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 {A_n} and C n {C_n} . For the case of A n {A_n} we show that there are no simple torsion-free A n {A_n} -modules of degree k for n ≥ 4 n \geq 4 and 2 ≤ k ≤ n − 2 2 \leq k \leq n - 2 . We conclude with some examples showing that there exist simple torsion-free A n {A_n} -modules of degrees 1 , n − 1 1,n - 1 , and n.