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...

## 1 Answer

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Recently I discovered that this result is true. In fact, it is a consequence of the Proposition 2.2 of paper https://link.springer.com/article/10.1007/BF00750760 written by Chari-Pressley.