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).