socle of tensor products

Conditions for complete irreducibility of tensor products

Abstract

A certain class of rank two pointed Hopf algebras is considered. The simple modules of their Drinfel'd double is described using Radford's method \cite{rad}. The socle of the tensor product of two such modules is computed and a formula similar to the one in \cite{one} is obtained in some conditions. Cases when such a tensor product is completely irreducible are also given in the last section.