Rational vertex operator algebras with small central charges and generators

We prove that rational vertex operator algebras are finitely generated. An abstract construction of C1-cofinite vertex operator algebras is introduced. The rational and C 2-cofinite simple vertex operator algebras whose effective central charges c˜ and central charges c are equal and less than 1 are classified. Such a vertex operator algebra is zero if c < 0 and C if c = 0. If c > 0, it is an extension of discrete Virasoro vertex operator algebra L( cp,q, 0) by its irreducible modules. We also prove that a simple vertex operator algebra generated by two weight 2 vectors is either a vertex operator algebra associated to a highest irreducible W(2, 2)-module or a tensor product of two irreducible Virasoro vertex operator algebras. Furthermore, a rational, C2-cofinite and simple vertex operator algebra whose weight 1 subspace is zero and weight 2 subspace is 2-dimensional, and with central charge c = 1 is isomorphic to L(½, 0) ⊗ L(½, 0).