Branching rules for level-zero extremal weight modules from $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_{n+1})$ to $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$

In this paper, we study the structure of a $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$-module ${\mathrm{\Psi }}_{\epsilon }^{\ast }V(\lambda )$, where $V(\lambda )$ is the extremal weight module of level-zero dominant weight $\lambda$ over the quantum affine algebra $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_{n+1})$ and ${\mathrm{\Psi }}_{\epsilon }:U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_n)\to U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_{n+1})$ is an injective algebra homomorphism. We establish a direct sum decomposition ${\mathrm{\Psi }}_{\epsilon }^{\ast }V(\lambda )\cong M_{0,\epsilon }\oplus \cdots \oplus M_{m,\epsilon }$, where $M_{0,\epsilon }$ and $M_{m,\epsilon }$ are isomorphic to a tensor product of an extremal weight module over $U_q({\widehat{\mathfrak{s}\mathfrak{l}}}_n)$ and a symmetric Laurent polynomial ring. Moreover, when $\lambda$ is a multiple of a level-zero fundamental weight, we show that ${\mathrm{\Psi }}_{\epsilon }^{\ast }V(\lambda )$ is isomorphic to a direct sum of extremal weight modules.