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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 Liealgebras $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ 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 moduleand that any cyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$--modules canbe constructed as quotient module of the restriction to$\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$ of some finite dimensionalirreducible $\mathfrak{sl}(n+2)$--modules. This explicit realization of thecyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$--modules plays a role intheir classification.

The classification of the cyclic $\mathfrak{sl}(n+1)\ltimes \mathbbm{C}^{n+1}$--modules