  In the study of 3d-3d correspondence occurs a natural $q$-Weyl algebra
associated to an ideal triangulation of a 3-manifold with torus boundary
components, and a module of it. We study the action of this module on the
(rotated) 3d-index of Dimofte-Gaiotto-Gukov and we conjecture some structural
properties: bilinear factorization in terms of holomorphic blocks, pair of
linear $q$-difference equations, the determination of the 3d-index in terms of
a finite size matrix of rational functions and the asymptotic expansion of the
$q$-series as $q$ tends to 1 to all orders. We illustrate our conjectures with
computations for the case of the three simplest hyperbolic knots.
