  We consider the algebraic structure of $\mathbb{N}$-graded vertex operator
algebras with conformal grading $V=\oplus_{n\geq 0} V_n$ and $\dim V_0\geq 1$.
We prove several results along the lines that the vertex operators $Y(a, z)$
for $a$ in a Levi factor of the Leibniz algebra $V_1$ generate an affine
Kac-Moody subVOA. If $V$ arises as a shift of a self-dual VOA of CFT-type, we
show that $V_0$ has a `de Rham structure' with many of the properties of the de
Rham cohomology of a complex connected manifold equipped with Poincar\'{e}
duality.
