\(|Z_{Kup}|=|Z_{Henn}|^{2}\) for lens spaces (Q2637474)

Connections between the topological quantum field theories for 3-manifolds due to Reshetikhin-Turaev and due to Turaev-Viro have been recognized almost from their inception, beginning with [\textit{K. Walker}, On Witten's 3-manifold invariants, preprint (1991)]; [\textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] and more recently [\textit{B. Balsam} and \textit{A. Kirilov jun.}, Turaev-Viro invariants as an extended TQFT, preprint (2010), {\url arXiv:1004.1533}] and [\textit{V. Turaev} and \textit{A. Virelizier}, On two approaches to 3-dimensional TQFTs, preprint (2010)]. Hennings invariants are non-semisimple generalizations of Reshetikhin-Turaev invariants, and Kuperberg invariants are non-semisimple generalizations of Turaev-Viro invariants. In this paper the authors extend a known identity for Reshetikhin-Tuaev and Turaev-Viro invariants to these generalizations for lens spaces, showing that \(|Z_{\text{Kup}}|= |Z_{\text{Henn}}|^2\) for lens spaces and factorizable finite dimensional ribbon Hopf algebras. The invariants are computed using a framed Heegaard diagram for the Kuperberg invariant and a chain-mail link for the Hennings invariant. The authors conjecture that this relation holds for more general 3-manifolds, but finding a choice of framing for the Kuperberg invariant allowing for direct comparison with the Hennings invariant for the general case is difficult, as noted by the authors.