On the representation theory of the Drinfeld double of the Fomin-Kirillov algebra $\mathcal{FK}_3$

TL;DR

This paper explores the detailed structure of modules over the Drinfeld double of the Fomin-Kirillov algebra, revealing its complex representation type and classifying module extensions.

Contribution

It provides a comprehensive classification of indecomposable modules, extensions, and projective modules for the Drinfeld double of the Fomin-Kirillov algebra, demonstrating its wild representation type.

Findings

Classified indecomposable summands of tensor products.

Described extensions of simple modules.

Showed the algebra has wild representation type.

Abstract

Let $\mathcal{D}$ be the Drinfeld double of $\mathcal{FK}_3\#\Bbbk{\mathbb S}_3$. The simple $\mathcal{D}$-modules were described in arXiv:1409.0438. In the present work, we describe the indecomposable summands of the tensor product between them. We classify the extensions of the simple modules and show that $\mathcal{D}$ is of wild representation type. We also investigate the projective modules and their tensor products.