Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ category grading-restricted generalized $V$-modules that admits the algebraic braided tensor structure Huang-Lepowsky-Zhang. We give conditions under which inherits semisimplicity from $A$-modules in $\mathcal{C}$, vice versa. The most important condition is $A$ rigid $V$-module with non-zero categoric...

The recent researchs show that C2-cofiniteness is a natural conditition to consider a vertex operator algebra with finitely many simple modules. Therefore, we extended the tensor product theory of vertex operator algebras developed by Huang and Lepowsky without assuming the compatibility condition nor the semisimplicity of grading operator so that we could apply it to all vertex operator algebr...

Since X is infinite it contains two distinct points x and y. Suppose there exist disjoint open sets A and B (in the cofinite topology) such that x ∈ A and y ∈ B. Then A ⊆ Bc, which is finite so A is also finite. This is a contradiction because Ac is finite but X = A ∪Ac is infinite. Hence, the cofinite topology on X is not T2. Suppose that X is countable, and let x ∈ X. Choose a surjection q : ...

We reformed the tensor product theory of vertex operator algebras developed by Huang and Lepowsky so that we could apply it to all vertex operator algebras satisfying C2-cofiniteness. We also showed that the tensor product theory develops naturally if we include not only ordinary modules, but also weak modules with a composition series of finite length (we call it an Artin module). In particula...

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V(0) = C1 and the contragredient module V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C2-cofinite. We announce a proof of the Verlinde conjecture for V , that is, of the statement that the matrices formed by the fusion rules amo...

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)) = 0 for n < 0, V(0)=C1, and the contragredient module V' is isomorphic to V as a V-module; (ii) every N-gradable weak V-module is completely reducible; (iii) V is C(2)-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among...

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0, V(0) = C1 and the contragredient module V ′ is isomorphic to V as a V -module. (ii) Every N-gradable weak V -module is completely reducible. (iii) V is C2-cofinite. We announce a proof of the Verlinde conjecture for V , that is, of the statement that the matrices formed by the fusion rules amo...