q-Analogue of $A_{m-1}\oplus A_{n-1}\subset A_{mn-1}$

A natural embedding $A_{m-1}\oplus A_{n-1}\subset A_{mn-1}$ for the corresponding quantum algebras is constructed through the appropriate comultiplication on the generators of each of the $A_{m-1}$ and $A_{n-1}$ algebras. The above embedding is proved in their $q$-boson realization by means of the isomorphism between the ${\mathcal{A}}_q^{-}$ (mn)$\sim {\otimes} ^n $\mathcal{A}$_q^{-}$(m)$\sim {\otimes}^m$\mathcal{A}$_q^{-}$(n) algebras.