Intertwining operators, the composability condition and the complete reducibility of generalized modules

TL;DR

This paper establishes that generalized modules in a suitable category satisfy the composability condition for a class of vertex algebras, enabling the application of a cohomological criterion for their complete reducibility.

Contribution

It proves that generalized modules meet the composability condition in a specific category, facilitating the use of cohomological methods for module reducibility.

Findings

Generalized modules satisfy the composability condition.

The criterion applies to grading-restricted vertex algebras.

Supports complete reducibility analysis of modules.

Abstract

A cohomological criterion for the complete reducibility of modules of finite length satisfying a composability condition for a meromorphic open-string vertex algebra $V$ has been given by Qi and the author. In order to apply this criterion, one needs to determine what types of $V$-modules satisfy the composability condition. In this paper, we prove that the composability condition is satisfied by generalized modules in a suitable category for a grading-restricted vertex algebra satisfying natural conditions.