G-links invariants, Markov traces and the semi-cyclic \(U_q\mathfrak{sl}(2)\)-modules (Q2872784)

\textit{R. Kashaev} and \textit{N. Reshetikhin} [Proc. Symp. Pure Math. 73, 151--172 (2005; Zbl 1083.57018)] proposed a generalization of the construction of the Reshetikhin-Turaev invariant from links to tangles with a flat connection in a principle \(G\)--bundle of the complement of the tangle. The paper under review is the beginning of a project to construct an invariant of \(3\)-manifolds from such an invariant. It concerns the case when \(G\) is the Borel subgroup of \(SL_2(\mathbb{C})\) and the category of semi-cyclic representations of \(U_\xi (\mathfrak{sl}_2)\) defined by De Concini and Kac. Using a modified Markov trace, the paper constructs a \(G\)-link invariant that can be interpreted as a natural extension of the Akutsu-Deguchi-Ohtsuki (ADO) invariant of links. Experimental computations suggest that this extension may be trivial.