W-Algebra W(2, 2) and the Vertex Operator Algebra $${L(\frac{1}{2},\,0)\,\otimes\, L(\frac{1}{2},\,0)}$$

Abstract

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 an irreducible highest weight W(2, 2)- module or a tensor product of two simple Virasoro vertex operator algebras. Furthermore, we show that any rational, C 2-cofinite and simple vertex operator algebra whose weight 1 subspace is zero, weight 2 subspace is 2-dimensional and with central charge c = 1 is isomorphic to $${L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}$$ .