  In this paper, we investigate representations of $\operatorname{At}(N)$, the
Atiyah algebroids of a holomorphic line bundles $N$ over a complex manifold
$Y$. In particular, we relate $\operatorname{At}(N)$-modules with logarithmic
connections through two functors. On the one hand, we use these functors to the
define invariants (monodromy) for representations of Atiyah algebroids. On the
other hand, this opens the way to use the theory of Lie algebroids to study
problems about logarithmic connections; we will give an example of this by
showing that the existence of Deligne's extensions of flat connections and the
Riemann-Hilbert correspondence for regular flat meromorphic connections may be
obtained as pull-back of similar results for $\operatorname{At}(N)$-modules,
and, at this level, these results are a direct consequence of the second
theorem of Lie.
