Logarithmic Hennings invariants for restricted quantum ${\mathfrak{sl}}(2)$

Abstract

We construct a Hennings-type logarithmic invariant for restricted quantum $\mathfrak{s}\mathfrak{l}\left(2\right)$ at a $2{\mathsf{p}}^{\text{ th}}$ root of unity. This quantum group $U$ is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a $3$–manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored by central elements and by trace classes, or elements in the $0^{\text{ th}}$ Hochschild homology of $U$, respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of $U$, and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.