Derived counterparts of fusion categories of quantum groups

In this paper, we study derived versions of the fusion category associated to Lusztig's quantum group U_q. The categories that arise in this way are not semisimple but recovers the usual fusion ring when passing to Grothendieck rings. On the derived level it turns out that it is possible to define fusion for U_q without using the notion of tilting modules. Hence, we arrive at a definition of the fusion ring that makes sense in any spherical category. We apply this new definition to the small quantum group and we relate it with some rings appearing in [22].