Rational vertex operator algebras and the effective central charge

We establish that the Lie algebra of weight 1 states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank 1 is bounded above by the effective central charge $\widetilde c$ . We show that lattice vertex operator algebras may be characterized by the equalities $\widetilde c=l=c$ , and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality l = c.