Determinant formula and a realization for the Lie algebra W (2, 2)

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra $W(2,2)$. We construct a natural realization of a certain vaccum module for the algebra $W(2,2)$ via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra $W(2,2)$.