Rigorous results for a hierarchy of generalized Heisenberg models

The Lieb-Schultz-Mattis theorem is extended to generalized Heisenberg models related to non-exceptional Lie algebras. It is shown that there are no energy gaps above the ground sates for SO(4), Sp(2) and SU(4) Heisenberg models, gaps are suspected to occur in SO(5) and SO(6) models. The nondegenerate ground state for these models is rigorously proven.