The classification of the perfect cyclic [formula omitted]-modules

Abstract

A perfect cyclic module of a perfect Lie algebra g with solvable radical r and Levi decomposition g=s⋉r is a finite dimensional module of g generated by an irreducible module of the semisimple Lie algebra s. In this paper we classify all the perfect cyclic finite dimensional indecomposable modules of the perfect Lie algebras sl(n+1)⋉Cn+1, given by the semidirect sum of the simple Lie algebra An with its standard representation. Furthermore, using the embedding of the Lie algebra sl(n+1)⋉Cn+1 in sl(n+2), we show that any finite dimensional irreducible module of sl(n+2) restricted to sl(n+1)⋉Cn+1 is a perfect cyclic module and that any perfect cyclic sl(n+1)⋉Cn+1-module can be constructed as quotient module of the restriction to sl(n+1)⋉Cn+1 of some finite dimensional irreducible sl(n+2)-module. This explicit realization of the perfect cyclic sl(n+1)⋉Cn+1-modules plays a role in their classification.