Classification of irreducible modules of the vertex algebra when L is a nondegenerate even lattice of an arbitrary rank

In this paper, we first classify all irreducible modules of the vertex algebra when L is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible -module is isomorphic to a submodule of an irreducible twisted VL-module. We then extend this result to a vertex algebra when L is a nondegenerate even lattice of finite rank.