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

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.