A New Proof of ${\mathbf{Z}}_{\mathbf{Kup}}=|{\mathbf{Z}}_{\mathbf{Henn}}{|}^2$ for Semisimple Hopf Algebras

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.