Irreducible representations of pointed Hopf algebras of type $A_2$

TL;DR

This paper classifies irreducible representations of certain finite-dimensional pointed Hopf algebras of type A2, revealing their infinite representation type and constructing indecomposable modules of all dimensions.

Contribution

It provides a complete classification of irreducible modules for a family of pointed Hopf algebras of type A2 and analyzes their representation type.

Findings

Algebras have infinite representation type.

Constructed indecomposable modules of all natural dimensions.

Studied a semisimple quotient category of representations.

Abstract

We classify the irreducible representations of a family of finite-dimensional pointed liftings $H_\lambda$ of the Nichols algebra associated with the diagram $A_2$ with parameter $q=-1$. We show that these algebras have infinite representation type and construct an indecomposable $H_\lambda$-module of dimension $n$ for each $n\in\mathbb{N}$. Finally, we study a semisimple category $\underline{\operatorname{Rep}} H_\lambda$ arising as a quotient of $\operatorname{Rep} H_\lambda$.