Regularity of Rational Vertex Operator Algebras

Rationality, regularity, and 𝐶₂-cofiniteness

“…The notion of regularity is given in [5] as a generalization of rationality to weak modules for vertex operator algebras. Regularity says that any weak module is a direct sum of irreducible ordinary modules.…”

“…Regularity says that any weak module is a direct sum of irreducible ordinary modules. It is proved in [5] that rational vertex operator algebras associated to highest weight modules for affine Kac-Moody algebras, Virasoro algebra, and positive definite even lattices are regular. Based on these results a stronger conjecture is proposed in [5].…”

$${\mathbb{Z}}$$ -Graded Weak Modules and Regularity

“…[B], [FLM]). We first recall notions of weak, admissible and ordinary modules from [FLM], [Z], [DLM1].…”

On certain categories of modules for affine Lie algebras

“…As in [DLM], in view of the complete reducibility theorem in [K1] we only need to show that every nonzero restricted integrableĝ-module W contains a highest weight integrable (irreducible) submodule. We now reformulate the proof of [DLM,Theorem 3.7] as follows: Claim 1: There exists a nonzero u ∈ W such that (g ⊗ tC[t])u = 0. For n ∈ Z, set g(n) = {a(n) | a ∈ g}.…”