The quantum double of the restricted quantum group ${\bar{\mathbf{u}}}_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{2}})$

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