# Highest-$\ell$-weight tensor products and diagram subalgebras

Let $$U_q(\mathcal{L}({\mathfrak{g}}))$$ be a quantum loop algebra and $$I$$ the set of indexes of Dynking diagram of $$\mathfrak{g}$$. Consider $$J\subset I$$ a connected subdiagram, so that $$U_q(\mathcal{L}({\mathfrak{g}})_J)$$ is a diagram subalgebra of $$U_q(\mathcal{L}({\mathfrak{g}}))$$. Suppose that $$V$$ (resp. $$W$$) is a finite-dimensional highest-$$\ell$$-weight $$U_q(\mathcal{L}({\mathfrak{g}}))$$-module, generated by the highest-$$\ell$$-weight vector $$v\in V$$ (resp. $$w\in W$$). Furthermore, suppose that the tensor product $$V\otimes W$$ is also highest-$$\ell$$-weight, that is, $$V\otimes W=U_q(\mathcal{L}({\mathfrak{g}}))v\otimes U_q(\mathcal{L}({\mathfrak{g}}))w=U_q(\mathcal{L}({\mathfrak{g}}))(v\otimes w).$$ Then, can I say that $$U_q(\mathcal{L}({\mathfrak{g}})_J)v\otimes U_q(\mathcal{L}({\mathfrak{g}})_J)w=U_q(\mathcal{L}({\mathfrak{g}})_J)(v\otimes w)$$ is also true? I am also interested in partial answers, for example, the cases where $$V$$ and $$W$$ are both irreducible...