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