On Projective Modules in Category $\mathcal{O}_{int}$ of Quantum $\mathfrak{sl}_{2}$

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The restriction of a Verma module of \({\bf U}(\mathfrak{sl}_3)\) to \({\bf U}(\mathfrak{sl}_2)\) is isomorphic to a Verma module tensoring with all the finite dimensional simple modules of \({\bf U}(\mathfrak{sl}_2)\). The canonical basis of the Verma module is compatible with such a decomposition. An explicit decomposition of the tensor product of the Verma module of highest weight 0 with a finite dimensional simple module into indecomposable projective modules in the category \(\mathcal O_{\rm{int}}\) of quantum \(\mathfrak{sl}_2\) is given.