Representations of the small quasi-quantum group

TL;DR

This paper explores the representation theory of small quasi-quantum groups, classifies all finite-dimensional indecomposable modules, and analyzes their tensor product decompositions and ring structures.

Contribution

It provides a complete classification of modules and tensor product rules for small quasi-quantum groups, extending understanding beyond traditional quantum groups.

Findings

All finite-dimensional indecomposable modules are classified.

Tensor product decomposition rules are established.

The structure of projective and Green rings is described.

Abstract

In this paper, we study the representation theory of the small quantum group $\overline{U}_q$ and the small quasi-quantum group $\widetilde{U}_q$, where $q$ is a primitive $n$-th root of unity and $n>2$ is odd. All finite dimensional indecomposable $\widetilde{U}_q$-modules are described and classified. Moreover, the decomposition rules for the tensor products of $\widetilde{U}_q$-modules are given. Finally, we describe the structures of the projective class ring $r_p(\widetilde{U}_q)$ and the Green ring $r(\widetilde{U}_q)$. We show that $r(\overline{U}_q)$ is isomorphic to a subring of $r(\widetilde{U}_q)$, and the stable Green rings $r_{st}(\widetilde{U}_q)$ and $r_{st}(\overline{U}_q)$ are isomorphic.