Representations of the small quasi-quantum group

Abstract

In this paper, we study the representation theory of the small quantum group ${\bar{U}}_q$ 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 $r_p\left({\tilde{U}}_q\right)$ and the Green ring $r\left({\tilde{U}}_q\right)$. We show that $r\left({\bar{U}}_q\right)$ is isomorphic to a subring of $r\left({\tilde{U}}_q\right)$, and the stable Green rings $r_{st}\left({\tilde{U}}_q\right)$ and $r_{st}\left({\bar{U}}_q\right)$ are isomorphic.