2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115148It is shown that a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V) is semisimple and each irreducible admissible V-module is ordinary. A contravariant form on a Verma type admissible V-module is constructed and the radical is exactly the maximal proper submodule. As an application the rationality of V_L^+ for any positive definite even lattice is obtained.33pagesQuantum AlgebraRepresentation Theory17B69text