Representations of the small quasi-quantum group

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.