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Borel-type subalgebras of the lattice vertex operator algebra

Jianqi Liu

TL;DR

This work introduces and analyzes conic-, Borel-, and parabolic-type subVOAs within lattice vertex operator algebras, establishing their nontrivial structural properties and Lie-algebraic analogies. It proves that rank-one Borel-type subVOA $V_B$ of $V_{\,\mathbb{Z}\alpha}$ has a precisely described Zhu algebra $A(V_B)$, namely $A(V_B)\cong\mathbb{C}[x]\oplus\mathbb{C} y$ with $y^2=0$, $yx=-Ny$, and $xy=Ny$, and that irreducible $V_B$-modules correspond to irreducible Heisenberg modules, with fusion rules mirroring those of the Heisenberg VOA. The paper further shows that conic-type subVOAs can be $C_1$-cofinite in low rank, while some Borel- and parabolic-type subVOAs are not, highlighting a nuanced landscape of finiteness properties. It also proves a normalizer property for parabolic-type subVOAs and clarifies when degree-one subalgebras realize Borel or parabolic Lie structures. Overall, these subVOAs provide new CFT-type, irrational, sometimes $C_1$-cofinite examples and connect VOA substructures to classical Lie-theoretic subalgebras through precise algebraic and representation-theoretic descriptions.

Abstract

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 $V_{\Z\al}$, we explicitly determine its Zhu's algebra $A(V_B)$ in terms of generators and relations.

Borel-type subalgebras of the lattice vertex operator algebra

TL;DR

This work introduces and analyzes conic-, Borel-, and parabolic-type subVOAs within lattice vertex operator algebras, establishing their nontrivial structural properties and Lie-algebraic analogies. It proves that rank-one Borel-type subVOA of has a precisely described Zhu algebra , namely with , , and , and that irreducible -modules correspond to irreducible Heisenberg modules, with fusion rules mirroring those of the Heisenberg VOA. The paper further shows that conic-type subVOAs can be -cofinite in low rank, while some Borel- and parabolic-type subVOAs are not, highlighting a nuanced landscape of finiteness properties. It also proves a normalizer property for parabolic-type subVOAs and clarifies when degree-one subalgebras realize Borel or parabolic Lie structures. Overall, these subVOAs provide new CFT-type, irrational, sometimes -cofinite examples and connect VOA substructures to classical Lie-theoretic subalgebras through precise algebraic and representation-theoretic descriptions.

Abstract

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 of , we explicitly determine its Zhu's algebra in terms of generators and relations.
Paper Structure (32 sections, 34 theorems, 112 equations, 2 figures)

This paper contains 32 sections, 34 theorems, 112 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Subalgebras of the lattice vertex operator algebra
  3. The main theorems
  4. Conventions
  5. Submonoids of an even lattice
  6. Definition of conic, Borel, and parabolic-type submonoids
  7. Definition and first properties
  8. Existence of Borel-type submonoid
  9. Borel-type submonoids and the root system
  10. Borel-type and parabolic-type subVOAs of $V_L$
  11. Basics of the lattice VOAs
  12. SubVOAs of $V_L$ associated to submonoids of $L$
  13. Basic properties of conic, Borel, and parabolic-type subVOAs
  14. Relations with the root system and degree-one Lie algebra of $V_L$
  15. Irrationality and non-simplicity
  16. ...and 17 more sections

Key Result

Theorem A

The conic-type subVOA $V_C=V_{\mathbb{Z}_{\geq 0}\alpha}$ of the rank-one lattice VOA $V_{\mathbb{Z}\alpha}$ is $C_1$-cofinite. Moreover, let $L=\mathbb{Z}\alpha\oplus \mathbb{Z}\beta$ be a rank-two positive-definite even lattice such that $(\alpha|\beta)=-n$, $(\beta|\beta)=2k$ and $(\alpha|\alpha)

Figures (2)

  • Figure 1: Borel-type submonoid of $A_2$
  • Figure 2: Fundamental domain of the rank-two lattice

Theorems & Definitions (81)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Example 2.5
  • ...and 71 more