On Integrality of Non-Semisimple Quantum Representations of Mapping Class Groups

For a root of unity $\zeta$ of odd prime order, we restrict coefficients of non-semisimple quantum representations of mapping class groups associated with the small quantum group ${\mathfrak{u}}_{\zeta }{\mathfrak{s}\mathfrak{l}}_2$ from $\mathbb{Q}(\zeta )$ to $\mathbb{Z}[\zeta ]$. We do this by exhibiting explicit bases of states spaces that span $\mathbb{Z}[\zeta ]$-lattices that are invariant under projective actions of mapping class groups.