Modified graded Hennings invariants from unrolled quantum groups and modified integral

TL;DR

This paper develops a new topological invariant for 3-manifolds using unrolled quantum groups and modified integrals, extending previous invariants and connecting to the CGP-invariant.

Contribution

It generalizes the construction of topological ribbon Hopf algebras from unrolled quantum groups and introduces a modified graded Hennings invariant.

Findings

Constructed topological ribbon Hopf algebras from unrolled quantum groups.

Defined a modified graded Hennings invariant using new algebraic tools.

Extended the invariant to empty manifolds, recovering the CGP-invariant.

Abstract

The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra $\mathfrak{sl}(2|1)$. 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. Finally, we use the notion of a modified integral to extend this invariant to empty manifolds and show that it recovers the CGP-invariant.