A CONSTRAINT ON THE EXISTENCE OF SIMPLE TORSION-FREE LIE MODULES

For any simple Lie algebra L with Cartan subalgebra H the classificati on of all simple H-diagonalizable L-modules having a finite-dimensiona l weight space is known to depend on determining the simple torsion-fr ee L-modules of finite degree. It is further known that the only simpl e Lie algebras which admit simple torsion-free modules of finite degre e are those of types A(n) and C-n. For the case of A(n) we show that t here are no simple torsion-free A(n)-modules of degree k for n greater than or equal to 4 and 2 less than or equal to k less than or equal t o n - 2. We conclude with some examples showing that there exist simpl e torsion-free A(n)-modules of degrees 1, n - 1, and n.