use the well known Frenkel-Kac-Segal vertex operator algebra construction of the standard ${\tilde{\mathfrak{g}}}$-modules of level 1, [@FK], [@S]. The brief outline of the construction follows and the details can be found in [@FLM], [@DL] or [@LL]. We use the notation from [@LL].

Let $M(1)$ denote the Fock space for the homogenous Heisenberg subalgebra and let ${\mathbb{C}}[P]$ be the group algebra of the weight lattice $P$ with a basis $e^{\lambda}, \lambda \in P$. Set $V_Q = M(1) \otimes {\mathbb{C}}[Q]$ and $V_P = M(1) \otimes {\mathbb{C}}[P]$. The action of Heisenberg subalgebra on the tensor product $V_P$ extends to the action of Lie algebra ${\tilde{\mathfrak{g}}}$ via the vertex operator formula $$\begin{aligned}
\label{vertex}
x_{\alpha}(z)= Y(e^{\alpha},z) = E^{-}(- \alpha,z) E^{+}(-\alpha,z) e_{\alpha} z^{\alpha}\end{aligned}$$ for properly chosen root vectors $x_{\alpha}$. Here $z^{\alpha} = 1 \otimes z^{\alpha}$, $z^{\alpha}e^{\lambda} = z^{\langle \alpha, \lambda \rangle}$ and $$E^{\pm}(\alpha,z) = E^{\pm}(\alpha,z) \otimes 1 = \textrm{exp} \left( \sum_{n>0} \frac{\alpha(\pm n) z^{\mp n}}{\pm n}  \right) \otimes 1.$$

On $V_P$ we have the following gradation: a homogenous vector $v = h(-n_1) \dots h(-n_r) \otimes e^{\lambda}$ is of degree $$\begin{aligned}
\label{gradacija}
\textrm{deg} (v) = -n_1 - n_2 - \dots -n_r - \frac{1}{2} \langle \lambda, \lambda \rangle .\end{aligned}$$

Then, as a ${\tilde{\mathfrak{g}}}$-module $$M(1) \otimes {\mathbb{C}}[P] = L(\Lambda_0) 