The classification of the cyclic $\mathfrak{s}\mathfrak{l}(n+1)⋉\text{mathbbm}{C}^{n+1}$--modules

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