  In this paper we classify all the cyclic finite dimensional indecomposable\\
modules of the perfect Lie algebras $\mathfrak{sl}(n+1)\ltimes
\mathbbm{C}^{n+1}$, given by the semidirect sum of the simple Lie algebra $A_n$
with its standard representation. Furthermore, using the embeddings of the Lie
algebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ in $\mathfrak{sl}(n+2)$,
we show that any finite dimensional irreducible module of $\mathfrak{sl}(n+2)$
restricted to $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ is a cyclic module
and that any cyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$--modules can
be constructed as quotient module of the restriction to
$\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ of some finite dimensional
irreducible $\mathfrak{sl}(n+2)$--modules. This explicit realization of the
cyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$--modules plays a role in
their classification.
