General non-semisimple relation between the Kuperberg invariant and the Hennings–Kauffman–Radford invariant

Establish the conjectured relationship between the Kuperberg invariant Kup(M, b; H), defined for a finite-dimensional Hopf algebra H and a framing b of a closed oriented 3-manifold M, and the Hennings–Kauffman–Radford invariant HKR(M, φ; D(H)) associated to the Drinfeld double D(H) and a 2-framing φ of M, in the non-semisimple setting. Precisely determine the hypotheses on H and the framing data under which these invariants coincide or otherwise relate for closed oriented 3-manifolds.

Background

In the semisimple setting, a well-known connection between the Witten–Reshetikhin–Turaev invariants and the Turaev–Viro construction has been established. Analogous structures in the non-semisimple setting include the Kuperberg invariant (for finite-dimensional Hopf algebras) and the Hennings–Kauffman–Radford invariant.

Several partial results support a relation between these invariants, for example equality under specific conditions (e.g., for double balanced Hopf algebras with suitable choices of framing data). The present paper proves a restricted, category-level equality between the chromatic spherical invariant derived from H-mod and HKR for D(H) when H is spherical, but does not resolve the full non-semisimple conjecture for the original Kuperberg invariant across all Hopf algebras.