Smash product construction of modular lattice vertex algebras

1.   Introduction

It is well known that there are three important classes of rational vertex operator algebras over the field of complex numbers; namely, affine vertex operator algebras, Virasoro vertex operator algebras and lattice vertex operator algebras (see ^[1]). It is natural to study the corresponding vertex algebra structures over fields of prime characteristic first.

There have already been some works on modular vertex algebras and their representations. For example, modular $A(V)$ theory and $A_n(V)$ theory were studied in ^[2,3], modular Virasoro vertex operator algebras were studied in ^[4], and framed vertex operator algebras were studied in ^[5]. Modular vertex algebras obtained from integral forms in some vertex operator algebras over the field of complex numbers were used to study modular moonshine in ^[6,7,8].

Dong and Griess introduced an integral form of the vertex algebras associated with positive definite even lattices over a field of characteristic zero in ^[9], and the related modular vertex algebras were studied in ^[10].

In a series of papers (see ^[11,12,13]), Li studied nonlocal vertex algebras over a field of characteristic zero. In particular, in ^[13] Li introduced a smash product construction of nonlocal vertex algebras and used smash product to give a different construction of lattice vertex algebras and their modules (cf. ^[1,14]). Motivated by ^[13], in this paper we study nonlocal vertex algebras and the smash product construction over fields of prime characteristic. As an application, modular vertex algebras associated with positive definite even lattices are reconstructed by using smash products. This gives another construction of the modular vertex algebras obtained from integral forms introduced by Dong and Griess of lattice vertex operator algebras over a field of characteristic zero.

This paper is organized as follows: In Section 2, we present some basic results on modular nonlocal vertex algebras and give the smash product construction of nonlocal vertex algebras. In Section 3, we use the smash product to construct the vertex algebras associated with positive definite even lattices.

2.   Nonlocal vertex algebras and smash products

In this section, we first present some basic results on modular nonlocal vertex algebras. Then we give the smash product construction of modular nonlocal vertex algebras. The proofs for most of the results in this section are the same as those for characteristic zero (see ^[11,12,13]).

Let $\mathbb{F}$ be an algebraically closed field of an odd prime characteristic $p$, which is fixed throughout this paper. All vector spaces, including algebras, are considered to be over $\mathbb{F}$. We use $\mathbb{Z}$ for the integers, $\mathbb{Z}_+$ for the positive integers, and $\mathbb{N}$ for the nonnegative integers.

Note that for any $m\in \mathbb{Z}, \ k\in \mathbb{N}$,

Then we shall also view $\binom{m}{k}$ as an element of $\mathbb{F}$. Furthermore, for $m\in \mathbb{Z}$ we have

The following definition is the same as in characteristic zero (see ^[11,12,13]).

Definition 2.1. A nonlocal vertex algebra is a vector space $V$ endowed with a distinguished vector 1, called the vacuum vector, and endowed with a linear map

such that for $u, v\in V$,

and for $u, v, w\in V$, there exists a nonnegative integer $l$ such that

As in ^[15], define $\mathcal{B}$ to be the bialgebra with a basis $\{ \mathcal{D}^{(r)}\mid r\in \mathbb{N}\}$, where

for $m, n\in \mathbb{N}$. Set

The bialgebra structure of $\mathcal{B}$ can be described in terms of the generating functions as

in particular, we have $e^{x \mathcal{D}}e^{-x \mathcal{D}} = 1$.

Remark 2.2. Let $U$ be any vector space. Then $U[[x, x^{-1}]]$ is a $\mathcal{B}$-module with $\mathcal{D}^{(n)}$ for $n\in \mathbb{N}$ acting as the $n$-th Hasse differential operator $\partial_x^{(n)}$ with respect to $x$, which is defined by

Set $e^{z\partial_{x}} = \sum_{n\in \mathbb{N}}z^{n}\partial_x^{(n)}$. Then

As in the case of characteristic zero, we have (cf. ^[13]):

Lemma 2.3. Let $V$ be a nonlocal vertex algebra.Then $V$ is naturally a $\mathcal{B}$-module with

for $u\in V$ and $n\in \mathbb{N}$.Furthermore,

Proof. Let $u\in V$. By (2.4) and (2.10), we have

and in particular, $\mathcal{D}^{(0)} = 1$ on $V$.

Let $l\in \mathbb{N}$ be such that

As $Y(\mathbf{1}, x) = 1$ and $Y(u, x)\mathbf{1}$ only involves nonnegative powers of $x$, we have

Multiplying both sides by $(x_2+x_0)^{-l}$ we obtain

Using (2.12) and the equation above, we have

Therefore $V$ is a $\mathcal{B}$-module.

Let $u, v\in V$. Then there is $l\in \mathbb{N}$ such that

We may assume that $x^l Y(u, x)v\in V[[x]]$ by replacing $l$ with a bigger integer if necessary, so that

Then

Multiplying both sides by $(x_2+x_0)^{-l}$ we have

Let $u, v\in V$ and let $l\in \mathbb{N}$ be such that

Since $Y(Y(u, x_0)v, x_2)\mathbf{1}$ involves only nonnegative powers of $x_2$, we can multiply both sides by $(x_0+x_2)^{-l}$ to get

Then

that is,

on $V$. Applying $e^{-x_2 \mathcal{D}}$ from left, as $e^{x_2 \mathcal{D}}e^{-x_2 \mathcal{D}} = 1$, we have

Thus (2.11) holds.

The following two results of ^[11] are valid here with the same proof.

Lemma 2.4. For any subset $S$ of a nonlocal vertex algebra $V$, the subalgebra $\langle S\rangle$ generated by $S$ is linearly spanned by vectors

for $r\in \mathbb{N}$, $v^{(i)}\in S$, $n_i\in \mathbb{Z}$.

Let $V$ be a nonlocal vertex algebra. For $u, v\in V$, we say $u, v$ are mutually local if there exists $k\in \mathbb{N}$ such that

A subset $S$ of $V$ is said to be local if every pair of elements of $S$ are mutually local.

Lemma 2.5. Let $V$ be a nonlocal vertex algebra and let $S$ be a local subset of $V$.Then the subalgebra $\langle S\rangle$ of $V$ generated by $S$ is a vertex algebra.

Definition 2.6. Let $V$ be a nonlocal vertex algebra. A $V$-module is a vector space $W$ endowed with a linear map

such that for $v \in V$ and $w \in W$,

and for $u, v\in V$ and $w\in W$, there exists a nonnegative integer $l$ such that

A unital associative algebra $A$ is called a $\mathcal{B}$-module algebra if $A$ is a $\mathcal{B}$-module such that

for $h\in \mathcal{B}$ and $a, b\in A$, where $\Delta(h) = \sum h_{(1)}\otimes h_{(2)}$ is in the Sweedler notation.

Example 2.7. Let $A$ be a $\mathcal{B}$-module algebra. Then $A$ has a nonlocal vertex algebra structure with $1$ as the vacuum vector and

Furthermore, on any module $W$ for $A$ as an associative algebra, there exists a module structure $Y_W$ for $A$ with $Y_W(a, x)w = (e^{x \mathcal{D}}a)w$ for $a\in A$, $w\in W$.

Remark 2.8. Let $A$ and $B$ be $\mathcal{B}$-module algebras. Then $A\otimes B$ is a $\mathcal{B}$-module algebra with $e^{x \mathcal{D}} = e^{x \mathcal{D}_A}\otimes e^{x \mathcal{D}_B}$.

Let $V$ and $U$ be nonlocal vertex algebras. A linear map $f$ from $V$ to $U$ is a homomorphism of nonlocal vertex algebras if

It is straightforward to show that $fe^{x \mathcal{D}_V} = e^{x \mathcal{D}_U}f$ if $f$ is a homomorphism from $V$ to $U$. A homomorphism of nonlocal vertex algebras from $V$ to $V$ is called an endomorphism of $V$.

Remark 2.9. Let $A$ and $B$ be $\mathcal{B}$-module algebras. Then a linear map $f$ from $A$ to $B$ is a homomorphism of nonlocal vertex algebras from $A$ to $B$ if and only if $f$ is a homomorphism of algebras and of $\mathcal{B}$-modules, that is, $fe^{x \mathcal{D}_A} = e^{x \mathcal{D}_B}f$.

As in the case of characteristic zero, we have:

Lemma 2.10. Let $V$ and $U$ be nonlocal vertex algebras and let $f$ be a linear map from $V$ to $U$.If

where $T$ is a generating subset of $V$, then $f$ is a homomorphism of nonlocal vertex algebras.

Proof. Set

We must show that $V = S$. Since $T\subset S$ and $T$ generates $V$, it suffices to show that $S$ is a nonlocal vertex subalgebra of $V$. As $\mathbf{1}\in S$, it remains to show that for $u, v\in S$ and $n\in \mathbb{Z}$ we have $u_n v\in S$, that is,

for $w\in V$. Let $l\in \mathbb{N}$ be such that

Since $u, v\in S$, we have

Multiplying both sides by $(x_2+x_0)^{-l}$, we obtain (2.19).

Let $V$ be a nonlocal vertex algebra and let $A$ be a $\mathcal{B}$-module algebra. Following ^[13], we consider $V$ as a subalgebra of $V\otimes A$ by the natural embedding, and consider $V\otimes A$ as an $A$-module with $A$ acting on the second factor. Then

as $A$-modules and as spaces. We also consider any linear map from $V$ to $V\otimes A$ or from $V$ to $V$ as an $A$-linear endomorphism of $V\otimes A$. Then we consider the vertex operator map $Y$ of $V$ as an $A$-linear map from $V\otimes A$ to $(\text{End}(V\otimes A))[[x, x^{-1}]]$, that is,

We denote by $Y_{\text{ten}}$ the vertex operator map of $V\otimes A$.

Lemma 2.11. Let $V$ be a nonlocal vertex algebra, let $A$ be a $\mathcal{B}$-module algebra, and let $f$ be a linear map from $V$ to $V\otimes A$.Then $f$ is a homomorphism of nonlocal vertex algebras if and only if

Proof. For any $u\in V$ and $a\in A$, we see that

It follows that

Then for $v\in V$, we see $fY(v, x) = Y_{\text{ten}}(f(v), x)f$ is equivalent to

Thus our assertion follows.

We denote by $\mathbb{F}((x))^-$ the $\mathcal{B}$-module algebra $\mathbb{F}((x))$ with $\mathcal{D}^{(n)}$ acting as $(-1)^n\partial_x^{(n)}$ for $n\in \mathbb{N}$. Now we consider the special case with $A = \mathbb{F}((x))^-$.

Definition 2.12. Let $V$ be a nonlocal vertex algebra. We define $\text{PEnd}(V)$ to be the subspace of $\text{Hom}_ \mathbb{F}(V, V\otimes \mathbb{F}((x))^-)$ consisting of the elements $f(x)$ such that

Lemma 2.13. Let $V$ be a nonlocal vertex algebra and let $f(x)\in\text{Hom}_ \mathbb{F}(V, V\otimes \mathbb{F}((x))^-)$.Then $f(x)\in \text{PEnd}(V)$ if and only if $f(x)$ is a homomorphism of nonlocal vertex algebrasfrom $V$ to $V\otimes \mathbb{F}((x))^-$.Furthermore,

Proof. Since $e^{-x_2\partial_x}g(x) = g(x-x_2)$ for $g(x)\in \mathbb{F}((x))^-$, we have

Thus the first assertion follows from Lemma 2.11.

For $f(x)\in \text{PEnd}(V)$, since $f(x)$ is a homomorphism of nonlocal vertex algebras, we have

Then (2.20) holds.

Let $V$ be a nonlocal vertex algebra. We say a subset $U$ of $\text{Hom}(V, V\otimes \mathbb{F}((x))^-)$ is $\Delta$-closed if for every $a(x)\in U$, there exist elements $a_{(1i)}(x), a_{(2i)}(x)\in U$, $i = 1, 2, \ldots, r$, such that

We denote by $B(V)$ the sum of all the $\Delta$-closed subspaces $U$ of $\text{Hom}(V, V\otimes \mathbb{F}((x))^-)$ such that

Using the same arguments in [13, Proposition 3.4] we have:

Lemma 2.14. Let $V$ be a nonlocal vertex algebra.Then $B(V)$ is $\Delta$-closed and $B(V)$ is a $\mathcal{B}$-module algebra with $\mathcal{D}^{(n)}$ acting as $\partial_x^{(n)}$ for $n\in \mathbb{N}$.Furthermore, $V$ is a module for $B(V)$ as a nonlocal vertex algebra with $Y_V(a(x), x_0) = a(x_0)$ for $a(x)\in B(V)$.

The following notion is the modular counterpart of the notion of differential bialgebra in ^[13].

Definition 2.15. A $\mathcal{B}$-module bialgebra is a bialgebra $(B, \Delta, \varepsilon)$ endowed with a $\mathcal{B}$-module structure such that $\varepsilon e^{x \mathcal{D}} = \varepsilon$ and $\Delta e^{x \mathcal{D}} = (e^{x \mathcal{D}}\otimes e^{x \mathcal{D}})\Delta$.

We shall need the following notion (see ^[13]).

Definition 2.16. A nonlocal vertex algebra $V$ endowed with a coalgebra structure $(V, \Delta, \varepsilon)$ is called a vertex bialgebra if $\Delta$ and $\varepsilon$ are homomorphisms of nonlocal vertex algebras.

Let $(B, \Delta, \varepsilon)$ be a $\mathcal{B}$-module bialgebra. Then $B$ is a $\mathcal{B}$-module algebra, and we have a nonlocal vertex algebra $B$ by Example 2.7. From definition we have $\Delta(1) = 1\otimes 1$ and $\varepsilon(1) = 1$. Furthermore,

for $a, b\in B$. Then $\Delta$ and $\varepsilon$ are nonlocal vertex algebra homomorphisms. Thus $B$ is a vertex bialgebra.

Definition 2.17. Let $H$ be a vertex bialgebra. A nonlocal vertex $H$-module-algebra is a nonlocal vertex algebra $V$ endowed with an $H$-module structure on $V$ such that

for $h\in H$, $u, v\in V$.

The following results of ^[13] hold with the same arguments.

Lemma 2.18. Let $H$ be a vertex bialgebra, let $T$ be a generating subset of $H$ as a nonlocal vertex algebra, let $V$ be a nonlocal vertex algebra, and let $(V, Y_V^H)$ be an $H$-module.Suppose that

for $h\in T$ and $u, v\in V$.Then $V$ is a nonlocal vertex $H$-module-algebra.

Theorem 2.19. Let $H$ be a vertex bialgebra, and let $V$ be a nonlocal vertex $H$-module-algebra.We define $V\sharp H = V\otimes H$ as a vector space and define

for $u, v\in V$ and $h, k\in H$.Then $(V\sharp H, Y^\sharp)$ is a nonlocal vertex algebra.Furthermore,

for $h\in H$ and $v\in V$.

3.   Lattice vertex algebras

In ^[9], Dong and Griess introduced an integral form of vertex operator algebras associated to even lattices over the complex field, from which one can define modular lattice vertex algebras. In this section, we construct the modular lattice vertex algebras through smash product.

Let $L$ be a positive definite even lattice with a basis $\{\gamma_1, \ldots, \gamma_d\}$ and let $L^{\circ}$ be the dual lattice of $L$. Let $A_L$ denote the $d\times d$ matrix $(\langle \gamma_i, \gamma_j\rangle)_{1\le i, j\le d}$. Note that $\det(A_L)$ is independent of the choice of a basis for $L$. Let $\epsilon:L\times L\to \mathbb{F}^\times$ be a map such that

for $\alpha, \beta, \gamma\in L$. Denote by $\mathbb{F}_\epsilon[L]$ the $\epsilon$-twisted group algebra of $L$ with $\mathbb{F}$-basis $\{e_\alpha\mid \alpha\in L\}$ and multiplication

Next, recall from ^[9] the ring $M(1)_{ \mathbb{Z}}$. Denote by $M(1)$ the polynomial algebra generated by $s_{\alpha, n}$ for $\alpha\in \{\gamma_1, \ldots, \gamma_d\}$ and $n\in \mathbb{Z}_+$. Set $s_{\alpha, 0} = 1$ for $\alpha\in \{\gamma_1, \ldots, \gamma_d\}$. For $\alpha\in \{\gamma_1, \ldots, \gamma_d\}$, we set

Note that $E^-(-\alpha, x)$ is an invertible element of $M(1)[[x]]$ as $s_{\alpha, 0} = 1$. For a general element $\alpha = k_1\gamma_1+k_2\gamma_2+\cdots+k_d\gamma_d\in L$, where $k_1, \ldots, k_d\in \mathbb{Z}$, we define

Then for $\alpha, \beta\in L$,

As $M(1)$ is isomorphic to the universal enveloping algebra of the abelian Lie algebra with basis $\{s_{\alpha, n}\mid \alpha\in \{\gamma_1, \ldots, \gamma_d\}, n\in \mathbb{Z}_+\}$, we see $M(1)$ is naturally a bialgebra with

for $\alpha\in L$.

Define a $\mathcal{B}$-action on $M(1)$ by $e^{z \mathcal{D}}\mathbf{1} = \mathbf{1}$ and

for $r\in \mathbb{Z}_+$, $\alpha_i\in \{\gamma_1, \ldots, \gamma_d\}$. Then (3.3) holds for $r\in \mathbb{Z}_+$, $\alpha_i\in L$. It is straightforward to check

Then $M(1)$ is a $\mathcal{B}$-module algebra. Furthermore, for $r\in \mathbb{Z}_+$, $\alpha_i\in L$, we have

and

Therefore, $(M(1), \Delta, \varepsilon)$ is a $\mathcal{B}$-module bialgebra.

For $\alpha\in\{\gamma_1, \gamma_2, \ldots, \gamma_d\}$, we inductively define linear operators $r_{\alpha, n}$ for $n\in \mathbb{N}$ on $M(1)$ by

for $\beta_i\in\{\gamma_1, \gamma_2, \ldots, \gamma_d\}$, $m_i\in \mathbb{N}$, where $r_{\alpha, m}$ is understood to be zero if $m < 0$. From (3.5), we see

for $\beta\in \{\gamma_1, \gamma_2, \ldots, \gamma_d\}$ and $m\in \mathbb{N}$. For $\alpha\in\{\gamma_1, \gamma_2, \ldots, \gamma_d\}$, we set

Furthermore, for $\alpha = k_1\gamma_1+k_2\gamma_2+\cdots+k_d\gamma_d\in L$, where $k_1, \ldots, k_d\in \mathbb{Z}$, we define

Then for $\alpha, \beta\in L$,

and furthermore,

Lemma 3.1. In the $\mathcal{B}$-module algebra $M(1)$, we have

for $\alpha\in L$ and $u\in M(1)$.

Proof. We use induction on $m$ to show (3.8) holds for $u = s_{\alpha_1, n_1}s_{\alpha_2, n_2}\ldots s_{\alpha_m, n_m}$ with $m\in \mathbb{N}$, $\alpha_i\in L$, $n_i\in \mathbb{N}$. For $u = \mathbf{1}$, we have

The induction step is given by

This completes the induction. As $M(1)$ is spanned by elements of the form

our assertion follows.

Set

an associative algebra.

Lemma 3.2. For $\alpha\in L$ and $u\in M(1)$, define

Then $B_{L, \epsilon}$ is a $\mathcal{B}$-module algebra.

Proof. For $e_\alpha\otimes u, e_\beta\otimes v\in B_{L, \epsilon}$, we have

Thus $B_{L, \epsilon}$ is a $\mathcal{B}$-module algebra.

Set

a unital commutatively associative algebra. As in the case of characteristic zero (see ^[13]), we have the following universal property of $B_L$:

Lemma 3.3. Let $A$ be a $\mathcal{B}$-module algebra and let $f: \mathbb{F}[L]\to A$ be a homomorphism of algebras.Then $f$ can be extended uniquely to a homomorphism of $\mathcal{B}$-module algebras from $B_L$ to $A$.

Proof. For $\alpha\in \{\gamma_1, \gamma_2, \ldots, \gamma_d\}$, define

Since $M(1)$ is freely generated by $s_{\alpha, n}$ for $\alpha\in \{\gamma_1, \gamma_2, \ldots, \gamma_d\}$ and $n\in \mathbb{Z}_+$, it follows that $f$ can be extended to a homomorphism of algebras.

Now, we show that in fact (3.11) holds for all $\alpha\in L$. Let $P$ be the subset of $L$ consisting of $\alpha$ such that (3.11) holds for all $n\in \mathbb{N}$. From definition, we have $\gamma_i\in P$ for $1\le i\le d$. Assume $\alpha, \beta\in P$. Then we get

proving $\alpha+\beta \in P$.

Now, assume $\alpha\in P$. As $E^-(\alpha, x)E^-(-\alpha, x) = 1$ and $(e^{x \mathcal{D}}fe^{\alpha})(e^{x \mathcal{D}}fe^{-\alpha}) = e^{x \mathcal{D}}1 = 1$, we have

proving $-\alpha\in P$. Thus $P = L$, that is, (3.11) holds for all $\alpha\in L$.

Next we show that $f$ is a homomorphism of $\mathcal{B}$-modules. For $\alpha\in L$, we have

Using the equation above and (3.3), we have

Since $B_L$ as an algebra is generated by $s_{\alpha, n}$ and $e^\alpha$ for $\alpha\in L$ and $n\in \mathbb{N}$, it follows that $f e^{x \mathcal{D}} = e^{x \mathcal{D}}f$. Thus $f$ is a $\mathcal{B}$-module homomorphism.

For $\alpha\in L$, we define $x^{\alpha}\in (\text{End} V_L)[x, x^{-1}]$ by

for $\beta\in L^{\circ}$ and $u\in M(1)$.

Lemma 3.4. For $\alpha\in L$, we have

Proof. For $\beta\in L$, we have

Since $L$ generates $V_{L, \epsilon}$ as a nonlocal vertex algebra, it follows from Lemma 2.10 that $E^+(-\alpha, x)x^{\alpha}$ is a homomorphism of nonlocal vertex algebras. By Lemma 2.13, we see that $E^+(-\alpha, x)x^{\alpha}\in\text{PEnd}(B_{L, \epsilon})$.

Lemma 3.5. There exists a unique $B_L$-module structure $Y_M$ on $B_{L, \epsilon}$ such that

and $(B_{L, \epsilon}, Y_M)$ is a nonlocal vertex $B_L$-module-algebra.

Proof. Denote $\Phi_{\alpha}(x) = E^+(-\alpha, x)x^{\alpha}$ for $\alpha\in L$. By Lemma 3.4, we have $\Phi_{\alpha}(x) \in\text{PEnd}(B_{L, \epsilon})$. Clearly $\Phi_0(x) = 1$ and

Let $A$ be the subalgebra of $B(B_{L, \epsilon})$ generated by $\partial_x^{(n)}\Phi_{\alpha}(x)$ for $n\in \mathbb{N}$, $\alpha\in L$. Clearly $A$ is a commutative $\mathcal{B}$-module algebra. By Lemma 3.3, there exists a homomorphism $f$ of $\mathcal{B}$-module algebras from $B_L$ to $A$ such that $f(e^{\alpha}) = \Phi_{\alpha}(x)$ for all $\alpha\in L$. Then by Lemma 2.18 we see that $B_{L, \epsilon}$ is a nonlocal vertex $B_L$-module-algebra.

As $B_{L, \epsilon}$ is a nonlocal vertex $B_L$-module-algebra by Lemma 3.5, we have the nonlocal vertex algebra $B_{L, \epsilon}\sharp B_L$ by Theorem 2.19.

Theorem 3.6. Denote

a subspace of $B_{L, \epsilon}\sharp B_L$.Then $U$ is a vertex subalgebra of the nonlocal vertex algebra $B_{L, \epsilon}\sharp B_L$, and the linear map

for $\alpha\in L$ and $u\in M(1)$ is a vertex algebra homomorphism.Furthermore, if $\det(A_L)\not\equiv 0\pmod{p}$, the map $\pi$ is an isomorphism.

Proof. This is a slight modification of the proof in ^[13]. As $\Delta$ is a homomorphism from $M(1)$ to $M(1)\otimes M(1)$, we see that $\pi$ is a linear homomorphism. We then show that $\pi$ is a vertex algebra homomorphism. Let $\alpha, \beta \in L$ and $u\in M(1)$. Then we have

By Lemma 3.1, we have

Since $\Delta(e^{\alpha}) = e^{\alpha}\otimes e^{\alpha}$, by Lemma 3.2 we have

Then

Therefore

for $\alpha, \beta\in L$ and $u\in M(1)$. Since $L$ generates $V_L$ as a vertex algebra by [10, Theorem 1], it follows from Lemma 2.10 that $\pi$ is a nonlocal vertex algebra homomorphism. As $V_L$ is a vertex algebra, we see that $\pi$ is a vertex algebra homomorphism. If $\det(A_L)\not\equiv 0\pmod{p}$, it follows from [10, Theorem 13] that $V_L$ is a simple vertex algebra, then $\pi$ is an isomorphism.

Extend $\epsilon$ to a map from $L\times L^{\circ}$ to $\mathbb{F}^\times$ such that

for $\alpha, \beta\in L$ and $\gamma\in L^{\circ}$ (see ^[14]). Define an $\mathbb{F}_\epsilon[L]$-module structure on $\mathbb{F}[L^{\circ}]$ by

Set

By the same proof of [13, Proposition 5.8], we have:

Proposition 3.7. There exists a unique $V_L$-module structure on $V_{L^{\circ}}$ such that

for $\alpha\in L$.

Acknowledgments

The author was supported by the China NSF (grant 11571391) and the Heilongjiang Provincial NSF (grant JQ2020A002).

Conflict of interest

The authors declare there is no conflicts of interest.