  Any triple $(W,L,\rho)$, where $W$ is a compact closed oriented 3-manifold,
$L$ is a link in $W$ and $\rho$ is a flat principal $B$-bundle over $W$ ($B$ is
the Borel subgroup of upper triangular matrices of $SL(2,\mc)$), can be encoded
by suitable {\it distinguished} and {\it decorated} triangulations ${\cal
T}=(T,H,{\cal D})$. For each $\cal T$, for each odd integer $N\geq 3$, one
defines a state sum $K_N({\cal T})$, based on the Faddeev-Kashaev quantum
dilogarithm at $\omega =\exp(2\pi i/N)$, such that $K_N(W,L,\rho)=K_N({\cal
T})$ is a well-defined complex valued invariant. The purely topological,
conjectural invariants $K_N(W,L)$ proposed earlier by Kashaev correspond to the
special case of the {\it trivial} flat bundle. Moreover, we extend the
definition of these invariants to the case of flat bundles on $W\setminus L$
with non necessarily trivial holonomy along the meridians of the link's
components, and also to 3-manifolds endowed with a $B$-flat bundle and with
\emph{arbitrary} non-spherical parametrized boundary components. We point out
some remarkable specializations of the invariants; among these, the so called
{\it Seifert-type} invariants, when $W=S^3$: these seem to be good candidates
in orther to fully reconduct the Jones polynomials in the main stream of
quantum hyperbolic invariants. Finally, we try to set our results against the
heuristic backgroud of the Euclidean analytic continuation of (2+1)-quantum
gravity with negative cosmological constant, regarded as a gauge theory with
the {\it non compact} group SO(3,1) as gauge group.
