  For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex
tensor category theory of Huang and Lepowsky to identify the category of
standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed
level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional
$\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard
$\hat{\mathfrak{g}}$-modules is the module category for the simple vertex
operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this
category is equivalent as an abelian category to
$\mathbf{D}(\mathfrak{g},\ell)$, the category of finite-dimensional modules for
the Zhu's algebra $A(L_{\hat{\mathfrak{g}}}(\ell,0))$, which is a quotient of
$U(\mathfrak{g})$. Our main result is a direct construction using
Knizhnik-Zamolodchikov equations of the associativity isomorphisms in
$\mathbf{D}(\mathfrak{g},\ell)$ induced from the associativity isomorphisms
constructed by Huang and Lepowsky in
$L_{hat{\mathfrak{g}}}(\ell,0)-\mathbf{mod}$. This construction shows that
$\mathbf{D}(\mathfrak{g},\ell)$ is closely related to the Drinfeld category of
$U(\mathfrak{g})[[\hbar]]$-modules used by Kazhdan and Lusztig to identify
categories of $\hat{\mathfrak{g}}$-modules at irrational and most negative
rational levels with categories of quantum group modules.
