Borel-type subalgebras of the lattice vertex operator algebra

In this paper, we introduce and study new classes of sub-vertex operator algebras of the lattice vertex operator algebras (VOAs), which we call the conic, Borel, and parabolic-type subVOAs. These CFT-type VOAs, which are not necessarily strongly finitely generated, satisfy properties similar to the usual Borel and parabolic subalgebras of a Lie algebra. For the lowest-rank nontrivial example of Borel-type subVOA $V_B$ of $V_{Z\alpha }$, we explicitly determine its Zhu's algebra $A\left(V_B\right)$ in terms of generators and relations.