Title: Borel and parabolic-type subalgebras of the lattice vertex operator algebra

Abstract: In this paper, we introduce and study some new classes of subalgebras of the lattice vertex operator algebras, which we call the Borel-type and parabolic-type subVOAs. For the lowest-rank examples of Borel-type subVOAs $V_{B}$ of $V_{\Z\al}$, and one nontrivial lowest-rank example of the parabolic-type subVOA $V_P$ of the lattice VOA $V_{A_2}$ associated to the root lattice $A_2$, we explicitly determine their Zhu's algebras $A(V_B)$ and $A(V_P)$ in terms of generators and relations. Using the descriptions of $A(V_B)$ and $A(V_P)$, we classify the irreducible modules over $V_B$ and $V_P$.