  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.
