Representations of vertex operator algebras

In this thesis we study the representation theory of vertex operator algebras. The thesis consists of two parts. The first part deals with the connection among rationality, regularity and $C_2$-cofiniteness of vertex operator algebras. It is proved that if any $Z$-graded weak module for a vertex operator algebra V is completely reducible, then V is rational and $C_2$-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras. Motivated by classification of rational vertex operator algebras with central charge c = 1. We compute the quantum dimensions of irreducible modules of the rational and C2-cofinite vertex operator algebra $V_{L_2}^{A_4}$ in the other part.This result will be used to determine the fusion rules for this algebra.