Derived representations of quantum character varieties

Quantum moduli algebras ${\mathcal{L}}_{g,n}^{\mathrm{i}\mathrm{n}\mathrm{v}}(H)$ were introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the context of quantization of character varieties of surfaces and exist for any quasitriangular Hopf algebra $H$. In this paper we construct representations of ${\mathcal{L}}_{g,n}^{\mathrm{i}\mathrm{n}\mathrm{v}}(H)$ on cohomology spaces ${\mathrm{E}\mathrm{x}\mathrm{t}}_H^m(X,M)$ for all $m\ge 0$, where $X$ is any $H$-module and $M$ is any ${\mathcal{L}}_{g,n}(H)$-module endowed with a compatible $H$-module structure. As a corollary and under suitable assumptions on $H$, we obtain projective representations of mapping class groups of surfaces on such Ext spaces. This recovers the projective representations constructed by Lentner-Mierach-Schweigert-Sommerhäuser from Lyubashenko theory, when the category $\mathcal{C}=H\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is used in their construction. Other topological applications are matrix-valued invariants of knots in thickened surfaces and representations of skein algebras on Ext spaces.