The Green rings of the generalized Taft Hopf algebras

TL;DR

This paper analyzes the structure of the Green ring of generalized Taft algebras, identifying nilpotent elements, the Jacobson radical, and indecomposable representations, extending previous algebraic results.

Contribution

It provides a detailed description of the nilpotent elements, radical, and indecomposable representations of the Green ring of generalized Taft algebras, extending prior work.

Findings

Nilpotent elements are sums of indecomposable projective representations.

The Jacobson radical is generated by a single element with rank n - n/d.

All finite-dimensional indecomposable representations over the complexified Green ring are classified.

Abstract

In this paper, we investigate the Green ring $r(H_{n,d})$ of the generalized Taft algebra $H_{n,d}$, extending the results of Chen, Van Oystaeyen and Zhang in \cite{Coz}. We shall determine all nilpotent elements of the Green ring $r(H_{n,d})$. It turns out that each nilpotent element in $r(H_{n,d})$ can be written as a sum of indecomposable projective representations. The Jacobson radical $J(r(H_{n,d}))$ of $r(H_{n,d})$ is generated by one element, and its rank is $n-n/d$. Moreover, we will present all the finite dimensional indecomposable representations over the complexified Green ring $R(H_{n,d})$ of $H_{n,d}.$ Our analysis is based on the decomposition of the tensor product of indecomposable representations and the observation of the solutions for the system of equations associated to the generating relations of the Green ring $r(H_{n,d})$.