# Rank-two parabolic-type VOAs and nilpotency of nil ideals

## TL;DR

This paper classifies rank-two parabolic-type sub-VOAs of lattice VOAs, analyzes their modules, explores rings with nilpotent but non-nil ideals, and studies properties of their simple quotients.

## Contribution

It provides a systematic classification of subVOAs, their modules, and investigates new examples of rings with nil ideals that are not nilpotent, along with properties of their simple quotients.

## Key findings

- Classified all types of rank-two parabolic subVOAs.
- Identified new rings with nil ideals that are not nilpotent.
- Showed simple quotients are C₁-cofinite irrational VOAs.

## Abstract

In this paper, we undertake a systematic study of the parabolic-type sub-vertex operator algebras (subVOAs) \(V_P\) of rank-two lattice VOAs \(V_L\), originally introduced by the first-named author. We first classify all possible types of such subVOAs by analyzing the corresponding submonoids \(P \subseteq L\). For each type of \(V_P\), we then classify its irreducible modules. Certain Zhu algebras \(A(V_P)\) provide new examples of rings with nil ideals that are not nilpotent. Finally, we show that the simple quotient \(V_H\) of any parabolic-type subVOA \(V_P\) is a \(C_1\)-cofinite irrational VOA satisfying the strongly unital property recently introduced by Damiolini--Gibney--Krashen.