The main objective of this paper is to use a specific Hopf superalgebra in order to get first a universal invariant of links and then a Hennings-type topological invariant of pairs \((M, \omega)\), where \(M\) is a 3-manifold and \(\omega\) is a cohomology class in \(H^1(M,G)\) (with \(G = ( \mathbb{C}/\mathbb{Z} \times \mathbb{C}/\mathbb{Z}, +)\).) The Hopf superalgebra in question is \(\mathcal{U}^H = \mathcal{U}^H_\xi \mathfrak{sl}(2|1)/(e_1^l,f_1^l)\), as defined and reviewed in Section 2.2.1.
In order to obtain the link invariant and then the invariant of manifolds, one first needs to construct a completion of \(\mathcal{U}^H\), the topological ribbon Hopf superalgebra \(\widehat{\mathcal{U}^H}\), whose topology is determined by the norm of uniform convergence on compact sets. This is done first in Section 2.2.2, where a series of propositions imply the compatibility between the Hopf and the topological structures, and then in Section 2.2.3., where an added element \(\theta\) is shown to be the twist required for the ribbon structure. Further on, in Section 2.3, the bosonization of \(\widehat{\mathcal{U}^H}\), denoted by \(\widehat{\mathcal{U}^{H_\sigma}}\) is obtained. This is a ribbon Hopf algebra, from which will come the wanted invariants.
For the constructed ribbon Hopf algebra, a well-known method is used in order to obtain the universal invariant of oriented framed links. This appears in Section 3, where the invariant is constructed (Section 3.2), with Theorem 3.2 explaining that the expression obtained is indeed a (topological) invariant of framed links.
In Section 4, the invariant of framed links is used to obtain an invariant of manifolds \(M\) with added \(\omega \in H^1(M,G)\). The method used is adapted from Hennings’s construction. Hennings uses a right integral for a given Hopf algebra, which is known to exist if the Hopf algebra is finite dimensional. Virelizier generalized this notion to finite type unimodular ribbon Hopf \(\pi\)-algebras with a right \(\pi\)-integral (\(\pi\) a group). None of these can be used directly for \(\mathcal{U}^H\), whose associated \(G\)-coalgebra is not of finite type. However, it induces a finite-type Hopf \(G\)-coalgebra by forgetting two elements, and from this the manifold invariant can be obtained. It appears in Section 4.3, in Theorem 4.15, and its construction needs a discrete Fourier transform, whose properties are explored in Section 4.2.