  Prototypical rational vertex operator algebras are associated to affine Lie
algebras at positive integer level k. These share the same representation
theory as quantum groups at roots of unity and physically correspond to the
Wess-Zumino-Witten theories. One would like to identify the full (bulk)
conformal field theories whose chiral halves are those VOAs. Mathematically,
these correspond to the module categories for their representation theory.
Until now, this has been done only for sl(2) (famously, the A-D-E
classification of Cappelli-Itzykson-Zuber) and sl(3). The problem reduces to
knowing the possible extensions of those VOAs, and the tensor equivalences
between those extensions. Recent progress puts the tensor equivalences in good
control, especially for sl(n). Our paper focuses on the extensions. We prove
that, for any simple g, there is a bound K(g) growing like the rank-cubed, such
that for any level k>K(g), the only extensions are generic (simple-current
ones). We use that bound to find all extensions (hence module categories) for
g=sl(4) and sl(5). In the sequel to this paper, we find all extensions for all
simple g of rank less than or equal to 6.
