N ov 2 00 7 W - algebra W ( 2 , 2 ) and the vertex operator algebra L ( 1 2 , 0 ) ⊗ L ( 1 2 , 0 )

In this paper the W -algebra W (2, 2) and its representation theory are studied. It is proved 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, any 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( 2 , 0) ⊗ L( 1 2 , 0).