  In this text, we study derived versions of the fusion category associated to
Lusztig's quantum group $\textbf{U}_q$. The categories that so arise are
non-semisimple but recovers the usual fusion ring when passing to complexified
Grothendieck rings. On the derived level it turns out that it is possible to
define fusion for $\textbf{U}_q$ 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 related with some rings of A. Lachowska.
