  Let $L$ be the basic (level one vacuum) representation of the affine
Kac-Moody Lie algebra $\hat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW
filtration on $L$ is a linear span of vectors of the form $x_1... x_lv_0$,
where $l\le m$, $x_i\in \hat{\mathfrak g}$ and $v_0$ is a highest weight vector
of $L$. In this paper we give two descriptions of the associated graded space
$L^{\rm gr}$ with respect to the PBW filtration. The "top-down" description
deals with a structure of $L^{\rm gr}$ as a representation of the abelianized
algebra of generating operators. We prove that the ideal of relations is
generated by the coefficients of the squared field $e_\theta(z)^2$, which
corresponds to the longest root $\theta$. The "bottom-up" description deals
with the structure of $L^{\rm gr}$ as a representation of the current algebra
${\mathfrak g}\otimes {\mathbb C}[t]$. We prove that each quotient
$F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of
$m$ copies of ${\mathfrak g}$.

Quantum Algebra;Representation Theory