Irreducible Modules for the Quantum Affine Algebra $U_q({\widehat{sl}}_2)$ and its Borel subalgebra $U_q({\widehat{sl}}_2{)}^{\ge 0}$

Let $U_q({\widehat{sl}}_2{)}^{\ge 0}$ denote the Borel subalgebra of the quantum affine algebra $U_q({\widehat{sl}}_2)$. We show that the following hold for any choice of scalars ${ϵ}_0,{ϵ}_1$ from the set $1,-1$. (i) Let $V$ be a finite-dimensional irreducible $U_q(\hat{sl}_2)^{\geq 0}$-moduleoftype$(\epsilon_0,\epsilon_1)$\mathrm{.}Thentheactionof$U_q(\hat{sl}_2)^{\geq 0}$on$V$extendsuniquelytoanactionof$U_q(\hat{sl}_2)$on$V$\mathrm{.}Theresulting$U_q(\hat{sl}_2)$-modulestructureon$V$isirreducibleandoftype$(\epsilon_0,\epsilon_1)$. (ii) Let $V$ be a finite-dimensional irreducible $U_q({\widehat{sl}}_2)$-module of type $({ϵ}_0,{ϵ}_1)$. When the $U_q({\widehat{sl}}_2)$-action is restricted to $U_q({\widehat{sl}}_2{)}^{\ge 0}$, the resulting $U_q({\widehat{sl}}_2{)}^{\ge 0}$-module structure on $V$ is irreducible and of type $({ϵ}_0,{ϵ}_1)$.