Rationality of the vertex algebra when L is a non-degenerate even lattice of arbitrary rank

In this paper we prove that the vertex algebra is rational if L is a negative definite even lattice of finite rank, or if L is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices L, we show that the Zhu algebras of the vertex algebras are semisimple. This extends the result of Abe from [T. Abe, Rationality of the vertex operator algebra for a positive definite even lattice L, Math. Z. 249 (2) (2005) 455–484] which establishes the rationality of when L is a positive definite even lattice of finite rank.