Logarithmic Hennings invariants for restrictedquantum $sl(2)$

Abstract

We construct a Hennings-type logarithmic invariant for restricted quantum $sl(2)$ at a$2p^{th}$ 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 $0^{th}$ 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.