Modified graded Hennings invariants from unrolled quantum groups and modified integral

Abstract The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra sl ( 2 | 1 ) . We generalize this fact to the context of unrolled quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms , a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant of 3-manifolds endowed with a cohomology class and which contains a ribbon graph. Finally, we use the notion of a modified integral to extend this invariant to manifolds without ribbon graphs inside and show that it recovers the invariant of [6] .