Embeddings among quantum affine ${\mathfrak{s}\mathfrak{l}}_n$

We establish an explicit embedding of a quantum affine ${\mathfrak{s}\mathfrak{l}}_n$ into a quantum affine ${\mathfrak{s}\mathfrak{l}}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine ${\mathfrak{s}\mathfrak{l}}_n$ and ${\mathfrak{g}\mathfrak{l}}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A ${\mathfrak{g}\mathfrak{l}}_n$-variant of the embedding is also established.