Derived representations of quantum character varieties

Quantum moduli algebras $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ 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 $\mathcal{L}_{g,n}^{\mathrm{inv}}(H)$ on cohomology spaces $\mathrm{Ext}_H^m(X,M)$ for all $m \geq 0$, 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-Sommerh\"auser from Lyubashenko theory, when the category $\mathcal{C} = H\text{-}\mathrm{mod}$ 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.