Vertex operator algebras associated to admissible representations of ŝl₂

The Kac-Wakimoto admissible modules for ŝl₂ are studied from the point of view of vertex operator algebras. It is shown that the vertex operator algebra L(l, 0) associated to irreducible highest weight modules at admissible level l = p/q - 2 is not rational if l is not a positive integer. However, a suitable change of the Virasoro algebra makes L(l, 0) a rational vertex operator algebra whose irreducible modules are exactly these admissible modules for ŝl₂ and for which the fusion rules are calculated. It is also shown that the q-dimensions with respect to the new Virasoro algebra are modular functions.