Modular An(V) theory

A series of associative algebras An(V) for a vertex operator algebra V over an arbitrary algebraically closed field and nonnegative integers n are constructed such that there is a one to one correspondence between irreducible An(V)-modules which are not An−1(V) modules and irreducible V-modules. Moreover, it is proved that V is rational if and only if An(V) are semisimple for all n. In particular, the homogeneous subspaces of any irreducible V-module are finite dimensional for rational vertex operator algebra V.