Chromatic spherical invariant and Hennings invariant of 3-dimensional manifolds

TL;DR

This paper reveals a fundamental relation between the chromatic spherical invariant and the Hennings invariant for 3D manifolds, demonstrating their equality under certain algebraic conditions involving Hopf algebras.

Contribution

It establishes that for a spherical Hopf algebra, the chromatic spherical invariant equals the Hennings-Kauffman-Radford invariant of its Drinfeld double.

Findings

The invariants are equal for spherical Hopf algebras.

The relation holds when using the Drinfeld double.

Provides a link between two topological invariants.

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.