The quantum double of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$

Abstract

In this paper, we construct the quantum double $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ of the restricted quantum group $\mathbf{\overline{u}}_q(\mathfrak{sl_2})$. We describe the algebraic structure of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ by generators and relations. Moreover, we give the comultiplication $\Delta$, the counit $\varepsilon$ and the antipode $S$, respectively. Finally, we classify all irreducible representations of $D(\mathbf{\overline{u}}_q(\mathfrak{sl_2}))$ when $p=2$.