The quantum double of the restricted quantum group ${\overline{\mathbf{u}}}_q(\mathfrak{s}{\mathfrak{l}}_2)$

Abstract

In this paper, we construct the quantum double $D({\overline{\mathbf{u}}}_q(\mathfrak{s}{\mathfrak{l}}_2))$ of the restricted quantum group ${\overline{\mathbf{u}}}_q(\mathfrak{s}{\mathfrak{l}}_2)$. We describe the algebraic structure of $D({\overline{\mathbf{u}}}_q(\mathfrak{s}{\mathfrak{l}}_2))$ by generators and relations. Moreover, we give the comultiplication $\hat{I}”$, the counit $\epsilon$ and the antipode $S$, respectively. Finally, we classify all irreducible representations of $D({\overline{\mathbf{u}}}_q(\mathfrak{s}{\mathfrak{l}}_2))$ when $p=2$.