Modified graded Hennings invariants from unrolled quantum groups and modified integral

Nathan Geer , Ngoc Phu Ha , Bertrand Patureau-Mirand (1)

1 LM - Laboratoire de mathématiques de Brest

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 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].

Mots clés: 57M27; 17B37; Unrolled quantum group; Topological ribbon Hopf algebra; Hennings type invariant; Discrete Fourier transform; Modified integral

Domaines: Physique mathématique [math-ph]

Nathan Geer, Ngoc Phu Ha, Bertrand Patureau-Mirand. Modified graded Hennings invariants from unrolled quantum groups and modified integral. J.Pure Appl.Algebra, 2022, 226, pp.106815. ⟨10.1016/j.jpaa.2021.106815⟩. ⟨hal-02892969⟩