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

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.