The Green Rings of Taft algebras

TL;DR

This paper computes the Green ring of Taft algebras, revealing it as a commutative ring generated by two elements with recursive relations, supported by explicit examples for small n.

Contribution

It provides the first explicit description of the Green ring structure for Taft algebras, including recursive relations and concrete examples.

Findings

Green ring is commutative and generated by two elements

Relations in the Green ring are defined recursively

Explicit examples for n=2 to 8 are provided

Abstract

We compute the Green ring of the Taft algebra $H_n(q)$, where $n$ is a positive integer greater than 1, and $q$ is an $n$-th root of unity. It turns out that the Green ring $r(H_n(q))$ of the Taft algebra $H_n(q)$ is a commutative ring generated by two elements subject to certain relations defined recursively. Concrete examples for $n=2,3,..., 8$ are given.