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

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.