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.

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.

Equations

$$V_P=\underset{\alpha \in P}{⨁}{M}_{\hat{\mathfrak{h}}}(1,\alpha )$$

$$P=({\mathbb{Z}}_{\ge 0}\alpha )\cup (P^{+}(\gamma )\cap L),$$

$$V_P=V_H\oplus V^{+}\text{as vector spaces,}\text{and}{V}_P/V^{+}\cong V_H\text{as VOAs.}$$

$$V_H=\underset{n\in \mathbb{Z}}{⨁}{M}_{\hat{\mathfrak{h}}}(1,n\alpha )\cong M_{\widehat{\mathbb{C}\beta }}(1,0)\otimes V_{\mathbb{Z}\alpha },$$

$$V_H\cong M_{\widehat{\mathbb{C}\beta }}(1,0)\otimes V_{\mathbb{Z}\alpha }$$

$$\{(M_{\hat{\mathfrak{h}}}(1,\lambda ),{\tilde{Y}}_M):\lambda \in \mathfrak{h}\}$$

$$\left\{\right(L^{(\mu ,i)}=M_{\widehat{\mathbb{C}\beta }}(1,\mu )\otimes V_{\mathbb{Z}\alpha +\frac{i}{2N}\alpha },{\tilde{Y}}_M):\mu \in \mathbb{C}\beta ,i=0,1,\dots ,2N-1\}$$

$$P(\gamma )$$

$$P^{+}(\gamma )$$

$$P^{\ge 0}(\gamma )$$

$$P^{-}(\gamma )$$

$$P^{\le 0}(\gamma )$$

$$C={\mathbb{Z}}_{\ge 0}{\alpha }_1\oplus \cdots \oplus {\mathbb{Z}}_{\ge 0}{\alpha }_r.$$

$$\mathrm{Cone}({\alpha }_1,{\alpha }_2):={\mathbb{Z}}_{\ge 0}{\alpha }_1+{\mathbb{Z}}_{\ge 0}{\alpha }_2=\{m_1{\alpha }_1+m_2{\alpha }_2:m_1,m_2\in {\mathbb{Z}}_{\ge 0}\}.$$

$$L=\mathrm{Cone}({\alpha }_1,{\alpha }_2)\cup \mathrm{Cone}({\alpha }_1,-{\alpha }_2)\cup \mathrm{Cone}(-{\alpha }_1,{\alpha }_2)\cup \mathrm{Cone}(-{\alpha }_1,-{\alpha }_2).$$

$$({\alpha }_1|\gamma )=a_1>0,({\alpha }_2|\gamma )=a_2>0,\mathrm{and}\alpha =m{\alpha }_1+n{\alpha }_2.$$

$$-{\alpha }_1=\alpha +(-m-1){\alpha }_1+(-n){\alpha }_2\in M,\mathrm{and}-{\alpha }_2=\alpha +(-m){\alpha }_1+(-n-1){\alpha }_2\in M.$$

$$A=\lambda {\alpha }^1+\mu {\alpha }^2,\alpha ,\mu \in \mathbb{R},\mathrm{with}(\alpha |A)=\lambda m+\mu n=0.$$

$$\frac{\lambda }{\mu }+\frac{n}{m}=0.$$

$$\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\left[\begin{array}{cc}m& n\\ x& y\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right],\mathrm{with}my-nx=1,\mathrm{and}(\beta |\gamma )=x{a}_1+y{a}_2>0.$$

$$(x,y)=(x_0+tm,y_0+tn),\mathrm{where}t\in \mathbb{Z}.$$

$$(x_0+tm)a_1+(y_0+tn)a_2=x_0{a}_1+y_0{a}_2+t(m{a}_1+n{a}_2)>0,\mathrm{and}{x}_0+tm>0.$$

$$\frac{y_1}{x_1}-\frac{n}{m}=\frac{1}{m{x}_1}.$$

$$\frac{(A|\beta )}{\mu x_1}=\frac{(\lambda {\alpha }^1+\mu {\alpha }^2|x_1{\alpha }_1+y_1{\alpha }_2)}{\mu x_1}=\frac{\lambda }{\mu }+\frac{y_1}{x_1}=\frac{1}{m{x}_1}<0.$$

$$\left[\begin{array}{c}\alpha \\ {\beta }^{\prime }\end{array}\right]=\left[\begin{array}{cc}m& n\\ x& y\end{array}\right]\left[\begin{array}{c}{\alpha }_1\\ {\alpha }_2\end{array}\right],\mathrm{with}my-nx=-1,\mathrm{and}({\beta }^{\prime }|\gamma )=x{a}_1+y{a}_2>0.$$

$$\frac{y_2}{x_2}-\frac{n}{m}=-\frac{1}{m{x}_2},x_2{a}_1+y_2{a}_2>0,x_2>0.$$

$$\frac{(A|{\beta }^{\prime })}{\mu x_2}=\frac{(\lambda {\alpha }^1+\mu {\alpha }^2|x_2{\alpha }_1+y_2{\alpha }_2)}{\mu x_2}=\frac{\lambda }{\mu }+\frac{y_2}{x_2}=-\frac{1}{m{x}_2}>0,$$

$$z_0^{-1}\delta \left(\frac{z_1-z_2}{z_0}\right)$$

$$=z_2^{-1}\delta \left(\frac{z_1-z_0}{z_2}\right)Y_M(Y(a,z_0)b,z_2)u.$$

$$[h_1(m),h_2(n)]=m{\delta }_{m+n,0}(h_1|h_2)K,[K,\hat{\mathfrak{h}}]=0,h_1,h_2\in \mathfrak{h},m,n\in \mathbb{Z},$$

$$M_{\hat{\mathfrak{h}}}(1,\lambda )=U(\hat{\mathfrak{h}}){\otimes }_{U({\hat{\mathfrak{h}}}_{\ge 0})}\mathbb{C}{e}^{\lambda }$$

$$h_1(-n_1-1)\dots h_k(-n_k-1)e^{\lambda }:k\ge 0,h_1,\dots ,h_k\in \mathfrak{h},n_1\ge \cdots \ge n_k\ge 0.$$

$$Y(h_1(-n_1-1)\dots h_k(-n_k-1)1,z)=\begin{array}{c}\circ \\ \circ \end{array}({\partial }_z^{(n_1)}{h}_1(z))\dots ({\partial }_z^{(n_k)}{h}_k(z))\begin{array}{c}\circ \\ \circ \end{array}$$

Full text

1. 1 Introduction

2. 2 The rank-two parabolic-type monoids

3. 2.1 Basics of parabolic-type submonoids of an even lattice

4. 2.2 Classification of the rank-two parabolic-type submonoids

5. 3 Structural theory of $V_{P}$

6. 3.1 Basics of the lattice vertex operator algebras

7. 3.2 Tensor product description of the rank-two hyperplane VOAs

8. 3.3 Structure of the rank-two parabolic-type VOAs $V_{P}$ and $C_{1}$-cofiniteness

9. 4 Representation theory of $V_{P}$

10. 4.1 Nil ideal in the Zhu’s algebra $A(V_{P})$

11. 4.2 Classification of irreducible modules over $V_{P}$

12. 5 Strongly unital property of the Cartan-part subVOA $V_{H}$

13. 5.1 The mode transition algebras $\mathfrak{A}_{d}$ associated to a VOA

14. 5.2 Strongly unital property of $V_{H}$

15. References

1. Introduction

Parabolic-type subVOAs of lattice VOAs, introduced by the first-named author [Liu25], arise naturally in the quasi-triangular decomposition of lattice VOAs and exhibit many properties analogous to parabolic subalgebras of semisimple Lie algebras. These subVOAs also provide natural solutions to the operator-form classical Yang–Baxter equations on VOAs [BGL, BGLW]. In [Liu25(1)], a standard parabolic-type subVOA $V_{P}$ of the rank-two lattice VOA $V_{A_{2}}$ was used as a motivating example to study degree-zero induction for the embedding $V_{P}\hookrightarrow V_{A_{2}}$. In this paper, we develop a complete structural and representation-theoretic theory for all parabolic-type subVOAs of rank-two lattice VOAs, generalizing the rank-one results [Liu25, Section 6] as well as the special rank-two type-A case in [Liu25(1)]. Our methods naturally extend to higher-rank lattices. Furthermore, the simple quotients $V_{H}$ of these rank-two parabolic-type VOAs form a natural class of CFT-type $C_{1}$-cofinite irrational VOAs whose mode transition algebras $\mathfrak{A}_{d}$, recently introduced by Damiolini–Gibney–Krashen [DGK25], satisfy the strongly unital property.

The lattice vertex operator algebra $V_{L}$, introduced by Frenkel–Lepowsky–Meurman [FLM88], is the first and most fundamental example of a VOA and plays a central role in the theory. For example, twisted modules and orbifold subVOAs of the Leech lattice VOA $V_{\Lambda}$ led to the original construction of the moonshine module $V^{\natural}$ [FLM88], and $\mathbb{Z}_{2}$- and $\mathbb{Z}_{3}$-orbifolds of lattice VOAs were crucial in the proof of Schellekens’ conjecture [S93] on holomorphic VOAs of central charge $24$ [vEMS20, Lam11, LS15, LS16]. Sub-lattice VOAs in affine VOAs also play key roles in the representation theory and rationality of parafermion VOAs [DR17].

Although the representation theory of lattice VOAs is well understood [D93, D94, DL93], their structural theory continues to yield interesting results. In studying solutions to the classical Yang–Baxter equations on VOAs [BGL, BGLW], we observed that a lattice VOA $V_{L}=\bigoplus_{\alpha\in L}M_{\hat{\mathfrak{h}}}(1,\alpha)$ is structurally analogous to a semisimple Lie algebra if we regard $L$ as an analog of the root system and the Heisenberg modules $M_{\hat{\mathfrak{h}}}(1,\alpha)$ as an analog of the Cartan subalgebra. By taking submonoids $M\subset L$, one obtains many subVOAs $V_{M}=\bigoplus_{\alpha\in M}M_{\hat{\mathfrak{h}}}(1,\alpha)$ of $V_{L}$. Some of these subVOAs are particularly notable. For instance, a subVOA $V_{B}\leq V_{A_{2}}$ of this kind provides an example of a CFT-type, $C_{1}$-cofinite subVOA whose Zhu algebra $A(V_{B})$ is not noetherian [Liu25, Section 5.2]; we call such a VOA a Borel-type subVOA of $V_{A_{2}}$, a special type of rank-two parabolic-type VOA. Another standard rank-two parabolic-type subVOA $V_{P}\leq V_{A_{2}}$ served as a motivating example for the definition of degree-zero induction for VOA embeddings [Liu25(1)], and its Zhu algebra $A(V_{P})$ was shown to be a nilpotent extension of a skew-polynomial ring. Motivated by these intriguing properties, we provide a systematic study of rank-two parabolic-type VOAs in this paper, extending the previous work of the first-named author in [Liu25, Section 6].

To state our main results and outline their proofs, we first introduce some notation. Analogous to a Borel subalgebra (resp. Borel subgroup) in a semisimple Lie algebra (resp. a reductive algebraic group), a Borel-type submonoid contains half of the “root spaces.” More precisely, it is a submonoid $B\leq L$ such that $B\cup(-B)=L$ and $B\cap(-B)=\{0\}$. One can show that there exists a unique hyperplane $P(\gamma)$ in the Euclidean space $E$ spanned by $L$ such that $B$ is contained in the positive half-space defined by $P(\gamma)$.

We say that a submonoid $P\leq L$ is of parabolic-type if it contains a Borel-type submonoid $B\leq L$. A parabolic-type subVOA of $V_{L}$ is then a subVOA

[TABLE]

associated with a parabolic-type submonoid $P\leq L$. In particular, the parabolic-type subVOAs of a rank-two lattice VOA $V_{L}$ are completely determined by the corresponding parabolic-type submonoids. The following theorem provides the classification of parabolic-type submonoids in the rank-two case; see Theorem 2.8 for details.

Theorem A**.**

Let $P$ be a parabolic-type submonoid of a rank-two even lattice $L$. Then it has one of the following two types:

1. (1)

$P=B$* is a rank-two Borel-type submonoid. In this case,*

[TABLE]

where $P(\gamma)$ is the unique hyperplane such that $B\subset P^{\geq 0}(\gamma)$ and $P(\gamma)\cap L=\mathbb{Z}\alpha$ for some $\alpha\in L$ (possibly zero). 2. (2)

$P=P^{\geq 0}(\gamma)\cap L$* for some hyperplane $P(\gamma)\subset E$ such that $P(\gamma)\cap L=\mathbb{Z}\alpha\neq\{0\}$.*

See Figures 2 and 3 for an illustration in a standard rank-two lattice. We refer to $P$ as type-I if $P=B$, and as type-II if $P=P^{\geq 0}(\gamma)\cap L$ with $P(\gamma)\cap L\neq\{0\}$.

The proof of Theorem A involves a technical argument regarding the existence of a basis of the rank-two lattice $L$ contained in a particular submonoid, see Proposition 2.7. In analogy with semisimple Lie algebras, where a parabolic subalgebra $\mathfrak{p}\leq\mathfrak{g}$ contains a “Cartan-part” $\mathfrak{h}$. For example, in type $A$, $\mathfrak{h}$ is generated by block-diagonal matrices. The representation theory of $\mathfrak{p}$ is governed by $\mathfrak{h}$. Our second main theorem shows that a similar Cartan-part subVOA $V_{H}$ exists in any rank-two parabolic-type subVOA $V_{P}$, further justifying the terminology of “parabolic-type subVOA”; see Proposition 3.8 and Theorem 3.9.

Theorem B**.**

Let $V_{P}$ be a rank-two parabolic-type VOA. Then there exists a unique maximal proper ideal $V^{+}\lhd V_{P}$ and a simple subVOA $V_{H}\leq V_{P}$ such that

[TABLE]

Moreover, the simple quotient $V_{H}$ admits the following characterization:

1. (1)

If $P=B$ is of type-I, then $V_{H}=M_{\hat{\mathfrak{h}}}(1,0)$ is the rank-two Heisenberg VOA. 2. (2)

If $P$ is of type-II, with $P(\gamma)\cap L=\mathbb{Z}\alpha$, then

[TABLE]

where $V_{\mathbb{Z}\alpha}$ is the rank-one lattice VOA associated to $\mathbb{Z}\alpha$, $\beta\in\mathfrak{h}=\mathbb{C}\otimes_{\mathbb{Z}}L$ satisfies $\mathbb{C}\beta=(\mathbb{C}\alpha)^{\perp}$, and $M_{\widehat{\mathbb{C}\beta}}(1,0)$ is the rank-one Heisenberg VOA associated to $\mathbb{C}\beta$.

We refer to $V_{H}$ as the Cartan-part of $V_{P}$.

It is straightforward to see that $V_{P}$ admits a split decomposition $V_{P}=V_{H}\oplus V^{+}$ into a simple subVOA and an ideal. However, in the type-II case, the isomorphism

[TABLE]

is not immediately obvious. This structure was strongly suggested by computations of the Zhu algebra for the standard rank-two type-$A$ parabolic-type subVOA $V_{P}$ in [Liu25(1)]. To rigorously establish the isomorphism of VOAs, we carefully analyze the behavior of lattice vertex operators, as defined by Frenkel-Lepowsky-Meurman [FLM88], under the decomposition $\mathfrak{h}=\mathbb{C}\otimes_{\mathbb{Z}}L=\mathbb{C}\alpha\oplus(\mathbb{C}\alpha)^{\perp}$, see Proposition 3.8. With the explicit structural description of $V_{P}$ in Theorem B, one can readily verify the $C_{1}$-cofiniteness of $V_{H}$ and, under mild conditions on the lattice $L$ (3.21), also of $V_{P}$ itself; see Theorem 3.14. In particular, $V_{H}$ is always $C_{1}$-cofinite, while $V_{P}$ is $C_{1}$-cofinite only under the specified condition.

Similar to the situation in classical Lie theory, the representation theory of $V_{P}=V_{H}\oplus V^{+}$ is governed by its Cartan part $V_{H}$. In particular, the maximal ideal $V^{+}$ acts trivially on any irreducible $V_{P}$-module $M$, and the irreducible $V_{P}$-modules are in one-to-one correspondence with irreducible $V_{H}$-modules. However, given an irreducible $V_{P}$-module $W$, it is not immediately clear that $V^{+}$ acts as zero on $W$. To establish this, we employ the crucial tool of Zhu’s algebra $A(V)$ [Z96]. We show that the image $A(V^{+})$ forms a nil ideal in $A(V_{P})$, which then acts as zero on any irreducible $A(V_{P})$-module $W(0)$, see Proposition 4.3 and Lemma 4.5. Conversely, given an irreducible $V_{H}$-module $(M,Y_{M})$, one can extend it to an irreducible $V_{P}$-module $(M,\tilde{Y}_{M})$ by defining $\tilde{Y}_{M}|_{V^{+}}=0$. The following theorem provides a complete classification of irreducible $V_{P}$-modules, extending the rank-two type-$A$ case in [Liu25(1)].

Theorem C**.**

Let $V_{P}=V_{H}\oplus V^{+}$ be a rank-two parabolic-type VOA, and let $M=\bigoplus_{n=0}^{\infty}M(n)$ be an irreducible admissible $V_{P}$-module. Then $M$ is an irreducible $V_{H}$-module on which $V^{+}$ acts trivially. In particular, every irreducible admissible $V_{P}$-module is ordinary. Furthermore,

1. (1)

If $V_{P}$ is of type-I, then

[TABLE]

form a complete list of irreducible $V_{P}$-modules up to isomorphism. 2. (2)

If $V_{P}$ is of type-II, with $P(\gamma)\cap L=\mathbb{Z}\alpha$, $(\alpha|\alpha)=2N$, and $\mathbb{C}\beta=(\mathbb{C}\alpha)^{\perp}$, then

[TABLE]

form a complete list of irreducible $V_{P}$-modules up to isomorphism.

In a recent study of the smoothing property of conformal blocks associated with VOA modules, Damiolini-Gibney-Krashen [DGK25] introduced a sequence of associative algebras, called the mode transition algebras $\mathfrak{A}_{d}$ for $d\in\mathbb{N}$. These algebras are defined as tensor products of certain subquotients of the universal enveloping algebra $\mathscr{U}(V)$ of a VOA $V$, and they satisfy the property that $\mathfrak{A}_{d}$ surjects onto the kernel of the canonical map $\mathsf{A}_{d}\to\mathsf{A}_{d-1}$ between higher-level Zhu’s algebras [DLM98(2)] (see Definition 5.2). It was shown in [DGK25, Remark 3.4.6] that if $V$ is rational, then $\mathfrak{A}_{d}$ possesses a strong identity for all $d\in\mathbb{N}$, which is equivalent to the smoothing property for VOA-conformal blocks [DGK25, Theorem 5.0.3]. However, rationality is not necessary for the existence of strong identity in $\mathfrak{A}_{d}$; for example, it was proved in [DGK24] that the Heisenberg VOA satisfies the strongly unital condition for mode transition algebras. Since rationality is preserved under tensor products of VOAs [DMZ94] and the Cartan-part subVOA $V_{H}$ of a type-II parabolic-type VOA $V_{P}$ is a tensor product of two VOAs, $M_{\widehat{\mathbb{C}\beta}}(1,0)$ and $V_{\mathbb{Z}\alpha}$, each satisfying the strongly unital condition, it is natural to expect that $V_{H}$ also satisfies this property. Our final main theorem confirms this expectation.

Theorem D**.**

Let $V_{P}=V_{H}\oplus V^{+}$ be a rank-two parabolic-type VOA. Then its Cartan-part subVOA $V_{H}$ satisfies the strongly unital condition for mode transition algebras.

This paper is organized as follows. In Section 2, we recall the definitions and basic properties of conic, Borel, and parabolic-type submonoids of an even lattice, and we prove the classification theorem for rank-two parabolic-type submonoids (Theorem A). In Section 3, we review the definition of lattice VOAs and establish the structural properties of rank-two parabolic-type subVOAs $V_{P}$ (Theorem B). Section 4 is devoted to the classification of irreducible $V_{P}$-modules and the determination of fusion rules among them (Theorem C). Finally, in Section 5, we prove that the mode transition algebras $\mathfrak{A}_{d}$ of the Cartan-part subVOA $V_{H}$ of type-II parabolic-type VOAs are strongly unital for all $d\geq 0$ (Theorem D). Throughout, unless otherwise stated, all vector spaces are defined over $\mathbb{C}$.

2. The rank-two parabolic-type monoids

In this Section, we recall the notions of Borel and parabolic-type submonoids of a positive-definite even lattice $L$ introduced by the first named author in [Liu25]. Then we classify all the rank-two parabolic-type monoids as a preparation for the study of rank-two parabolic-type VOAs.

2.1. Basics of parabolic-type submonoids of an even lattice

Let $\gamma$ be a nonzero vector in a Euclidean space $(E,(\cdot|\cdot))$. Let $P(\gamma)$ denote the hyperplane passing through the origin and perpendicular to $\gamma$. We define the positive and non-negative sides of $P(\gamma)$ by $P^{+}(\gamma)$ and $P^{\geq 0}(\gamma)$, respectively, and the opposite sides by $P^{-}(\gamma)$ and $P^{\leq 0}(\gamma)$. More precisely,

[TABLE]

An even lattice $L$ in $(E,(\cdot|\cdot))$ is a free abelian group of rank $r=\dim_{\mathbb{R}}E$ such that $(\alpha|\alpha)\in 2\mathbb{Z}$ for all $\alpha\in L$. A submonoid $M\leq L$ is an additive sub-abelian group of $L$ containing [math].

Definition 2.1**.**

[Liu25, Definition 2.1] Let $L$ be an even lattice in a Euclidean space $(E,(\cdot|\cdot))$.

1. (1)

A submonoid $C\leq L$ is said to be of conic-type if there exists a basis $\{\alpha_{1},\dots,\alpha_{r}\}$ of $L$ such that

[TABLE] 2. (2)

A submonoid $B\leq L$ is said to be of Borel-type if it satisfies:

1. (a)

$B\cup(-B)=L$, 2. (b)

$B\cap(-B)=\{0\}$, 3. (c)

There exists a hyperplane $P(\gamma)\subset E$ such that $B\subset P^{\geq 0}(\gamma)$. 3. (3)

A proper submonoid $P\leq L$ is said to be of parabolic-type if there exists a Borel-type submonoid $B\leq L$ such that $B\subset P$.

Recall the following basic facts about Borel and parabolic-type monoids in [Liu25, Propositions 2.2, 2.4].

Proposition 2.2**.**

Let $L$ be an even lattice in the Euclidean space $(E,(\cdot|\cdot))$. Then

1. (1)

Borel and parabolic-type submonoids of $L$ exist. They are invariant under the action of $\mathrm{Aut}(L)$. 2. (2)

The hyperplane $P(\gamma)$ in the definition of Borel-type submonoid is unique. In other words, given a Borel-type submonoid $B$, with $B\subset P^{\geq 0}(\gamma)$, if there exists another $\gamma^{\prime}\in E\backslash\{0\}$ such that $B\subset P^{\geq 0}(\gamma^{\prime})$, then $P(\gamma)=P(\gamma^{\prime})$.

From now on, we fix a rank-two even lattice $L=\mathbb{Z}\alpha_{1}\oplus\mathbb{Z}\alpha_{2}$ in a two-dimensional Euclidean space $(E,(\cdot|\cdot))$. The following notion is useful for our later discussion.

Definition 2.3**.**

Let $L$ be a rank-two even lattice, and $\alpha_{1},\alpha_{2}\in L\backslash\{0\}$. Write

[TABLE]

We call $\mathrm{Cone}(\alpha_{1},\alpha_{2})$ a basis cone if $\{\alpha_{1},\alpha_{2}\}$ is a basis of the lattice $L$. In this case, $C=\mathrm{Cone}(\alpha_{1},\alpha_{2})$ is the conic-type submonid spanned by $\alpha_{1}$ and $\alpha_{2}$.

The following properties about $\mathrm{Cone}(\alpha_{1},\alpha_{2})$ are obvious, we omit the proof.

Lemma 2.4**.**

Let $L$ be a rank-two even lattice, and $\alpha_{1},\alpha_{2}\in L\backslash\{0\}$ be $\mathbb{Z}$-linearly independent.

1. (1)

If $M\subseteq L$ is a submonoid that contains $\alpha_{1},\alpha_{2}$, then $\mathrm{Cone}(\alpha_{1},\alpha_{2})\subseteq M$. 2. (2)

If $\mathrm{Cone}(\alpha_{1},\alpha_{2})$ is a basis cone, then it contains all the lattice points of $L$ in the conic area $\mathbb{R}_{\geq 0}\alpha_{1}+\mathbb{R}_{\geq 0}\alpha_{2}$ of the Euclidean space $E$. 3. (3)

If $\{\alpha_{1},\alpha_{2}\}$ a basis of $L$, then $L$ is a union of basis cones:

[TABLE]

Lemma 2.5**.**

The submonoid $M=P^{+}(\gamma)\cap L$ contains a basis $\{\alpha_{1},\alpha_{2}\}$ of $L$.

Proof.

Let $\{\beta_{1},\beta_{2}\}$ be any basis of $L$. Note that $\beta_{1},\beta_{2}$ cannot be both lying on $P(\gamma)$, since otherwise the rank of $L$ would be one. If $\beta_{1},\beta_{2}\in P^{\pm}(\gamma)$, then the basis $\{\alpha_{1}=\pm\beta_{1},\alpha_{2}=\pm\beta_{2}\}$ is contained in $P^{+}(\gamma)\cap L$ of $L$, we are done. Otherwise, we may assume $\beta_{1}\in P^{+}(\gamma)$ and $\beta_{2}\in P^{\leq 0}(\gamma)$. Then the basis $\{\alpha_{1}=\beta_{1},\alpha_{2}=\beta_{1}-\beta_{2}\}$ is contained in $P^{+}(\gamma)\cap L$. ∎

Definition 2.6**.**

A nonzero lattice point $\alpha\in L\backslash\{0\}$ is called primitive if $\alpha$ is not a multiple of any other lattice points in $L$. i.e., if $\alpha=k\beta$ for some $k\in\mathbb{Z}$ and $\beta\in L$, then $k=\pm 1$.

2.2. Classification of the rank-two parabolic-type submonoids

We need the following elementary result about the rank-two lattices for our classification theorem.

Proposition 2.7**.**

Let $P(\gamma)$ be a hyperplane in $E$, $\alpha\in P^{-}(\gamma)\cap L$ be a primitive lattice point, and $M$ be the submonoid in $L$ such that $\alpha\in M$ and $P^{+}(\gamma)\cap L\subset M$. Then $M=L$.

Proof.

By Lemma 2.5, there exists a basis $\{\alpha_{1},\alpha_{2}\}$ of $L$ that is contained in $P^{+}(\gamma)\cap L\subset M$. Let

[TABLE]

Note that $(\alpha|\gamma)=ma_{1}+na_{2}<0$. If $m<0$ and $n<0$, then

[TABLE]

It follows from Lemma 2.4 that $\mathrm{Cone}(\pm\alpha_{1},\pm\alpha_{2})\subseteq M$, and so $L=M$ by (2.3). Since $a_{1},a_{2}>0$ and $ma_{1}+na_{2}<0$, the remaining cases for the coefficients $m,n$ of $\alpha$ in (2.4) are $m<0,n\geq 0$ or $m\geq 0,n<0$. Without loss of generality, we assume $m<0,n\geq 0$.

Assume the hyperplane $\mathrm{span}_{\mathbb{R}}\{\alpha\}$ is given by $P(A)$ for some $A\in E\backslash\{0\}$. Then $\alpha\perp A$. Since $\{\alpha_{1},\alpha_{2}\}$ is an $\mathbb{R}$-basis of $E$, we may choose its dual basis $\{\alpha^{1},\alpha^{2}\}\subset E$ with respect to the inner product $(\cdot|\cdot)$ and assume

[TABLE]

Since $m<0$ and $A\neq 0$, we have $\mu\neq 0$. Replace $A$ by $-A$, if necessary, we assume $\mu>0$. Then

[TABLE]

We claim that there exists two lattice point $\beta,\beta^{\prime}\in L$ such that

1. (1)

$\beta,\beta^{\prime}\in P^{+}(\gamma)\cap L\subset M$. 2. (2)

$\{\alpha,\beta\}$ is a basis of $L$, $\{\alpha,\beta^{\prime}\}$ is also a basis of $L$. 3. (3)

$(A|\beta)<0$ and $(A|\beta^{\prime})>0$ i.e., $\beta,\beta^{\prime}$ are on different sides of the hyperplane $P(A)$.

See Figure 1 for an illustration of these vectors and hyperplanes in a standard rank-two lattice, where the shadow area represents $P^{+}(\gamma)$.

If $\beta$ and $\beta^{\prime}$ exists, then $\mathrm{Cone}(\pm\alpha,\beta)$ and $\mathrm{Cone}(\pm\alpha,\beta^{\prime})$ are basis cones by (2). By conditions (1), (3), and Lemma 2.4, $P^{\leq 0}(A)\cap L=\mathrm{Cone}(\alpha,\beta)\cup\mathrm{Cone}(-\alpha,\beta)\subset M$, and $P^{\geq 0}(A)\cap L=\mathrm{Cone}(\alpha,\beta^{\prime})\cup\mathrm{Cone}(-\alpha,\beta^{\prime})\subset M$. Hence $L=M$.

To prove the existence of $\beta$ and $\beta^{\prime}$, our idea is to first find some lattice points satisfying conditions (1) and (2), then pick out the ones that satisfy condition (3). We first prove the existence of $\beta$. Let $x,y\in\mathbb{Z}$ be integral variables such that

[TABLE]

Any solution $\beta=x\alpha_{1}+y\alpha_{2}$ to the system (2.6) would satisfy conditions (1) and (2) for $\beta$.

Since $\alpha\in L$ is primitive, and we assumed $m<0$ and $n\geq 0$, then we have $\gcd(m,n)=1$ if $m,n\neq 0$; and $m=-1$ if $n=0$. In particular, there exists integers $(x_{0},y_{0})$ such that $my_{0}-nx_{0}=1$. Fix an integral solution $(x_{0},y_{0})$ to $my-nx=1$, the general integral solutions to the Diophantine equation $my-nx=1$ are given by

[TABLE]

Since $ma_{1}+na_{2}<0$ and $m<0$, we may choose a $t\in\mathbb{Z}_{<0}$ small enough so that

[TABLE]

Fix such a $t$, and let $(x_{1},y_{1})=(x_{0}+tm,y_{0}+tn)$. Then $\beta=x_{1}\alpha_{1}+y_{1}\alpha_{2}$ is a solution to (2.6), with $x_{1}>0$. Since $my_{1}-nx_{1}=1$, $x_{1}>0$ and $m<0$, we have

[TABLE]

Now we show $\beta$ satisfies condition (3). By (2.5) and (2.7), together with the facts that $m<0$ and $x_{1}>0$, we have the following estimate:

[TABLE]

Since $\mu x_{1}>0$, we have $(A|\beta)<0$.

Now to show the existence of $\beta^{\prime}$, we consider another system similar to (2.6):

[TABLE]

A similar argument shows that there exists integers $(x_{2},y_{2})$ such that

[TABLE]

Then $\beta^{\prime}=x_{2}\alpha_{1}+y_{2}\alpha_{2}$ is a solution to (2.8), and

[TABLE]

since $m<0$ and $x_{2}>0$. Thus we have $(A|\beta^{\prime})>0$ since $\mu x_{2}>0$. ∎

Theorem 2.8**.**

Let $P$ be a parabolic-type submonoid of a rank-two even lattice $L$. Then it has two possible types:

1. (1)

$P=B$* is a rank-two Borel-type submonoid. In this case, $P=(\mathbb{Z}_{\geq 0}\alpha)\cup(P^{+}(\gamma)\cap L)$, where $P(\gamma)$ is the unique hyperplane such that $B\subset P^{\geq 0}(\gamma)$ and $P(\gamma)\cap L=\mathbb{Z}\alpha$ for some $\alpha\in L$ (possibly zero).* 2. (2)

$P=P^{\geq 0}(\gamma)\cap L$* for some hyperplane $P(\gamma)\subset E$ such that $P(\gamma)\cap L=\mathbb{Z}\alpha\neq\{0\}$.*

See Figures 2 and 3 for an illustration in a standard rank-two lattice. We say that $P$ is of type-I if $P=B$; and $P$ is of type-II if $P=P^{\geq 0}(\gamma)\cap L$ with $P(\gamma)\cap L\neq\{0\}$.

Proof.

By Definition 2.1, there exists a Borel-type submonoid $B$ such that $B\subseteq P$. By Proposition 2.2, there exists a unique hyperplane $P(\gamma)$ such that $B\subset P^{\geq 0}(\gamma)$. Since $B\cap(-B)=\{0\}$ and $P^{+}(\gamma)\cap P^{\leq 0}(\gamma)=\emptyset$, we have $(-B)\subseteq P^{\leq 0}(\gamma)$ and $P^{+}(\gamma)\cap L\subset B$.

Assume $P=B$ is of Type I. Note that $P(\gamma)\cap L$ is a discrete additive subgroup in the one-dimensional Euclidean space $P(\gamma)$. Then $P(\gamma)\cap L=\mathbb{Z}\alpha$ for some $\alpha\in L$. Since $B\cup(-B)=L$, replace $\alpha$ by $-\alpha$ if necessary, we may assume $\alpha\in B$. Then the monoid $M=(\mathbb{Z}_{\geq 0}\alpha)\cup(P^{+}(\gamma)\cap L)\subseteq B$. Clearly, $M\cap(-M)=\{0\}=B\cap(-B)$, and $M\cup(-M)=L$. Hence $M=B$.

Now assume $B\subsetneq P$. We claim that $P$ is of Type II. Indeed, by our argument above, $B=(\mathbb{Z}_{\geq 0}\alpha)\cup(P^{+}(\gamma)\cap L)$. In particular $P^{+}(\gamma)\cap L\subset P$. Let $\beta\in P\backslash B$. Then $\beta$ is either in $\mathbb{Z}_{<0}\alpha$ or $P^{-}(\gamma)\cap L$. If $\beta\in P^{-}(\gamma)\cap L$, write $\beta=k\alpha^{\prime}$ for some $k\geq 1$ and primitive vector $\alpha^{\prime}\in P^{-}(\gamma)\cap L$. Apply Proposition 2.7 to $M=P$, we have $P=L$, which contradicts the assumption that $P\subset L$ is a proper submonoid. Thus $\beta\in\mathbb{Z}_{<0}\alpha$. Since $\alpha\in P$, it follows that $-\alpha\in P$ and so $(\mathbb{Z}\alpha)\cup(P^{+}(\gamma)\cap L)=P^{\geq 0}(\gamma)\cap L\subseteq P$. Since $P$ contains no vectors in $P^{-}(\gamma)\cap L$, we must have $P=P^{\geq 0}(\gamma)\cap L$. ∎

3. Structural theory of $V_{P}$

In this Section, we determine the structure of all parabolic-type VOAs $V_{P}$ associated to the rank-two parabolic-type submonoids $P$ based on our classification theorem for the parabolic-type monoids in the previous Section.

3.1. Basics of the lattice vertex operator algebras

For the general definitions of vertex operator algebras (VOAs) and modules over VOAs, we refer to the classical texts [FLM88, FHL93, DL93, LL04, FZ92, Z96]. Here we recall the notion of modules over a VOA.

Definition 3.1**.**

Let $V$ be a VOA. An admissible $V$-module is a $\mathbb{N}$-graded vector space $M=\bigoplus_{n=0}^{\infty}M(n)$, equipped with a linear map $Y_{M}(\cdot,z):V\rightarrow\mathrm{End}(M)[\![z,z^{-1}]\!],\ Y_{M}(a,z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}$ called the module vertex operator, satisfying

1. (1)

(truncation property) For any $a\in V$ and $u\in M$, $Y_{M}(a,z)u\in M((z))$. 2. (2)

(vacuum property) $Y_{M}(\mathbf{1},z)=\mathrm{Id}_{M}$. 3. (3)

(Jacobi identity for $Y_{M}$) for any $a,b\in V$ and $u\in M$,

[TABLE] 4. (4)

($L(-1)$-derivative property) $Y_{M}(L(-1)a,z)=\frac{d}{dz}Y_{M}(a,z)$ for any $a\in V$. 5. (5)

(grading property) For any $a\in V$, $m\in\mathbb{Z}$, and $n\in\mathbb{N}$, $a_{m}M(n)\subseteq M(n+\mathrm{wt}a-m-1)$. i.e., $\mathrm{wt}(a_{m})=\mathrm{wt}a-m-1$.

We write $\deg v=n$ if $v\in M(n)$, and call it the degree of $v$. Submodules, quotient modules, and irreducible modules are defined in the usual categorical sense. Denote the category of admissible $V$-modules by $\mathsf{Adm}(V)$. $V$ is called rational if the category $\mathsf{Adm}(V)$ is semisimple [DLM98, Z96].

An admissible $V$-module $M$ is called ordinary if each degree-$n$ subspace $M(n)=M_{n+h}$ is a finite-dimensional eigenspace of $L(0)$ of eigenvalue $n+h$, where $h\in\mathbb{C}$ is called the conformal weight of $M$. In particular, if we write $L(0)v=(\mathrm{wt}v)\cdot v$ for $v\in M(n)$, then $\mathrm{wt}v=\deg v+h$.

More generally, a weak $V$-module is vector space $M$, together with a module vertex operator $Y_{M}(\cdot,z)$, satisfying conditions $(1),(2)$, and $(3)$ above.

We briefly recall the construction of Heisenberg and lattice VOAs in [FLM88]. Some of the formulas here will be used later.

Let $\mathfrak{h}$ be a $r\geq 1$-dimensional vector space, equipped with a nondegenerate bilinear form $(\cdot|\cdot):\mathfrak{h}\times\mathfrak{h}\rightarrow\mathbb{C}$. The affinization $\hat{\mathfrak{h}}=\mathfrak{h}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}K$ has Lie bracket

[TABLE]

where $h\otimes t^{m}\in\hat{\mathfrak{h}}$ is denoted by $h(m)$, together with a triangular decomposition $\hat{\mathfrak{h}}=(\hat{\mathfrak{h}})_{+}\oplus(\hat{\mathfrak{h}})_{0}\oplus(\hat{\mathfrak{h}})_{-}$, where $(\hat{\mathfrak{h}})_{\pm}=\bigoplus_{n\in\mathbb{Z}_{\pm}}\mathfrak{h}\otimes\mathbb{C}t^{n}$, and $(\hat{\mathfrak{h}})_{0}=\mathfrak{h}\otimes\mathbb{C}1\oplus\mathbb{C}K$. Let $(\hat{\mathfrak{h}})_{\geq 0}=(\hat{\mathfrak{h}})_{+}\oplus(\hat{\mathfrak{h}})_{0}$, which is a Lie sub-algebra of $\hat{\mathfrak{h}}$.

For each $\lambda\in\mathfrak{h}$, let $e^{\lambda}$ be a formal symbol associated to $\lambda$. Then $\mathbb{C}e^{\lambda}$ is a module over $(\hat{\mathfrak{h}})_{\geq 0}$, with $h(0)e^{\lambda}=(h|\lambda)e^{\lambda}$, $K.e^{\lambda}=e^{\lambda}$, and $h(n)e^{\lambda}=0$, for all $h\in\mathfrak{h}$ and $n>0$. The induced module

[TABLE]

is an irreducible module over $\hat{\mathfrak{h}}$. It has a basis

[TABLE]

It was proved in [FLM88, FZ92] that $M_{\hat{\mathfrak{h}}}(1,0)$ is a CFT-type simple VOA, with $\mathbf{1}=e^{0}$ and $\omega=\frac{1}{2}\sum_{i=1}^{r}u^{i}(-1)^{2}\mathbf{1}$, where $\{u^{1},\dots,u^{r}\}$ is an orthonormal basis of $\mathfrak{h}$, called the Heisenberg VOA of level-one, and $M_{\hat{\mathfrak{h}}}(1,\lambda)$, with $\lambda\in\mathfrak{h}$, are all the irreducible modules over the VOA $M_{\hat{\mathfrak{h}}}(1,0)$ up to isomorphism. The (module) vertex operator is defined by normal ordering

[TABLE]

where the labels $(i_{1},\dots,i_{p},j_{1},\dots,j_{q})$ are permutations of $(1,2,\dots,k)$ and

[TABLE]

for a field $a(z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}$.

Next, we recall the lattice VOAs. Let $L$ be an even lattice of rank $r\geq 1$ in a Euclidean space $E$, equipped with $\mathbb{Z}$-bilinear form $(\cdot|\cdot):L\times L\rightarrow\mathbb{Z}$. Let $\mathfrak{h}:=\mathbb{C}\otimes_{\mathbb{Z}}L$, extend $(\cdot|\cdot)$ to a $\mathbb{C}$-bilinear form $(\cdot|\cdot):\mathfrak{h}\times\mathfrak{h}\rightarrow\mathbb{C}$. Let $\epsilon:L\times L\rightarrow\langle\pm 1\rangle$ be a $2$-cocycle of the abelian group $L$ such that $\epsilon(\alpha,\beta)\epsilon(\beta,\alpha)=(-1)^{(\alpha|\beta)}$, for any $\alpha,\beta\in L$. Write $\mathbb{C}^{\epsilon}[L]=\bigoplus_{\alpha\in L}\mathbb{C}e^{\alpha}$, where $e^{\alpha}$ is a formal symbol associated to $\alpha$ for each $\alpha\in L$ ($e^{\alpha}$ is denoted by $\iota(\alpha)$ in [FLM88]). Let

[TABLE]

where we identify $M_{\hat{\mathfrak{h}}}(1,\alpha)$ with $M_{\hat{\mathfrak{h}}}(1,0)\otimes\mathbb{C}e^{\alpha}$ for each $\alpha\in L$. Define the lattice vertex operator $Y:V_{L}\rightarrow\mathrm{End}(V_{L})[[z,z^{-1}]]$ as follows

[TABLE]

where $\alpha,\beta\in L$, $h\in\mathfrak{h}$, and the operators $E^{\pm}$, $e^{\alpha}$, and $z^{\alpha}$ in (3.7), (3.8) are given by

[TABLE]

Moreover, the general lattice vertex operator is defined by normal ordering similar to (3.5)

[TABLE]

where $k\geq 1$, $n_{1}\geq\dots\geq n_{k}\geq 0$, and $h_{1},\dots h_{k}\in\mathfrak{h}$.

It was proved in [FLM88, Appendix A.2] that $V_{L}$ is a VOA, called the lattice VOA. The Heisenberg VOA $M_{\hat{\mathfrak{h}}}(1,0)$ is a subVOA of $V_{L}$ with the same Virasoro element $\omega=\frac{1}{2}\sum_{i=1}^{r}u^{i}(-1)^{2}\mathbf{1}$. Moreover, the lattice vertex operator $Y$ given by (3.7)–(3.10) are intertwining operators among the Heisenberg modules $M_{\hat{\mathfrak{h}}}(1,\alpha)$ in the decomposition (3.6) [FLM88, D93, DL93]. In particular, it satisfies

[TABLE]

The irreducible modules over a lattice VOA were classified by Dong.

Lemma 3.2**.**

[D93, Theorem 3.1]** Let $L$ be a even lattice in a Euclidean space $E$, and let $L^{\circ}=\{\lambda\in E:(\lambda|\alpha)\in\mathbb{Z},\ \forall\alpha\in L\}$ be the dual lattice. Assume $L^{\circ}/L=\bigsqcup_{i=0}^{r}(L+\lambda_{i})$ is the coset decomposition, then $\{V_{L+\lambda_{i}}:i=0,\dots,r\}$ are all the irreducible $V_{L}$-modules up to isomorphism. Moreover, $V_{L}$ is a rational VOA.

The following fact follows from (3.11).

Lemma 3.3**.**

[Liu25, Proposition 3.2]**. Let $L$ be an even lattice in a Euclidean space $E$, $M\leq L$ be a submonoid, and $S\subset L$ be a sub-semigroup. Let $V_{M}=\bigoplus_{\alpha\in M}M_{\hat{\mathfrak{h}}}(1,\alpha)$ and $V_{S}=\bigoplus_{\alpha\in S}M_{\hat{\mathfrak{h}}}(1,\alpha)$. Then

1. (1)

$(V_{M},Y,\omega,\mathbf{1})$* is a CFT-type subVOA of the lattice VOA $(V_{L},Y,\omega,\mathbf{1})$.* 2. (2)

$(V_{S},Y,L(-1))$* is a sub-vertex algebra without vacuum of $(V_{L},Y,L(-1))$ [HL96]. If, furthermore, $S\subset M$ and $M+S\subseteq S$, then $V_{S}$ is an ideal of $V_{M}$.*

By choosing particular kinds of submonoid $M$ in a lattice $L$, we have different types of subVOAs in $V_{L}$. The following subVOAs in $V_{L}$ are our main objects to study in this paper.

Definition 3.4**.**

Let $L=\mathbb{Z}\alpha_{1}\oplus\mathbb{Z}\alpha_{2}$ be a rank-two even lattice in a Euclidean space $E$.

1. (1)

Let $P\leq L$ be a parabolic-type submonoid, see Definition 2.1. We call the subVOA $V_{P}=\bigoplus_{\alpha\in P}M_{\hat{\mathfrak{h}}}(1,\alpha)$ of $V_{L}$ a rank-two parabolic-type VOA. We say that the VOA $V_{P}$ is of type-I (resp. type-II) if the rank-two submonoid $P$ is of type-I (resp. type-II) in Theorem 2.8. 2. (2)

Let $P(\gamma)\subset E$ be a hyperplane. Then the submonoid $H=P(\gamma)\cap L$ is a rank-one even lattice $\mathbb{Z}\alpha$ or $\{0\}$, see Theorem 2.8. When $H\neq\{0\}$, we call the subVOA

[TABLE]

a rank-two hyperplane VOA, see Figure 4 for an illustration.

Remark 3.5**.**

We observe the following facts from Definition 3.4.

1. (1)

The rank-two hyperplane VOA $V_{H}$ (3.12) is not isomorphic to the rank-one lattice VOA $V_{\mathbb{Z}\alpha}$ since $\dim\mathfrak{h}=2$ and $M_{\widehat{\mathbb{C}\alpha}}(1,n\alpha)\neq M_{\hat{\mathfrak{h}}}(1,n\alpha)$, see (3.4) and (3.6). 2. (2)

One can define parabolic-type VOAs in a higher rank lattice VOA, see [Liu25, Definition 3.4]. The notion of hyperplane VOA is also generalizable to the higher rank case.

3.2. Tensor product description of the rank-two hyperplane VOAs

Recall the notion of tensor product VOAs introduced by Frenkel-Huang-Lepowsky.

Lemma 3.6**.**

[FHL93, Section 2.5, Proposition 4.7.2, Corollary 4.7.3]** Let $(V_{1},Y_{V_{1}},\omega_{1},\mathbf{1}_{1})$ and $(V_{2},Y_{V_{2}},\omega_{2},\mathbf{1}_{2})$ be two VOAs. The tensor product space $V_{1}\otimes V_{2}$ has a VOA structure with

[TABLE]

$\mathbf{1}_{V_{1}\otimes V_{2}}=\mathbf{1}_{1}\otimes\mathbf{1}_{2}$* and $\omega_{V_{1}\otimes V_{2}}=\omega_{1}\otimes\mathbf{1}_{2}+\mathbf{1}_{1}\otimes\omega_{2}$.*

Furthermore, the irreducible ordinary modules over $V_{1}\otimes V_{2}$ are of the form $M_{1}\otimes M_{2}$, where $M_{i}$ are irreducible ordinary modules over $V_{i}$ for $i=1,2$, respectively. In particular, if $V_{1}$ and $V_{2}$ are both simple VOAs, then $V_{1}\otimes V_{2}$ is a simple VOA.

Lemma 3.7**.**

Let $L$ be an even lattice in a Euclidean space $E$ of rank at least two. Let $\alpha,\beta\in L\backslash\{0\}$ such that $(\alpha|\beta)=0$. Then for any $u=\alpha(-n_{1}-1)\dots\alpha(-n_{r}-1)e^{p\alpha}\in M_{\hat{\mathfrak{h}}}(1,p\alpha)\subset V_{L}$, with $r\geq 0$, $n_{1}\geq\dots\geq n_{r}\geq 0$, and $p\in\mathbb{Z}$, the following relation holds in $\mathrm{End}(V_{L})[\![z,z^{-1}]\!]$

[TABLE]

Proof.

By (3.2) we have $[\beta(m),\alpha(n)]=0$ for any $m,n\in\mathbb{Z}$. Hence $\beta(m)$ commutes with $(\partial^{(n)}\alpha(z))_{\pm}$ and $E^{\pm}(-p\alpha,z)$ for any $n\geq 1$ and $p\in\mathbb{Z}$. Then by (3.10), it suffices to show that $\beta(m)$ commutes with the operator $e_{p\alpha}z^{p\alpha}$ in $\mathrm{End}(V_{L})[\![z,z^{-1}]\!]$.

Indeed, if $m\neq 0$, by (3.7) and (3.8), $\beta(m)=\beta(m)\otimes 1$ clearly commutes with $e_{p\alpha}z^{p\alpha}=1\otimes e_{p\alpha}z^{p\alpha}$ as operators on $M_{\hat{\mathfrak{h}}}(1,\theta)\cong M_{\hat{\mathfrak{h}}}(1,0)\otimes\mathbb{C}e^{\theta}\subset V_{L}$ for any $\theta\in L$. On the other hand, if $m=0$, then $\beta(0)=1\otimes\beta(0)$ as an operator on $M_{\hat{\mathfrak{h}}}(1,0)\otimes\mathbb{C}e^{\theta}$. Since $(\beta|p\alpha)=0$, we have

[TABLE]

for any $\theta\in L$. This proves (3.14). ∎

Proposition 3.8**.**

The rank-two hyperplane VOA $V_{H}$ (3.12) is simple and is isomorphic to the tensor product of a rank-one Heisenberg VOA and a rank-one lattice VOA:

[TABLE]

where $\mathbb{C}\beta=(\mathbb{C}\alpha)^{\perp}$ in $\mathfrak{h}=\mathbb{C}\otimes_{\mathbb{Z}}L$ with respect to the extended $\mathbb{C}$-bilinear form $(\cdot|\cdot):\mathfrak{h}\times\mathfrak{h}\rightarrow\mathbb{C}$.

Proof.

Since $\mathfrak{h}=\mathbb{C}\alpha\oplus\mathbb{C}\beta$ and $[\beta(n),\alpha(m)]=0$ for all $m,n\in\mathbb{Z}$, the rank-two Heisenberg irreducible module $M_{\hat{\mathfrak{h}}}(1,p\alpha)$ has a basis $\beta(-n_{1})\dots\beta(-n_{r})\alpha(-m_{1})\dots\alpha(-m_{s})e^{p\alpha}$, where $r,s\geq 0$, $n_{1}\geq\dots\geq n_{r}\geq 1$ and $m_{1}\geq\dots\geq m_{s}\geq 1$, see (3.4). Then we have an isomorphism of vector spaces:

[TABLE]

which extends to a linear isomorphism

[TABLE]

Denote $M_{\widehat{\mathbb{C}\beta}}(1,0)$ by $V_{1}$ and $V_{\mathbb{Z}\alpha}$ by $V_{2}$ for short. We claim that

[TABLE]

for $u=\beta(-n_{1})\dots\beta(-n_{r})\alpha(-m_{1})\dots\alpha(-m_{s})e^{p\alpha}$ and $v=\beta(-n^{\prime}_{1})\dots\beta(-n^{\prime}_{r^{\prime}})\alpha(-m^{\prime}_{1})\dots\alpha(-m^{\prime}_{s^{\prime}})e^{q\alpha}$.

Indeed, by the commutativity of the lattice operators $\beta(n)$ and $Y(\alpha(-m_{1})\dots\alpha(-m_{s})e^{p\alpha},z)$ as in Lemma 3.7, together with (3.10), it is easy to see that

[TABLE]

Then it follows from Lemma 3.6 and (3.15), together with the definition of vertex operators (3.5) and (3.10) that

[TABLE]

This proves (3.16). Finally, let $\beta_{1}=\beta/\|\beta\|$ and $\alpha_{1}=\alpha/\|\alpha\|$, where we define $\|h\|$ to be the principal branch of $(h|h)^{\frac{1}{2}}$ for $h\in\mathfrak{h}$. Then $\{\beta_{1},\alpha_{1}\}$ is an orthonormal basis of $\mathfrak{h}$ and $\omega_{V_{H}}=\frac{1}{2}\beta_{1}(-1)^{2}\mathbf{1}+\frac{1}{2}\alpha_{1}(-1)^{2}\mathbf{1}$. Moreover, $\frac{1}{2}\beta_{1}(-1)^{2}\mathbf{1}$ is the Virasoro element of $M_{\widehat{\mathbb{C}\beta}}(1,0)$, and $\frac{1}{2}\alpha_{1}(-1)^{2}\mathbf{1}$ is the Virasoro element of $V_{\mathbb{Z}\alpha}$. Then by Lemma 3.6 and (3.15), we have

[TABLE]

Hence $\varphi:V_{H}\rightarrow M_{\widehat{\mathbb{C}\beta}}(1,0)\otimes V_{\mathbb{Z}\alpha}$ is an isomorphism of VOAs. Since the Heisenberg VOA $M_{\widehat{\mathbb{C}\beta}}(1,0)$ and the lattice VOA $V_{\mathbb{Z}\alpha}$ are both simple, by Lemma 3.6, $V_{H}$ is a simple VOA. ∎

3.3. Structure of the rank-two parabolic-type VOAs $V_{P}$ and $C_{1}$-cofiniteness

Recall that the rank-two parabolic-type submonoids have the following classification, see Theorem 2.8.

[TABLE]

Now we determine the structures of rank-two parabolic-type VOAs.

Theorem 3.9**.**

Let $V_{P}$ be a rank-two parabolic-type VOA. Then there exists a unique maximum proper ideal $V^{+}\lhd V_{P}$ and a simple subVOA $V_{H}\leq V_{P}$ such that $V_{P}=V_{H}\oplus V^{+}$ as vector spaces and $V_{P}/V^{+}\cong V_{H}$ as VOAs. Moreover, we have the following characterization for the simple quotient VOA $V_{H}$:

1. (1)

If $P=B$ is of type-I, then $V_{H}=M_{\hat{\mathfrak{h}}}(1,0)$ is the rank-two Heisenberg VOA. 2. (2)

If $P$ is of type-II, then $V_{H}\cong M_{\widehat{\mathbb{C}\beta}}(1,0)\otimes V_{\mathbb{Z}\alpha}$ is the hyperplane VOA (see (3.12) and Proposition 3.8), where $\beta\in\mathfrak{h}=\mathbb{C}\otimes_{\mathbb{Z}}L$ such that $\mathbb{C}\beta=(\mathbb{C}\alpha)^{\perp}.$

We call the subVOA $V_{H}$ the Cartan-part of $V_{P}$.

Proof.

Assume $P$ is of type-I. Let $S:=P\backslash\{0\}=\mathbb{Z}_{>0}\alpha\cup(P^{+}(\gamma)\cap L)$ in view of (3.17). Clearly, $S\subset P$ is a sub-semigroup such that $P+S\subseteq S$. Let $V^{+}:=V_{S}=\bigoplus_{\eta\in S}M_{\hat{\mathfrak{h}}}(1,\eta)$. Then by Proposition 3.3, $V^{+}\lhd V_{P}$ is an ideal. Moreover, we have $V_{P}=M_{\hat{\mathfrak{h}}}(1,0)\oplus V^{+}$, with $V_{H}:=M_{\hat{\mathfrak{h}}}(1,0)$ being a subVOA of $V_{P}$. Hence $V_{P}/V^{+}\cong V_{H}$ as VOAs. It remains that show that $V^{+}$ is the maximum proper ideal of $V_{P}$.

Indeed, let $J\lhd V_{P}$ be a proper ideal. It suffices to show $J\subseteq V^{+}$. Since $J$ is closed under the action of the field $Y(h(-1)\mathbf{1},z)=h(z)=\sum_{n\in\mathbb{Z}}h(n)z^{-n-1}$, we have

[TABLE]

If $J_{0}=M_{\hat{\mathfrak{h}}}(1,0)\cap J=0$, then $J\subseteq V^{+}$, we are done. Otherwise, $J_{0}=M_{\hat{\mathfrak{h}}}(1,0)$ since $M_{\hat{\mathfrak{h}}}(1,0)$ is a simple VOA. Then $\mathbf{1}\in M_{\hat{\mathfrak{h}}}(1,0)\in J$ and $J=V_{P}$, which contradicts the properness of $J$.

Now assume $P$ is of type-II. Let $S:=P^{+}(\gamma)\cap L$ and $V^{+}:=V_{S}$. Then by (3.17), we have $P=S\sqcup H$, with $H=P(\gamma)\cap L=\mathbb{Z}\alpha$ and $P+S\subseteq S$. it follows that $V_{P}=V_{H}\oplus V^{+}$, $V_{H}\leq V_{P}$ is a subVOA, and $V^{+}\lhd V_{P}$ is an ideal. Hence $V_{P}/V^{+}\cong V_{H}$ as VOAs. To show $V^{+}$ is the maximum proper ideal of $V_{P}$, we let $J\lhd V_{P}$ be a proper ideal. Then by (3.18), $J\cap V_{H}=\bigoplus_{\tau\in H}J_{\tau}$ is an ideal of the subVOA $V_{H}$. Since $\mathbf{1}\in V_{H}$, we have $J\cap V_{H}\neq V_{H}$. But $V_{H}$ is a simple VOA by Proposition 3.8. Hence $J\cap V^{+}=0$ and $J=J\cap V^{+}\subseteq V^{+}$. ∎

Remark 3.10**.**

We note that the direct sum decomposition $V_{P}=V_{H}\oplus V^{+}$ in Theorem 3.9 can be viewed as a direct sum of a CFT-type VOA $V_{H}$ with a $V_{H}$-module $V^{+}$. However, this direct sum is not the semi-direct product of the VOA $V_{H}$ with the module $V^{+}$ introduced by Li in [L94] since $Y_{V_{P}}(V^{+},z)V^{+}\neq 0$.

With the structure theorem for $V_{P}$, we discuss its $C_{1}$-cofiniteness. The subspace $C_{1}(V)\subset V$ was introduced by Li in [Li99]

[TABLE]

$V$ is called $C_{1}$-cofinite if $\dim V/C_{1}(V)<\infty$.

A related notion is the strongly generation property of VOAs introduced by Kac in [K97]. A CFT-type VOA $V$ is called strongly generated by a subset $U\subseteq V$ if $V$ is spanned by

[TABLE]

Karel and Li proved that a VOA $V$ is strongly generated by a finite-dimensional subspace $U\subset V$ if and only if $V$ is $C_{1}$-cofinite [Li99, KL99].

Proposition 3.11**.**

Let $V_{P}$ be a rank-two parabolic-type VOA. Then its Cartan-part subVOA $V_{H}$ is always $C_{1}$-cofinite.

Proof.

It is well-known that the Heisenberg VOA $M_{\hat{\mathfrak{h}}}(1,0)$ and the lattice VOA $V_{L}$ are $C_{1}$-cofinite [KL99]. By Theorem 3.9, to show $V_{H}$ is $C_{1}$-cofinite, it suffices to show if two CFT-type VOAs $V_{1}$ and $V_{2}$ are $C_{1}$-cofinite, then their tensor product $V_{1}\otimes V_{2}$ is also $C_{1}$-cofinite. By (3.13), it is clear that $C_{1}(V_{1})\otimes V_{2}+V_{1}\otimes C_{1}(V_{2})\subseteq C_{1}(V_{1}\otimes V_{2})$. Conversely, for $a^{1}\otimes a^{2},b^{1}\otimes b^{2}\in(V_{1}\otimes V_{2})_{+}$, since $L(-1)(a^{1}\otimes a^{2})=L(-1)a^{1}\otimes a^{2}+a^{1}\otimes L(-1)a^{2}\in C_{1}(V_{1})\otimes V_{2}+V_{1}\otimes C_{1}(V_{2})$ and

[TABLE]

We have $C_{1}(V_{1}\otimes V_{2})=C_{1}(V_{1})\otimes V_{2}+V_{1}\otimes C_{1}(V_{2})$ by (3.19), and so

[TABLE]

is a finite-dimensional vector space. ∎

Now we discuss the $C_{1}$-cofiniteness of the rank-two parabolic-type VOAs $V_{P}$. The following Lemma was proved in [Liu25, Lemma 5.1, Theorem 5.2, Remark 5.3].

Lemma 3.12**.**

Let $L=\mathbb{Z}\alpha\oplus\mathbb{Z}\beta$ be a rank-two even lattice, and $C=\mathrm{Cone}(\alpha,\beta)=\mathbb{Z}_{\geq 0}\alpha+\mathbb{Z}_{\geq 0}\beta$.

1. (1)

If $(\alpha|\beta)\geq 0$, then the conic-type VOA $V_{C}$ is $C_{1}$-cofinite. 2. (2)

If $(\alpha|\beta)=-n$ for some $n\geq 1$, then the conic-type VOA $V_{C}$ is $C_{1}$-cofinite if the following condition holds: Let $(\beta|\beta)=2k$ and $(\alpha|\alpha)=2\ell$, with $k\geq\ell\geq 1$, the integers $n,k,\ell$ need to satisfy

[TABLE]

In particular, if $(\alpha|\alpha)=2$, then $V_{C}$ is $C_{1}$-cofinite.

Lemma 3.13**.**

Let $V$ be a CFT-type VOA, and $V_{1},V_{2}$ be two CFT-type subVOAs that have the same vacuum and Virasoro elements with $V$ itself. If $V=V_{1}+V_{2}$ and $V_{1}$, $V_{2}$ are both $C_{1}$-cofinite, then $V$ is also $C_{1}$-cofinite.

Proof.

By (3.19), it is clear that $C_{1}(V_{1})+C_{2}(V_{2})\subseteq C_{1}(V)$. Then there exists canonical surjective linear maps

[TABLE]

Hence $\dim(V/C_{1}(V))\leq\dim(V_{1}/C_{1}(V_{1}))+\dim(V_{2}/C_{1}(V_{2}))<\infty$. ∎

Theorem 3.14**.**

Let $V_{P}$ be a rank-two parabolic-type VOA.

1. (1)

Assume $P=B$ is of type-I, see Theorem 2.8. Then $V_{P}$ is not $C_{1}$-cofinite. 2. (2)

Assume $P=P^{\geq 0}(\gamma)\cap L$ is of type-II, with $P(\gamma)\cap L=\mathbb{Z}\alpha\neq\{0\}$, see Theorem 2.8.

If there exists $\beta\in P$ such that $\{\alpha,\beta\}$ is a basis of $L$, and condition (3.21) is satisfied for the ordered basis $\{\alpha,\beta\}$ or $\{\beta,\alpha\}$, then $V_{P}$ is $C_{1}$-cofinite. In particular, if $(\alpha|\alpha)=2$ or $(\beta|\beta)=2$, then $V_{P}$ is $C_{1}$-cofinite.

Proof.

The proof of (1) is similar to the proof of [Liu25, Proposition 5.7] for the non-$C_{1}$-cofiniteness of the standard Borel-type subVOA $V_{B}$ of $V_{A_{2}}$. We briefly sketch the idea. Assume $P=B=(\mathbb{Z}_{\geq 0}\alpha)\cup(P^{+}(\gamma)\cap L)$, with $P(\gamma)\cap L=\mathbb{Z}\alpha$. Choose $\beta\in P$ such that $\{\alpha,\beta\}$ is a basis of $L$. Then $\beta-m\alpha\in P$ for any $m\geq 1$. One can show that $\{e^{\beta-m\alpha}\in V_{P}:m=1,2,\dots\}$ is not contained in $C_{1}(V_{P})$, and the image of this set is linearly independent in $V_{P}/C_{1}(V_{P})$. We omit the further details.

Now assume $P=P^{\geq 0}(\gamma)\cap L$ is of type-II, and there exists $\beta\in P$ such that $\{\alpha,\beta\}$ is a basis of $L$ and condition (3.21) is satisfied for $\{\alpha,\beta\}$. It follows from (2.3) that $P=\mathrm{Cone}(\alpha,\beta)\cup\mathrm{Cone}(-\alpha,\beta)$. Let $C_{1}=\mathrm{Cone}(\alpha,\beta)$ and $C_{2}=\mathrm{Cone}(-\alpha,\beta)$. Then $V_{P}=V_{C_{1}}+V_{C_{2}}$ as a VOA, see Lemma 3.3. Then $V_{C_{1}}$ is $C_{1}$-cofinite by Lemma 3.12 (2). On the other hand, since $(\alpha|-\beta)=n\geq 0$, then $V_{C_{2}}$ is $C_{1}$-cofinite by Lemma 3.12 (1). Hence $V_{P}=V_{C_{1}}+V_{C_{2}}$ is $C_{1}$-cofinite by Lemma 3.13. ∎

4. Representation theory of $V_{P}$

In this Section, we present a detailed investigation of the representation theory of the rank-two parabolic-type VOA $V_{P}$. In particular, we classify all the irreducible modules over $V_{P}$ and determine the fusion rules among them. Our main results indicate that the representation theory of the parabolic-type VOA $V_{P}$ is governed by its Cartan-part $V_{H}$, generalizing the representation theoretical results of the rank-one Borel-type VOAs [Liu25, Theorem 6.13].

4.1. Nil ideal in the Zhu’s algebra $A(V_{P})$

We want to show that the maximum ideal $V^{+}$ of $V_{P}$ (see Theorem 3.9) acts as [math] on any irreducible $V_{P}$-module. So that the irreducible modules of $V_{P}$ are in one-to-one correspondence with the irreducible modules over its Cartan-part $V_{H}$. However, this is not obvious from the definition of $V_{P}$. We invoke the important tool of Zhu’s algebra $\mathsf{A}=A(V)$ [Z96] to prove this claim.

First, we recall the definition and basic properties $A(V)$. Let $(V,Y,\mathbf{1},\omega)$ be a VOA. For homogeneous elements $a,b\in V$, define

[TABLE]

Let $O(V)=\mathrm{span}\{a\circ b:a,b\in V\}$, and let $A(V):=V/O(V)$. By [Z96, Theorem 2.1.1], $O(V)$ is a two-sided ideal with respect to $\ast$, and $A(V)$ is an associative algebra with respect to $\ast$, with the unit element $[\mathbf{1}]$. The following formulas can be found in [Z96, Lemma 2.1.3]:

[TABLE]

where $a,b\in V$ are homogeneous, and $m\geq n\geq 0$. Recall the following fundamental facts about Zhu’s algebra.

Lemma 4.1**.**

[Z96, Theorem 2.2.2]** Let $W$ be a weak $V$-module. Then the subspace

[TABLE]

is a left $A(V)$-module via $o:A(V)\rightarrow\mathrm{End}(\Omega(W)),[a]\mapsto o(a)=a_{\mathrm{wt}a-1}$. In particular, the degree-zero subspace $M(0)$ of an admissible $V$-module $M=\bigoplus_{n=0}^{\infty}M(n)$ a left $A(V)$-module.

Moreover, $M\mapsto M(0)$ is a one-to-one correspondence between the isomorphism class of irreducible admissible $V$-modules and the isomorphism class of irreducible $A(V)$-modules. In particular, $A(V)$ is a finite-dimensional semisimple algebra if $V$ is rational.

Lemma 4.2**.**

[FZ92, Proposition 1.4.2]** Let $V$ be a VOA and $I\lhd V$ be an ideal. Then $A(I)=(I+V)/O(V)$ is a two-sided ideal of $A(V)$.

In particular, if $V_{P}=V_{H}\oplus V^{+}$ is a rank-two parabolic-type VOA, then $A(V^{+})=\mathrm{span}\{[a]:a\in V^{+}\}$ is a two-sided ideal of $A(V_{P})$.

Proposition 4.3**.**

Let $V_{P}$ be a rank-two parabolic-type VOA. Then $A(V^{+})$ is a nil ideal of the Zhu’s algebra $A(V_{P})$. Furthermore, if $V_{P}$ is $C_{1}$-cofinite, see Theorem 3.14 for a sufficient condition, then $A(V^{+})$ is a nilpotent ideal of $A(V_{P})$.

Proof.

By Theorem 3.9, $A(V^{+})=\mathrm{span}\{[M_{\hat{\mathfrak{h}}}(1,\beta)]:\beta\in S\}$, where $S=P\backslash\{0\}$ if $P$ is of type-I, and $S=P^{+}(\gamma)\cap L$ if $P$ is of type-II.

We claim that $[M_{\hat{\mathfrak{h}}}(1,2\beta)]=0$ in $A(V_{P})$ for any $\beta\in S$. Indeed, since $0\notin S$, we may assume $(\beta|\beta)=2N$ for some $N\geq 1$. Then $\mathrm{wt}(e^{\beta})=N$. By (3.8) and (3.9),

[TABLE]

Similarly, we can show that $e^{\beta}_{-m}e^{\beta}=0$ for $m\leq 2N$. Then it follows from (4.4) that

[TABLE]

Moreover, it follows from (4.1) and (4.3) that $h(-n-2)u\equiv-h(-n-1)u\pmod{O(V_{P})}$ and $[h(-1)u]=[u]\ast[h(-1)\mathbf{1}]$ in $A(V_{P})$, for any $u\in V_{P}$, $h\in\mathfrak{h}$, and $n\geq 0$. Then we have

[TABLE]

for any $h^{i}\in\mathfrak{h}$ and $n_{1}\geq\dots\geq n_{r}\geq 0$, since $[e^{2\beta}]=0$. This shows $[M_{\hat{\mathfrak{h}}}(1,2\beta)]=0$ in $A(V_{P})$ for any $\beta\in S$.

Now let $\alpha\in[M_{\hat{\mathfrak{h}}}(1,\beta)]\subset A(V^{+})$. By (3.11) and (4.2) we have

[TABLE]

Hence each element in $A(V^{+})$ is nilpotent, and so $A(V^{+})$ is a nil ideal of $A(V_{P})$.

If $V_{P}$ is $C_{1}$-cofinite, then it follows from [Liu21, Theorem 3.1] that $A(V_{P})$ is a Noetherian algebra. Hence the nil ideal $A(V^{+})$ is nilpotent by Levitzky’s theorem [Le45]. ∎

Corollary 4.4**.**

Let $V_{P}$ be a rank-two parabolic-type VOA, and let $M=\bigoplus_{n=0}^{\infty}M(n)$ be an irreducible admissible $V_{P}$-module. Then the action of the ideal $A(V^{+})$ on $M(0)$ via $o:A(V_{P})\rightarrow\mathrm{End}(M(0))$ is zero.

Proof.

By Lemma 4.1, $M(0)$ is an irreducible module over $A(V_{P})$. Since $A(V^{+})\lhd A(V_{P})$ is a nil ideal by Proposition 4.3, it is contained in the Jacobson radical $J(A(V_{P}))$ which acts as zero on any irreducible $A(V_{P})$-module. ∎

4.2. Classification of irreducible modules over $V_{P}$

By Theorem 3.9, $V_{P}=V_{H}\oplus V^{+}$, with $V_{H}\leq V_{P}$ and $V^{+}\lhd V_{P}$. Let $(M,Y_{M})$ be an irreducible admissible $V_{H}$-module. Define

[TABLE]

It is clear that $(M,\tilde{Y}_{M})$ is an irreducible admissible $V_{P}$-module.

Lemma 4.5**.**

Let $V_{P}$ be a rank-two parabolic-type VOA and $M=\bigoplus_{n=0}^{\infty}M(n)$ be an irreducible admissible $V_{P}$-module. Then the action of the ideal $V^{+}$ on $M$ must be zero.

Proof.

Since $V^{+}$ is an ideal of $V_{P}$, its action on $M$ gives rise to an admissible submodule $N=V^{+}.M$ which has spanning elements

[TABLE]

We need to show that $N=0$.

Indeed, suppose $N\neq 0$. Since $M$ is irreducible, then $N=M$. We can also fix a nonzero element $u\in M(0)$ and write $M=\mathrm{span}\{b_{m}u:b\in V_{P},m\in\mathbb{Z}\}$, see [LL04, Proposition 4.5.7]. Then it follows from (4.6) that $M(0)=N(0)$ has spanning elements

[TABLE]

We use induction on $r\geq 1$ to show that elements of the form (4.7) are [math].

Consider the base case $r=1$. Note that $b_{m}u=0$ if $\deg(b_{m})<0$. Hence we may assume $\deg(a^{1}_{n_{1}})\leq 0$ in $a^{1}_{n_{1}}b_{m}u$. By Corollary 4.4 we have $o(V^{+})M(0)=0$. If $\deg(a^{1}_{n_{1}})=0$, then $\deg(b_{m})=0$ and $a^{1}_{n_{1}}b_{m}u=o(a^{1})o(b)u=0$ since $a^{1}\in V^{+}$ and $o(b)u\in M(0)$. If $\deg(a^{1}_{n_{1}})<0$, then $a^{1}_{n_{1}}u=0$ and

[TABLE]

since $\deg(a^{1}_{n_{1}})+\deg(b_{m})=0$ and $a^{1}_{j}b\in V^{+}$ for all $j\geq 0$. Now assume (4.7) holds for smaller $r$. If $\deg(a^{1}_{n_{1}})=\dots=\deg(a^{r}_{n_{r}})=0$ then $a^{1}_{n_{1}}\dots a^{r}_{n_{r}}b_{m}u=o(a^{1})\dots o(a^{r})o(b)u=0$. Otherwise, there exists $i$ such that $\deg(a^{i}_{n_{i}})<0$. Then

[TABLE]

Since each summand on the right-hand-side has shorter length, $a^{1}_{n_{1}}\dots a^{r}_{n_{r}}b_{m}u=0$ by the induction hypothesis. Thus, $M(0)=N(0)=0$, which is a contradiction. ∎

We also need the following result about representation of tensor product associative algebras.

Lemma 4.6**.**

Let $A$ be a finite-dimensional semisimple algebra over $\mathbb{C}$, with irreducible modules $S^{1},\dots,S^{r}$ up to isomorphism, and let $B=\mathbb{C}[x]\otimes A$ be the tensor product associative algebra. For any $\lambda\in\mathbb{C}$, let $e^{\lambda}$ be a formal symbol. Then $\mathbb{C}e^{\lambda}\otimes S^{i}$ is an irreducible $B$-module with

[TABLE]

Furthermore, any irreducible $B$-module is isomorphic to $\mathbb{C}e^{\lambda}\otimes S^{i}$ for some $\lambda\in\mathbb{C}$ and $1\leq i\leq r$.

Proof.

Clearly, $\mathbb{C}e^{\lambda}\otimes S^{i}$ is an irreducible $B$-module via (4.8). However, it is not straightforward that any irreducible $B$-module has the form $\mathbb{C}e^{\lambda}\otimes S^{i}$, since the subalgebra $\mathbb{C}[x]$ of $B$ is not finite-dimensional.

Since $A=M_{n_{1}}(\mathbb{C})\times\dots\times M_{n_{r}}(\mathbb{C})$ by the Artin-Wedderburn’s theorem, it suffices to show any irreducible module over the tensor product algebra $\mathbb{C}[x]\otimes M_{n}(\mathbb{C})\cong M_{n}(\mathbb{C}[x])$ is isomorphic to $\mathbb{C}e^{\lambda}\otimes S$, where $S=\mathbb{C}^{n}$ is the standard irreducible $M_{n}(\mathbb{C})$-module.

Let $M\neq 0$ be an irreducible $M_{n}(\mathbb{C}[x])$-module. Since $M$ is a module over the semisimple subalgebra $M_{n}(\mathbb{C})$, there exists a nonzero copy of $S=\mathbb{C}^{n}\subset M$. Let $N\subseteq M$ be the $\mathbb{C}[x]$-submodule generated by $S$. i.e., $N=\mathrm{span}\{f(x).u\in M:f(x)\in\mathbb{C}[x],\ u\in S\}$. Clearly, $N\leq M$ is a nonzero $M_{n}(\mathbb{C}[x])$-submodule, and so $N=M$. Moreover, $\mathbb{C}[x]\otimes S\rightarrow M$, $f(x)\otimes u\mapsto f(x).u$ is an epimorphism of $M_{n}(\mathbb{C}[x])$-modules. Then $M=(\mathbb{C}[x]\otimes S)/J$, where $J$ is a maximal proper submodule of $\mathbb{C}[x]\otimes S\cong\mathbb{C}[x]^{n}=\mathbb{C}[x]\oplus\dots\oplus\mathbb{C}[x]$. We claim there exists $\lambda\in\mathbb{C}$ such that

[TABLE]

Indeed, if there exists two elements $w_{1}=[\dots f_{i}(x)\dots]^{t}$ and $w_{2}=[\dots g_{j}(x)\dots]^{t}$ in $J$ such that $\gcd(f_{i}(x),g_{j}(x))=1$, assume $a(x)f_{i}(x)+b(x)g_{j}(x)=1$ for some $a(x),b(x)\in\mathbb{C}[x]$, then $(a(x)E_{1i}).w_{1}+(b(x)E_{1j}).w_{2}=[1,0,\dots,0]^{t}\in J$, where $E_{ij}\in M_{n}(\mathbb{C})$ is the matrix unit. It follows that $\mathbb{C}[x]^{n}=M_{n}(\mathbb{C}[x]).[1,0,\dots,0]^{t}\subseteq J$, which contradicts the properness of $J$. Hence there exists a nonzero polynomial $p(x)$ such that for any $[f_{1}(x),\dots,f_{n}(x)]^{t}\in J$, we have $p(x)|f_{i}(x)$ for all $i$. It follows that $J\subseteq\langle p(x)\rangle^{n}=\langle p(x)\rangle\oplus\dots\oplus\langle p(x)\rangle$. By the maximality of $J$, we have $p(x)=x-\lambda$ for some $\lambda\in\mathbb{C}$, and $J=\langle p(x)\rangle^{n}$. This proves (4.9).

Now it follows from (4.9) that $M\cong\mathbb{C}[x]^{n}/\langle x-\lambda\rangle^{n}\cong(\mathbb{C}[x]/\langle x-\lambda\rangle)^{n}\cong\mathbb{C}e^{\lambda}\otimes\mathbb{C}^{n}$ as a module over $M_{n}(\mathbb{C}[x])\cong\mathbb{C}[x]\otimes M_{n}(\mathbb{C})$. ∎

Theorem 4.7**.**

Let $V_{P}=V_{H}\oplus V^{+}$ be a rank-two parabolic-type VOA and $M=\bigoplus_{n=0}^{\infty}M(n)$ be an irreducible admissible $V_{P}$-module. Then $M$ is an irreducible $V_{H}$-module on which $V^{+}$ acts as zero. In particular, any irreducible admissible $V_{P}$-module is ordinary. Furthermore,

1. (1)

If $V_{P}$ is of type-I, then

[TABLE]

are all the irreducible $V_{P}$-modules up to isomorphism, where $\tilde{Y}_{M}$ is defined by (4.5); 2. (2)

If $V_{P}$ is of type-II, with $P(\gamma)\cap L=\mathbb{Z}\alpha$, $(\alpha|\alpha)=2N$, and $\mathbb{C}\beta=(\mathbb{C}\alpha)^{\perp}$, then

[TABLE]

are all the irreducible $V_{P}$-modules up to isomorphism, where $\tilde{Y}_{M}$ is defined by (4.5).

Proof.

By Lemma 4.5, $M$ is an irreducible admissible $V_{H}$-module on which $V_{+}$ acts as [math]. The non-trivial part is to show that $M$ is ordinary. i.e., $\dim M(n)<\infty$ for all $n\geq 0$.

If $V_{P}$ is of type-I, we have $V_{H}=M_{\hat{\mathfrak{h}}}(1,0)$ by Theorem 3.9. Then $M(0)$ is an irreducible module over $A(M_{\hat{\mathfrak{h}}}(1,0))\cong\mathbb{C}[x,y]$ by Lemma  4.1 and [FZ92, Theorem 3.1.2]. By Hilbert’s Nullstellensatz, $M(0)\cong\mathbb{C}e^{\lambda}$ for some $\lambda\in\mathfrak{h}$. Since the irreducible Heisenberg module $M_{\hat{\mathfrak{h}}}(1,\lambda)$ is a Verma $\hat{\mathfrak{h}}$-module by its construction (3.3), we have $M_{\hat{\mathfrak{h}}}(1,\lambda)\cong M$, and so $M$ is ordinary. This also proves (4.10).

If $V_{P}$ is of type-II, then $V_{H}\cong M_{\widehat{\mathbb{C}\beta}}(1,0)\otimes V_{\mathbb{Z}\alpha}$ by Theorem 3.9. It was proved in [DMZ94, Lemma 2.8] that $A(V_{H})\cong A(M_{\widehat{\mathbb{C}\beta}}(1,0))\otimes A(V_{\mathbb{Z}\alpha})\cong\mathbb{C}[x]\otimes A(V_{\mathbb{Z}\alpha})$ as associative algebras. Then by Lemma 3.2 and Lemma 4.1, $A(V_{\mathbb{Z}\alpha})$ is a semisimple algebra, and $U=M(0)$ is an irreducible module over $A(V_{H})\cong\mathbb{C}[x]\otimes A(V_{\mathbb{Z}\alpha})$. Hence $\dim U<\infty$ and $U\cong\mathbb{C}e^{\mu}\otimes S$ by Lemma 4.6, where $\mu\in\mathbb{C}\beta$ and $S$ is an irreducible $A(V_{\mathbb{Z}\alpha})$-module. Since $(\mathbb{Z}\alpha)^{\circ}/\mathbb{Z}\alpha=\bigsqcup_{i=0}^{2N-1}(\mathbb{Z}\alpha+\frac{i}{2N}\alpha)$, then $S$ is the bottom degree of $V_{\mathbb{Z}\alpha+\frac{i}{2N}\alpha}$ for some $0\leq i\leq 2N-1$ by Lemma 3.2. In particular, $U=M(0)$ is isomorphic to the bottom degree of the irreducible ordinary $V_{H}$-module $W=M_{\widehat{\mathbb{C}\beta}}(1,\mu)\otimes V_{\mathbb{Z}\alpha+\frac{i}{2N}\alpha}$. Note that both $M$ and $W$ are the unique irreducible quotients of the generalized Verma module $\bar{M}(U)$ associated to the irreducible $A(V_{H})$-module $U$ [DLM98, Theorems 6.2, 6.3]. Hence $M\cong W=M_{\widehat{\mathbb{C}\beta}}(1,\mu)\otimes V_{\mathbb{Z}\alpha+\frac{i}{2N}\alpha}$ is an ordinary module. This also proves (4.11). ∎

Remark 4.8**.**

In the argument of the type-II case of Theorem 4.7, we cannot directly apply Frenkel-Huang-Lepowsky’s result for irreducible ordinary modules over tensor product VOA (see Lemma 3.6) to the irreducible admissible $V_{H}$-module $M=\bigoplus_{n=0}^{\infty}M(n)$ without knowing $M$ is ordinary. In fact, there exists VOAs (e.g. affine VOAs of admissible level) whose irreducible admissible modules are not necessarily ordinary. Hence we have to show $M$ is an ordinary $V_{H}$-module first. In fact, (4.11) agrees with Lemma 3.6 once we have the ordinarity of $M$.

5. Strongly unital property of the Cartan-part subVOA $V_{H}$

In this Section, we prove that the Cartan-part subVOA $V_{H}$ of a type-II parabolic-type VOA $V_{P}$ (3.12) satisfies the strongly unital property [DGK25, DGK24]. These are new natural examples of CFT-type $C_{1}$-cofinite irrational simple VOA that satisfies the strongly unital property adding to the Heisenberg VOA example.

5.1. The mode transition algebras $\mathfrak{A}_{d}$ associated to a VOA

We refer to [FZ92, FBZ04, DGK25] for the definition of universal enveloping algebra $\mathscr{U}=\mathscr{U}(V)$ of a VOA $V$.

$\mathscr{U}=\bigoplus_{d\in\mathbb{Z}}\mathscr{U}_{d}$ is a canonically seminorm graded associative algebra. It has left and right neighborhoods at [math]:

[TABLE]

such that $\mathrm{N}^{n+1}_{\mathsf{L}}\mathscr{U}_{0}=\sum_{j\geq n+1}\mathscr{U}_{j}\cdot\mathscr{U}_{-j}=\mathrm{N}^{n+1}_{\mathsf{R}}\mathscr{U}_{0}$ for any $n\geq 0$, see [DGK25, Appendix A]. The Zhu algebra $\mathsf{A}=A(V)$ has the following identification as an associative algebra.

[TABLE]

Denote the category of $\mathsf{A}=A(V)$-modules by $\mathsf{Mod}(\mathsf{A})$.

On the other hand, let $W$ be a left module over the associative algebra $\mathscr{U}$ equipped with discrete topology. If the module action map $\mathscr{U}\times W\rightarrow W$ is continuous, then $W$ must be a filtered module

[TABLE]

We call such a $\mathscr{U}$-module $W$ exhaustive or continuously discrete. In particular, any admissible $V$-module $M=\bigoplus_{n=0}^{\infty}M(n)$ is naturally an exhaustive $\mathscr{U}$-module, with $M(n)\subseteq\Omega_{n}(M)$ for all $n$, see Definition 3.1. Denote the category of exhaustive left $\mathscr{U}$-modules by $\mathsf{ExMod}(\mathscr{U})$. Then $\mathsf{Adm}(V)$ is a full subcategory of $\mathsf{ExMod}(\mathscr{U})$.

It was observed in [DGK25] that $\Omega_{n}(W)$ agrees with the space of degree-$n$ highest-weight vectors in the weak $V$-module $W$ introduced in [DLM98]. i.e.,

[TABLE]

In particular,

[TABLE]

is a functor between abelian categories.

On the other hand, Dong-Li-Mason’s generalized Verma module functor $\bar{M}(\cdot)$ [DLM98, Theorem 6.2] can be identified with the following (left) induced module functor:

[TABLE]

The functors $\Omega$ and $\Phi^{\mathsf{L}}$ in (5.4) and (5.5) form an adjoint pair $\Phi^{\mathsf{L}}\dashv\Omega$ between abelian categories

[TABLE]

see [DGK25, Proposition 3.1.2]. The following Lemma is useful for our later discussion.

Lemma 5.1**.**

Let $V=V_{1}\otimes V_{2}$ be a tensor product VOA. Then the functors $\Phi^{\mathsf{L}}$ and $\Omega$, with $\mathsf{A}=A(V_{1}\otimes V_{2})$ and $\mathscr{U}=\mathscr{U}(V_{1}\otimes V_{2})$, have the following properties

1. (1)

$\Phi^{\mathsf{L}}(U)=\Phi^{\mathsf{L}}_{V_{1}}(\Phi^{\mathsf{L}}_{V_{2}}(U))=\Phi^{\mathsf{L}}_{V_{2}}(\Phi^{\mathsf{L}}_{V_{1}}(U))$* for any $U\in\mathsf{Mod}(\mathsf{A})$.* 2. (2)

$\Omega(W)=\Omega_{V_{1}}(W)\cap\Omega_{V_{2}}(W)$* for any $W\in\mathsf{Adm}(V).$*

Proof.

Both (1) and (2) are straightforward consequences of the fact that $(a_{1}\otimes a_{2})_{[n]}$ is a linear combination of operators of the form $(a_{1}\otimes\mathbf{1})_{[p]}(\mathbf{1}\otimes a_{2})_{[q]}=(\mathbf{1}\otimes a_{2})_{[q]}(a_{1}\otimes\mathbf{1})_{[p]}$, see [FHL93, eq. (4.7.3)]. More precisely, given any weak $V=V_{1}\otimes V_{2}$-module $W$, by (3.13),

[TABLE]

Clearly, (5.7) makes $W=\Phi^{\mathsf{L}}_{V_{1}}(\Phi^{\mathsf{L}}_{V_{2}}(U))$ an admissible $V$-module generated by $U$. Then there exists a canonical epimorphism $\pi:\Phi^{\mathsf{L}}(U)\rightarrow\Phi^{\mathsf{L}}_{V_{1}}(\Phi^{\mathsf{L}}_{V_{2}}(U))$. On the other hand, since $V_{1},V_{2}$ are subVOAs of $V_{1}\otimes V_{2}$ and the actions of $V_{1}$ and $V_{2}$ are commutative on a $V$-module, there exists a $V_{2}$-module homomorphism $\iota:\Phi^{\mathsf{L}}_{V_{2}}(U)\rightarrow\Phi^{\mathsf{L}}(U)$ such that $\imaginary\iota\subseteq\Omega_{V_{1}}(\Phi^{\mathsf{L}}(U))$. Note that $\iota$ also extends to a $V_{1}$-module homomorphism $\tau:\Phi_{V_{1}}^{\mathsf{L}}(\Phi_{V_{2}}^{\mathsf{L}}(U))\rightarrow\Phi^{\mathsf{L}}(U)$ by [DLM98, Theorem 6.2]. By (5.7), $\tau$ is also a $V$-homomorphism and is clearly an inverse of $\pi$. This proves (1).

Clearly, $\Omega(W)\subseteq\Omega_{V_{1}}(W)\cap\Omega_{V_{2}}(W)$ since $V_{1}$ and $V_{2}$ are subVOAs of $V$. On the other hand, assume $\deg((a_{1}\otimes a_{2})_{[n]})=\mathrm{wt}a_{1}+\mathrm{wt}a_{2}-n-1<0$, then for any $w\in\Omega_{V_{1}}(W)\cap\Omega_{V_{2}}(W)$ and any $i\in\mathbb{Z}$, we have

[TABLE]

since either $\deg((a_{1}\otimes\mathbf{1})_{[i]})=\mathrm{wt}a_{1}-i-1<0$ or $\deg((\mathbf{1}\otimes a_{2})_{[n-i-1]})=\mathrm{wt}a_{2}-n+i<0$. Hence $w\in\Omega(W)$ by (5.7). ∎

Using the right neighborhoods in (5.1), one can define a right induced module functor

[TABLE]

Note that the neighborhood $\mathrm{N}^{1}_{\mathsf{L}}\mathscr{U}$ (resp. $\mathrm{N}^{1}_{\mathsf{R}}\mathscr{U}$) contains all the negatively (resp. positively) graded subspaces of $\mathscr{U}$, see (5.1). Hence the quotient algebras in (5.5) and (5.8) have the gradations

[TABLE]

Definition 5.2**.**

[DGK25, DGK24] The mode transition algebra $\mathfrak{A}=\mathfrak{A}(V)$ associated to a VOA $V$ is given by the following vector space

[TABLE]

Write $\mathfrak{A}_{n,-m}=(\mathscr{U}/\mathrm{N}^{1}_{\mathsf{L}}\mathscr{U})_{n}\otimes_{\mathscr{U}_{0}}\mathsf{A}\otimes_{\mathscr{U}_{0}}(\mathscr{U}/\mathrm{N}^{1}_{\mathsf{R}}\mathscr{U})_{-m}$, then $\mathfrak{A}=\bigoplus_{n,m\in\mathbb{N}}\mathfrak{A}_{n,-m}$. The product on $\mathfrak{A}$ is denoted by $\star$, which satisfies

[TABLE]

see [DGK25, Appendix B, Definition 3.2.1] for more details of the product. $\mathfrak{A}_{d}=\mathfrak{A}_{d,-d}$ is an associative algebra under $\star$, called the $d$-th mode transition algebra of $V$.

An element $1_{d}\in\mathfrak{A}_{d}$ is called a strong unit if it satisfies

[TABLE]

where $\mathfrak{A}_{d}\cong\mathfrak{A}_{d,0}\otimes_{\mathsf{A}}\mathfrak{A}_{0,-d}$ and

[TABLE]

are right and left $\mathsf{A}$-modules, respectively. $\mathfrak{A}_{d}$ is said to be strongly unital if it admits a strong identity element. We say that the VOA $V$ satisfies the strongly unital condition for mode transition algebras, or simply $V$ is strongly unital, if its mode transition algebras $\mathfrak{A}_{d}$ are strongly unital for all $d\in\mathbb{N}$.

Lemma 5.3**.**

[DGK24, Theorem 4.0.10]**[DGK25, Theorem 6.1.1]** Any rational VOA satisfies the strongly unital condition for mode transition algebras. Moreover, any Heisenberg VOA $V=M_{\hat{\mathfrak{h}}}(k,0)$ of rank $r\geq 1$ satisfies the strongly unital condition for mode transition algebras.

5.2. Strongly unital property of $V_{H}$

The following theorem that characterizes the strongly unital condition was proved in an ongoing work [GL26].

Lemma 5.4**.**

Let $V$ be a $C_{1}$-cofinite VOA such that $\mathfrak{A}_{d,0}$ is a projective right $\mathsf{A}$-module for all $d\geq 0$. The following conditions are equivalent:

1. (1)

The VOA $V$ satisfies the strongly unital condition for mode transition algebras. 2. (2)

The adjoint pair $\Phi^{\mathsf{L}}\dashv\Omega$ in (5.6) is an adjoint equivalence.

*In particular, any admissible $V$-module $W$ is a generalized Verma module $\Phi^{\mathsf{L}}(\Omega(W))$. *

As a consequence, it is clear that the rank-two parabolic-type VOA $V_{P}$ does not satisfy the strongly unital property since it is not a simple VOA and so it is not a generalized Verma module.

Theorem 5.5**.**

Let $V_{P}=V_{H}\oplus V^{+}$ be a rank-two parabolic-type VOA. Then its Cartan-part subVOA $V_{H}$ satisfies the strongly unital condition for mode transition algebras.

Proof.

By Proposition 3.11, $V_{H}$ is $C_{1}$-cofinite. If $V_{P}$ is of type-I, then $V_{H}\cong M_{\hat{\mathfrak{h}}}(1,0)$ by Theorem 3.9, and the conclusion follows from Lemma 5.3.

Assume $V_{P}$ is of type-II, with $V_{H}\cong M_{\widehat{\mathbb{C}\beta}}(1,0)\otimes V_{\mathbb{Z}\alpha}=V_{1}\otimes V_{2}$ as VOAs, see Theorem 3.9. Both $V_{1}=M_{\hat{\mathfrak{h}}}(1,0)$ and $V_{2}=V_{\mathbb{Z}\alpha}$ are strongly unital by Lemma 5.3 and Lemma 3.2. Moreover, it follows from the definition of universal enveloping algebra and tensor product vertex operators (3.13) that

[TABLE]

for any left $A(V_{H})$-module $W$. Then by (5.10),

[TABLE]

as a right $A(V_{H})\cong A(V_{1})\otimes_{\mathbb{C}}A(V_{2})$-module. Since $\mathfrak{A}_{i,0}(V_{1})$ (resp. $\mathfrak{A}_{j,0}(V_{2})$) is a projective right $A(V_{1})$ (resp. $A(V_{2})$) module, respectively, it is clear that $\mathfrak{A}_{d,0}(V_{H})$ is a projective right $A(V_{H})$-module. Now let $U\in\mathsf{Mod}(A(V_{H}))$. By Lemma 5.1,

[TABLE]

since the unit maps for $V_{1}$ and $V_{2}$ are both bijective. Suppose there exists some element

[TABLE]

where $n_{1}\geq\dots\geq n_{k}\geq 1$, $a^{1},\dots,a^{r}\in V_{\mathbb{Z}\alpha}$, and $u^{i},v^{j}\in U$. Choose the index $i$ such that $n_{1}$ is largest among all the summands $\beta(-n_{1})\dots\beta(-n_{k})u^{i}$. Then there exists a nonzero scalar $c\in\mathbb{C}$ such that

[TABLE]

in view of (3.2). Then $u^{i}=0$, which is a contradiction. Thus, $\Omega\Phi^{\mathsf{L}}(U)=\Phi^{\mathsf{L}}_{V_{2}}(U)\cap\Phi^{\mathsf{L}}_{V_{1}}(U)=U$. On the other hand, given $W\in\mathsf{Ext}(\mathscr{U}(V_{H}))$, since the action of $V_{1}$ and $V_{2}$ are commutative on $W$, $\Omega_{V_{2}}(W)$ is a $V_{1}$-submodule in $W$, and $\Omega(W)=\Omega_{V_{1}}(\Omega_{V_{2}}(W))$. Since $V_{2}$ is strongly unital, $\Phi^{\mathsf{L}}_{V_{2}}\Omega_{V_{2}}(W)=W$ by Lemma 5.4. Then $\Omega_{V_{1}}(\Phi^{\mathsf{L}}_{V_{2}}\Omega_{V_{2}}(W))=\Omega_{V_{1}}(W)$ and

[TABLE]

in view of Lemma 5.1. Hence $V_{H}=V_{1}\otimes V_{2}$ satisfies the strongly unital condition for mode transition algebras by Lemma 5.4. ∎