Chromatic spherical invariant and Hennings invariant of 3-dimensional manifolds

Abstract

This paper establishes a relation between two invariants of $3$-dimensional manifolds: the chromatic spherical invariant $\mathcal{K}$ and the Hennings-Kauffman-Radford invariant $\operatorname{HKR}$. We show that, for a spherical Hopf algebra $H$, the invariant $\mathcal{K}$ associated to the pivotal category of finite-dimensional $H$-modules is equal to the invariant $\operatorname{HKR}$ associated to the Drinfeld double $D(H)$ of the same Hopf algebra.