Is the vertex algebra associated to a negative definite even lattice a vertex operator algebra?

Question

Is the lattice vertex algebra associated to a negative definite even lattice a vertex operator algebra? In particular if $L=\mathbb{Z}\alpha$ with $<\alpha, \alpha>=-2$, does $V_L=\bigoplus_{n \in \mathbb{Z}} V_n$ satisfy dim$V_n<\infty$ and $V_n=0$ for $n$ sufficiently small?

Answer 1

$V_L$ is graded, it does have an action of Virasoro (the usual formula), but the vertex operator $\Gamma_\alpha$ has conformal weight $-1$ and there are vectors of arbitrarily large negative conformal weight.