# Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittaker vectors

## TL;DR

This paper classifies certain simple weak modules over singlet vertex operator algebras, focusing on modules with specific eigenvector properties related to the conformal vector, advancing understanding of their module structure.

## Contribution

It provides a classification of simple weak modules for singlet vertex operator algebras with particular eigenvector conditions, revealing new module structures.

## Key findings

- Classification of simple weak modules with specified eigenvector properties
- Identification of modules with eigenvalues for conformal vector modes
- Enhanced understanding of module structure for singlet vertex algebras

## Abstract

Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $\omega$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$, $\omega_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s} u\in{\mathbb C}^{\times} u$, and $\omega_i u=0$ for all $i>s$.