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

Abstract

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 ${\epsilon }_0,{\epsilon }_1$ from the set $1,-1$. (i) Let $V$ be a finite-dimensional irreducible $U_q({\widehat{sl}}_2{)}^{\ge 0}$-module of type $({\epsilon }_0,{\epsilon }_1)$. Then the action of $U_q({\widehat{sl}}_2{)}^{\ge 0}$ on $V$ extends uniquely to an action of $U_q({\widehat{sl}}_2)$ on $V$. The resulting $U_q({\widehat{sl}}_2)$-module structure on $V$ is irreducible and of type $({\epsilon }_0,{\epsilon }_1)$. (ii) Let $V$ be a finite-dimensional irreducible $U_q({\widehat{sl}}_2)$-module of type $({\epsilon }_0,{\epsilon }_1)$. 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 $({\epsilon }_0,{\epsilon }_1)$.

27 pages

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