$|Z_{\mathrm{Kup}}|=|Z_{\mathrm{Henn}}{|}^2$ for lens spaces

Abstract

M. Hennings and G. Kuperberg defined quantum invariants $Z_{\mathrm{Henn}}$ and $Z_{\mathrm{Kup}}$ of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that $|Z_{\mathrm{Kup}}|=|Z_{\mathrm{Henn}}{|}^2$ for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.