Abstract:
This survey article covers some of the results contained in the
papers by Costantino, Geer and Patureau
and by Blanchet, Costantino, Geer and Patureau. In the first one the authors construct two families of
Reshetikhin–Turaev-type invariants of 3-manifolds, ${\mathrm N}_r$ and ${\mathrm N}^0_r$, using non-semisimple categories of representations of a
quantum version of ${\mathfrak{sl}_2}$ at a $2r$-th root of unity with $r \geq 2$. The secondary invariants ${\mathrm N}^0_r$ conjecturally
extend the original Reshetikhin–Turaev quantum ${\mathfrak{sl}_2}$ invariants.
The authors also provide a machinery to produce invariants out of more general ribbon categories which can lack the semisimplicity
condition. In the second paper a renormalized version of ${\mathrm N}_r$ for
$r \neq 0 \; (\mathrm{mod} \; 4)$ is extended to a TQFT, and connections with classical invariants such as the
Alexander polynomial and the Reidemeister torsion are found. In particular, it is shown that the use of richer categories
pays off, as these non-semisimple invariants are strictly finer than the original semisimple ones:
indeed they can be used to recover the classification of lens spaces, which Reshetikhin–Turaev invariants
could not always distinguish.