A New Proof of $\bf{Z_{Kup}=|Z_{Henn}|^2}$ for Semisimple Hopf Algebras

TL;DR

This paper provides a new proof that for semisimple Hopf algebras, the Kuperberg invariant equals the square of the Hennings invariant, clarifying their relationship in 3-manifold topology.

Contribution

The paper introduces a novel proof of the equality between Kuperberg and Hennings invariants for semisimple Hopf algebras, enhancing understanding of their connection.

Findings

Proves $Z_{Kup}=|Z_{Henn}|^2$ for semisimple Hopf algebras

Clarifies the relationship between Kuperberg and Hennings invariants

Provides a new, simplified proof of a known equality

Abstract

Hennings and Kuperberg defined quantum invariants $Z_{Henn}$ and $Z_{Kup}$ for closed oriented $3$-manifolds based on certain Hopf algebras, respectively. When the Hopf algebras are semisimple, it is shown that $Z_{Kup}=|Z_{Henn}|^2$. In this paper, we present a new proof of this equality.