A New Proof of ${\mathbf{Z}}_{\mathbf{K}\mathbf{u}\mathbf{p}}=\mathbf{\mid }{\mathbf{Z}}_{\mathbf{H}\mathbf{e}\mathbf{n}\mathbf{n}}{\mathbf{\mid }}^{\mathbf{2}}$ for Semisimple Hopf Algebras

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 $Z_{Kup}=\mathrm{\mid }{Z}_{Henn}{\mathrm{\mid }}^2$. In this paper, we present a new proof of this equality.