The Green Ring of Drinfeld Double $D(H_4)$

TL;DR

This paper analyzes the structure of the Green ring of the Drinfeld double of Sweedler's Hopf algebra, providing tensor product decompositions and describing the ring's generators and relations.

Contribution

It offers the first detailed description of the Green ring of $D(H_4)$, including tensor product decompositions and its algebraic structure.

Findings

Decomposition of tensor products of indecomposable modules

Structure of the Green ring $r(D(H_4))$ described

Green ring generated by infinitely many elements with relations

Abstract

In this paper, we study the Green ring (or the representation ring) of Drinfeld quantum double $D(H_4)$ of Sweedler's 4-dimensional Hopf algebra $H_4$. We first give the decompositions of the tensor products of finite dimensional indecomposable modules into the direct sum of indecomposable modules over $D(H_4)$. Then we describe the structure of the Green ring $r(D(H_4))$ of $D(H_4)$ and show that $r(D(H_4))$ is generated, as a ring, by infinitely many elements subject to a family of relations.