The trace on projective representations of quantum groups.

For certain roots of unity, we consider the categories of weight modules over three quantum groups: small, unrestricted and unrolled. The first main theorem of this paper is to show that there is a modified trace on the projective modules of the first two categories. The second main theorem is to show that category over the unrolled quantum group is ribbon. Partial results related to these theorems were known previously. [ABSTRACT FROM AUTHOR]