Quantum moduli algebras \(\mathcal {L}_{g,n}^{\textrm{inv}}(H)\) 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 \(\mathcal {L}_{g,n}^{\textrm{inv}}(H)\) on cohomology spaces \(\textrm{Ext}_H^m(X,M)\) 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 obtained in Lentner et al. (Springer Briefs in Mathematical Physics, 2023) from Lyubashenko theory, when the category \(\mathcal {C} = H\text {-mod}\) 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.
