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Any finite-dimensional Hopf algebra H is Frobenius and the stable category ofH-modules is triangulated monoidal. To H-comodule algebras we assigntriangulated module-categories over the stable category of H-modules. Thesemodule-categories are generalizations of homotopy and derived categories ofmodules over a differential graded algebra. We expect that, for suitable H, ourconstruction could be a starting point in the program of categorifying quantuminvariants of 3-manifolds.

Hopfological algebra and categorification at a root of unity: The first steps