The Green ring of a family of copointed Hopf algebras

TL;DR

This paper classifies the Green ring of certain copointed Hopf algebras related to the Fomin-Kirillov algebra, revealing their Morita equivalence to well-known quantum groups and describing their indecomposable modules and tensor product decompositions.

Contribution

It provides a detailed description of the Green ring for a family of copointed Hopf algebras, including module classifications and tensor product structures, extending understanding of their representation theory.

Findings

Morita equivalence to Drinfeld doubles of Taft algebras and small quantum groups

Classification of indecomposable modules over these algebras

Explicit presentation of the Green ring by generators and relations

Abstract

The copointed liftings of the Fomin-Kirillov algebra $\mathcal{FK}_3$ over the algebra of functions on the symmetric group $\mathbb{S}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a generic parameter are Morita equivalent to the non-simple blocks of well-known Hopf algebras: the Drinfeld doubles of the Taft algebras and the small quantum groups $u_{q}(\mathfrak{sl}_2)$. The indecomposable modules over these were classified independently by Chen, Chari--Premet and Suter. Consequently, we obtain the indocomposable modules over the generic liftings of $\mathcal{FK}_3$. We decompose the tensor products between them into the direct sum of indecomposable modules. We then deduce a presentation by generators and relations of the Green ring.