A constraint on the existence of simple torsion free Lie modules

Abstract:

For any simple Lie algebra L with Cartan subalgebra H the classification of all simple H-diagonalizable L-modules having a finite-dimensional weight space is known to depend on determining the simple torsion-free L-modules of finite degree. It is further known that the only simple Lie algebras which admit simple torsion-free modules of finite degree are those of types ${A_n}$ and ${C_n}$. For the case of ${A_n}$ we show that there are no simple torsion-free ${A_n}$-modules of degree k for $n \geq 4$ and $2 \leq k \leq n - 2$. We conclude with some examples showing that there exist simple torsion-free ${A_n}$-modules of degrees $1,n - 1$, and n.