Ordered spanning sets for vertex operator algebras and their modules

TL;DR

This paper explores the structure of vertex operator algebras related to Moonshine, focusing on ordered spanning sets and their implications for understanding algebraic and modular properties.

Contribution

It introduces ordered spanning sets with difference-one conditions for vertex operator algebras and modules, advancing the structural understanding of these algebras.

Findings

Existence of finite generating sets under C2-cofiniteness

Construction of Poincare-Birkhoff-Witt-like spanning sets

Ordered spanning sets with difference-one conditions

Abstract

Moonshine relates three fundamental mathematical objects: the Monster sporadic simple group, the modular function j, and the moonshine module vertex operator algebra. Examining the relationship between modular functions and the representation theory of vertex operator algebras reveals rich structure. In particular, C2-cofiniteness (also called Zhu's finiteness condition) implies the existence of finite generating sets and Poincare-Birkhoff-Witt-like spanning sets for vertex operator algebras and their modules. These spanning sets feature desirable ordering restrictions, e.g., a difference-one condition.