Reshetikhin-Turaev (RT) invariants are constructed in the context of semi-simple categories of representations of quantum groups with some extra structure and satisfying some extra conditions (modular categories). The paper under review extends this RT construction to the case of non-semi-simple categories of representations of a quantum group \(\bar{U}^H_q(\mathfrak{sl}(2))\) (the construction for other quantum groups is briefly discussed in Section 6). For a closed oriented \(3\)-manifold \(M\), a coloured link \(T\) in \(M\), and an element \(\omega\in H^1(M\setminus T;\mathbb{C}/2\mathbb{Z})\), the families of invariants \(N_r(M,T,\omega)\) and \(N_r^0(M,T,\omega)\) that they obtain are computable and new, and are stronger than the RT invariants (Sections 2.1 and 2.3). They also admit a statement of the Volume Conjecture which the authors prove for an infinite class of links (Section 2.2). The invariants \(N_r(S^3,T,\omega)\) contain the multivariable Alexander polynomial, Kashaev’s invariant, and the ADO invariant of the link \(T\) much in the same way that the RT invariants contain quantum invariants of links. The construction contains several subtleties and rests on the notion of quantum dimension as defined by the last two authors and V. Turaev [Compos. Math. 145, No. 1, 196–212 (2009; Zbl 1160.81022)].
Several directions are suggested for future research, including extending these invariants to TQFTs and relating them to non-semi-simple analogues of Turaev-Viro invariants [N. Geer et al., Adv. Math. 228, No. 2, 1163–1202 (2011; Zbl 1237.57011)].