On Two Invariants of Three Manifolds from Hopf Algebras

TL;DR

This paper proves a long-standing conjecture that two quantum invariants of three manifolds derived from finite dimensional Hopf algebras are equal, extending known relations from semisimple to non-semisimple cases.

Contribution

It establishes the equivalence of the Kuperberg and Hennings-Kauffman-Radford invariants for non-semisimple Hopf algebras, resolving a 20-year-old open problem.

Findings

Confirmed the invariants are equal for non-semisimple Hopf algebras.

Used special Heegaard diagrams to relate the invariants.

Extended classical relations to more general algebraic structures.

Abstract

We prove a 20-year-old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional Hopf algebras, namely, the Kuperberg invariant and the Hennings-Kauffman-Radford invariant. The two invariants can be viewed as a non-semisimple generalization of the Turaev-Viro-Barrett-Westbury $(\text{TVBW})$ invariant and the Witten-Reshetikhin-Turaev $(\text{WRT})$ invariant, respectively. By a classical result relating $\text{TVBW}$ and $\text{WRT}$, it follows that the Kuperberg invariant for a semisimple Hopf algebra is equal to the Hennings-Kauffman-Radford invariant for the Drinfeld double of the Hopf algebra. However, whether the relation holds for non-semisimple Hopf algebras has remained open, partly because the introduction of framings in this case makes the Kuperberg invariant significantly more complicated to handle. We give an affirmative answer to this question. An important ingredient in the proof involves using a special Heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the Heegaard diagram.