1. Introduction
This paper is concerned with finiteness property of (twisted) Zhu’s algebra and its bimodules. The Zhu’s algebra and its twisted higher-level versions carry representation theoretical information about modules and twisted modules over vertex operator algebras (VOAs). Bimodules of Zhu’s algebra have been used to compute (twisted) fusion rules. It has been observed in examples that Zhu’s algebras are often Noetherian and even finite-dimensional. This is unexpected given the analogy between Zhu’s algebra and universal enveloping algebra of a Lie algebra since it is an open problem whether the latter is Noetherian when the Lie algebra is not finite-dimensional [Reference Goodearl and Warfield21]. By leveraging the relationships between the twisted Zhu’s algebra and Zhu’s C 2-algebra, we show the Notherianity for a large class.
In the study of the modular invariance property of VOAs, Zhu introduced an associative algebra A(V) attached to a VOA of conformal field theory CFT-type [Reference Zhu41]. Associated with an admissible V-module M, an A(V)-bimodule A(M) was introduced by Frenkel and Zhu in order to compute the fusion rules among irreducible modules over affine VOAs [Reference Frenkel and Zhu18]. The main result in this paper is that A(V), together with its g-twisted analog 
$A_g(V)$ [Reference Dong, Li and Mason12] and bimodule 
$A_g(M)$ [Reference Gao, Liu and Zhu20, Reference Qifen Jiang and Jiao37], is left (or right) Noetherian if V is C 1-cofinite [Reference Karel and Li27, Reference Li30] and M is (weakly) C g1-cofinite. If, in addition, V is C 2-cofinite [Reference Zhu41], then the g-twisted higher order generalizations 
$A_{g,n}(V)$ [Reference Dong, Li and Mason10, Reference Dong, Li and Mason11] and 
$A_{g,n}(M)$ [Reference Qifen Jiang and Jiao37] are finite-dimensional for all 
$n\geq 0$. These algebraic structures encode important information about the representation theory of the VOAs including the fusion rules. Noetherianity is one of the most important finiteness properties, which gives tools for their study, for instance from (non-commutative) algebraic geometry [Reference Damiolini, Gibney and Krashen6].
Zhu proved in [Reference Zhu41] that there is a one-to-one correspondence between irreducible V-modules and irreducible A(V)-modules, which leads to an equivalency between the categories of V-modules and A(V)-modules for rational VOAs. Zhu’s result was generalized by Dong, Li and Mason to the g-twisted case in [Reference Dong, Li and Mason12], and higher order (twisted) cases in [Reference Dong, Li and Mason10, Reference Dong, Li and Mason11], wherein the notions of g-twisted Zhu’s algebra 
$A_g(V)$, higher order Zhu’s algebra 
$A_n(V)$ for 
$n\geq 0$ and g-twisted higher order Zhu’s algebra 
$A_{g,n}(V)$ were introduced, and the one-to-one correspondences between irreducible (g-twisted) V-modules and irreducible modules over these generalized Zhu’s algebra were established. From this point of view, Zhu’s algebra and its generalizations tell us about the representation theory of VOAs.
Dong, Li and Mason proved that A(V) is finite-dimensional if V is C 2-cofinite [Reference Dong, Li and Mason12, Reference Dong, Li and Mason13]. For the classical non-C 2-cofinite VOAs like the vacuum module VOA 
$V_{\hat{\mathfrak{g}}}(\ell,0)$, the Heisenberg VOA 
$M_{\hat{\mathfrak{h}}}(\ell,0)$ and the universal Virasoro VOA 
$\bar{V}(c,0)$ [Reference Frenkel and Zhu18, Reference Lepowsky and Li28], their Zhu’s algebra is isomorphic to 
$U(\mathfrak{g})$, 
$\mathbb{C}[x_1,\dots, x_n]$ and 
$\mathbb{C}[x]$, respectively. Although these associative algebras are infinite-dimensional, they are all Noetherian. Moreover, in numerous calculations for concrete examples [Reference Addabbo and Barron2, Reference algebra3, Reference Dong, Li and Mason9, Reference Dong and Nagatomo15, Reference Frenkel and Zhu18, Reference Wang39], we see that Zhu’s algebra is close to a quotient algebra of certain universal enveloping algebra of a Lie algebra 
$\mathfrak{g}$. In fact, it was proved by He in [Reference He22] that the higher order Zhu’s algebra 
$A_n(V)$ is isomorphic to a subquotient algebra of the degree zero part of the universal enveloping algebra U(V) of a VOA defined by Frenkel and Zhu [Reference Frenkel and Zhu18]. VOAs generalize Lie algebras and so the Noetherian property is unexpected given what is known about Lie algebras. For instance, if 
$\mathfrak{g}$ is the Witt algebra, it was proved by Sierra and Walton that 
$U(\mathfrak{g})$ is not Noetherian [Reference Sierra and Walton38]. Adding to the unexpectedness of the result, 
$A_g(V)$ is Noetherian for all C 1-cofinite V, an unrestricted class, encompassing what are considered all reasonable examples, including the non-C 2-cofinite VOAs mentioned above.
The Noetherianity for Zhu’s algebra has been established as an ingredient for the study of the representation theory of C 1-cofinite VOAs. It was used in a recent work of Damiolini, Gibney and Krashen in [Reference Damiolini, Gibney and Krashen6].
To state our main results, and describe how they are proved, we introduce some notation. Let V be a VOA of CFT-type, 
$g\in \mathrm{Aut}(V)$ be an automorphism of order 
$T \lt \infty$, and 
$R_2(V)=V/C_{2}(V)$ be the C 2-algebra [Reference Zhu41]. It was observed by Zhu that A(V) has a filtration 
$\{F_pA(V)\}_{p=0}^\infty $ obtained by the grading 
$V=\oplus_{p=0}^\infty V_p$. The associated graded algebra 
$\mathrm{gr} A(V)$ is commutative and unital. Arakawa, Lam and Yamada observed that there is an epimorphism 
$R_2(V)\rightarrow \mathrm{gr} A(V)$ of commutative algebras [Reference algebra3]. It turns out that this epimorphism is quite useful for the study of the structure theory of A(V). Using this morphism, Yang and the author proved a Schur’s lemma for C 1-cofinite VOAs over an arbitrary field [Reference Yang and Liu40]. The twisted Zhu’s algebra 
$A_g(V)$ carries a similar level filtration 
$\cup_{p=0}^\infty F_p A_g(V)$, and there exists epimorphism from 
$R_2(V)$ to the associated graded algebra 
$\mathrm{gr} A_g(V)$ as well. It was proved by Li that 
$R_2(V)$ is a finitely generated algebra if V is C 1-cofinite [Reference Li30, Reference Li32]. Combining these facts together, we can prove our first main theorem (see Theorem 3.1):
Theorem A.
Let V be a CFT-type VOA that is C 1-cofinite, and let 
$g\in \mathrm{Aut}(V)$ be a finite order automorphism. Then 
$A_g(V)$ is left and right Noetherian as an associative algebra.
The A(V)-bimodule A(M) and its twisted analog 
$A_g(M)$ were introduced to compute the fusion rules among (g-twisted)-modules over V [Reference Frenkel and Zhu18, Reference Gao, Liu and Zhu20, Reference Li31, Reference Liu33]. Li introduced a cofinite condition for V-modules, which we call the weakly C 1-cofinite condition, and proved that the fusion rule among three irreducible untwisted V-modules 
$M^1,M^2$ and M 3 is finite if M 1 is weakly C 1-cofinite, see [Reference Li30]. In order to handle the g-twisted case, we modify Li’s confinite condition and introduce a subspace 
$\widetilde{C}^g_1(M)$ associated with M. We say that M is weakly 
$C_1^g$-cofinite if 
$\dim M/\widetilde{C}^g_1(M) \lt \infty$, see Definition 2.5. Huang independently introduced another C 1-cofinite condition for modules in [Reference Huang23], which is slightly stronger than Li’s C 1-condition, to guarantee the convergence of iterated intertwining operators. As an application, Huang also proved that the fusion rule among V-modules 
$M^1,M^2$ and M 3 is finite if M 1 is C 1-cofinite. These C 1-cofinite conditions for V-modules correspond to finite generation properties of the twisted bimodule 
$A_g(M)$ over twisted Zhu’s algebra 
$A_g(V)$. The following is our second main theorem (see Theorem 3.3):
Theorem B.
Let M be an untwisted admissible V-module. Then
(1)
$A_g(M)$ is finitely generated as a left or right
$A_g(V)$-module if M is C 1-cofinite.(2)
$A_g(M)$ is finitely generated as an
$A_g(V)$-bimodule if M is weakly C g1-cofinite.
In particular, for a C 1-cofinite VOA V, 
$A_g(M)$ is Noetherian as a left or right 
$A_g(V)$-module if M is C 1-cofnite; 
$A_g(M)$ is Noetherian as an 
$A_g(V)$-bimodule if M is weakly C g1-cofinite.
As a Corollary of Theorem B, using the g-twisted fusion rules theorem proved by Gao, the author and Zhu in [Reference Gao, Liu and Zhu20], we can prove following finiteness property for fusion rules among g-twisted modules, which simultaneously generalizes both Li and Huang’s result about finiteness of fusion rules under C 1 condition to the g-twisted case (see Corollary 3.4):
Corollary C.
Let M 1 be an untwisted ordinary V-module, and 
$M^2,M^3$ be g-twisted ordinary V-modules. If the M 1 is (weakly) C 1-cofinite, then the fusion rule 
$N{\binom{M^3}{M^1\;M^2}}$ is finite.
Theorem A gives us a sufficient condition for the Noetherianity of 
$A_g(V)$. In § 4, by giving a counter-example, we show that 
$A_g(V)$ is not Noetherian in general if the CFT-type VOA V is not C 1-cofinite. Since the classical examples of CFT-type VOAs (rational or not) are all C 1-cofinite [Reference Dong, Li and Mason14], it is not trivial to find a CFT-type non-C 1-cofinite VOA. The example we construct is a sub-VOA 
$V_M=\bigoplus_{\gamma\in M} M_{\hat{\mathfrak{h}}}(1,\gamma)$ of the lattice VOA 
$V_{A_2}$, where 
$M=\{m\alpha+n\beta: m\geq n\geq 1\}\cup \{0\}$ is an abelian submonoid of the root lattice A 2, see Figure 1. This example is a modification of the Borel-type sub-VOA of a lattice VOA defined by the author in [Reference Liu34]. Using a similar method as in [Reference Liu34], we give an explicit description of the Zhu’s algebra 
$A(V_M)$ of VM. The non-Noetherianity follows from the description. The following is our third main theorem (see Theorem 4.2, Theorem 4.7 and Corollary 4.8):
Theorem D.
Let 
$V_M=M_{\hat{\mathfrak{h}}}(1,0)\oplus\bigoplus_{m\geq n\geq 1} M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)$. Then
(1)
$V_M/C_1(V_M)$ has a basis
$\{\mathbf{1}+C_1(V_M),e^{m\alpha+\beta}+C_1(V_M):m\geq 1\}$. In particular, the CFT-type VOA VM is not C 1-cofinite.(2)
$A(V_M)\cong \mathbb{C}[x,y]\oplus \left(\bigoplus_{m=1}^\infty z_m\mathbb{C}[y]\right),$ where
$J=\bigoplus_{m=1}^\infty z_m\mathbb{C}[y]$ is a two-sided ideal of
$A(V_M)$ which is not finitely generated. In particular,
$A(V_M)$ is not Noetherian.
The higher level generalization of Zhu’s algebra 
$A_n(V)$ was introduced by Dong, Li and Mason in [Reference Dong, Li and Mason11] to study the rationality of VOAs. They proved that V is rational if and only if 
$A_n(V)$ are semisimple for all 
$n\geq 0$. 
$A_n(V)$ was further generalized to the g-twisted case in [Reference Dong, Li and Mason10]. The g-twisted higher Zhu’s algebra 
$A_{g,n}(V)$ controls the first n level of a g-twisted admissible V-module M, where 
$n=l+\frac{i}{T}$ with 
$l\in \mathbb{N}$ and 
$0\leq i\leq T-1$. In § 5.1, we introduce a shifted level-filtration 
$\cup_{p=2l}^\infty F_p A_{g,n}(V)$ on 
$A_{g,n}(V)$ which is compatible with the product on 
$A_{g,n}(V)$, see Lemma 5.1. On the other hand, Zhu’s C 2-algebra 
$R_2(V)$ also has a higher order generalization 
$R_{2l+2}(V)=V/C_{2l+2}(V)$. However, unlike 
$R_2(V)$, the associative algebra 
$R_{2l+2}(V)$ is not commutative in general. In § 5.2, we show that there is a surjective linear map 
$\varphi_n: R_{2l+2}(V)\rightarrow \mathrm{gr} A_{g,n}(V)$, which is a homomorphism of associative algebras if 
$i \lt \lfloor T/2\rfloor$, see Theorem 5.5.
Gaberdiel and Neitzke proved that the C 2-cofinite condition is strong enough so that it implies 
$\dim R_{2l+2}(V) \lt \infty$ for all 
$l\geq 0$, see [Reference Gaberdiel and Neitzke19]. Using this fact, Miyamoto proved that 
$A_n(V)$ are finite-dimensional for all 
$n\geq 0$ if V is C 2-cofinite, which is a key property for the modular invariance of pseudo trace functions of C 2-cofinite VOAs [Reference Miyamoto36]. Buhl found a module version of Gaberdiel and Neitzke’s theorem and proved that 
$A_n(M)$ are finite-dimensional for all 
$n\geq 0$ if V is C 2-cofinite and M is C 2-cofinite [Reference Buhl4]. The finiteness of 
$A_n(M)$ could be useful in generalizing Huang’s modular invariance of logarithmic intertwining operators [Reference Huang25] to C 2-cofinite but not necessarily rational VOAs. With the surjective linear map 
$\varphi_n: R_{2l+2}(V)\rightarrow \mathrm{gr} A_{g,n}(V)$, we can prove our last main theorem, which is a twisted version of Miyamoto and Buhl’s finiteness results about 
$A_n(V)$ and 
$A_n(M)$ (see Corollaries 5.6 and 5.10):
Theorem E.
Let M be an untwisted irreducible admissible V-module, and let 
$n=l+\frac{i}{T}\in \frac{1}{T}\mathbb{Z}$, where 
$l\in \mathbb{N}$ and 
$0\leq i\leq T-1$.
(1) If V is C 2-cofinite, then
$ A_{g,n}(V)$ is a finite-dimensional associative algebra, and
$A_{g,n}(M)$ is a finite-dimensional
$A_{g,n}(V)$-bimodule.(2) If
$i \lt \lfloor T/2\rfloor$, and
$R_{2l+2}(V)$ is a finitely generated associative algebra, then
$ A_{g,n}(V)$ is left and right Noetherian. If, furthermore, M is
$C_{2l+1}$-cofinite, then
$ A_{g,n}(M)$ is left and right Noetherian.
We conjecture that 
$A_{g,n}(V)$ are left and right Noetherian for all 
$n\geq 0$ if V is C 1-cofinite. According to a recent structural result about the higher order Zhu’s algebra of the Heisenberg VOA by Damiolini, Gibney and Krashen in [Reference Damiolini, Gibney and Krashen5], we know that this conjecture is true if V is the Heisenberg VOA and 
$g=\mathrm{Id}_V$.
This paper is organized as follows: In § 2, we recall the definitions of g-twisted modules, twisted Zhu’s algebra 
$A_g(V)$ and its bimodule 
$A_g(M)$, the C 2-algebra 
$R_2(V)$ and its relation with the C 1-cofinite condition. In § 3, we prove Theorem A, Theorem B and Corollary C. In § 4, we introduce the CFT-type VOA VM and prove that it is not C 1-cofinite. Then we determine the Zhu’s algebra 
$A(V_M)$ and show that it is not Noetherian as claimed in Theorem D. In § 5, we first introduce a shifted level filtration on 
$A_{g,n}(V)$ and discuss its relations with the 
$C_{2l+2}$-algebra 
$R_{2l+2}(V)$, then we use it to prove Theorem E.
Convention: All vector spaces are defined over 
$\mathbb{C}$, the field of complex number. 
$\mathbb{N}$ represents the set of natural numbers including 0.
2. Preliminaries
2.1. g-twisted modules over VOAs
For the definitions of VOAs, untwisted modules over VOAs, Zhu’s algebra and its bimodule, we refer to [Reference Dong and Lepowsky8, Reference Frenkel, Huang and Lepowsky16–Reference Frenkel and Zhu18, Reference Lepowsky and Li28, Reference Zhu41]. Throughout this paper, we assume a VOA 
$(V,Y,\mathbf{1},\omega)$ is of CFT-type: 
$V=V_{0}\oplus V_{+}$, where 
$V_{0}=\mathbb{C}\mathbf{1}$ and 
$V_{+}=\bigoplus_{n=1}^{\infty}V_{n}$.
Let 
$g:V\rightarrow V$ be an automorphism of V finite order T [Reference Frenkel, Lepowsky and Meurman17]. Then V has a g-eigenspace decomposition [Reference Dijkgraaf, Vafa, Verlinde and Verlinde7, Reference Dong and Lepowsky8]:
\begin{equation}
V=\bigoplus_{r=0}^{T-1}V^r,\quad \mathrm{where}\quad V^r=\{a\in V: g(a)=\mathrm{e}^{\frac{2\pi i r}{T}}a\}.
\end{equation} In the rest of this paper, we fix an automorphism 
$g\in \mathrm{Aut}(V)$ of order T.
Definition 2.1. [Reference Dong and Lepowsky8, Reference Dong, Li and Mason12, Reference Huang24]
A g -twisted weak V -module is a pair 
$(M,Y_M)$, where M is a vector space, and YM a linear map
\begin{align*}
Y_M:V&\rightarrow \mathrm{End}(M)[[z^{1/T},z^{-1/T}]]\\
a&\mapsto Y_M(a,z)=\sum_{n\in \mathbb{Z}} a_n z^{-n-1-\frac{r}{T}},\quad \mathrm{for}\ a\in V^r,
\end{align*}satisfying the following properties:
(a) (truncation property) For any
$a\in V$ and
$v\in M$, we have
$a_nv=0$ for
$n\in \frac{1}{T}\mathbb{Z}$ and
$n\gg 0$.(b) (g -twisted Jacobi identity) For any
$a\in V^r$ with
$0\leq r\leq T-1$, and
$b\in V$, we have
(2.2)\begin{equation}
\begin{aligned}
&z_{0}^{-1}\delta\left(\frac{z_1-z_2}{z_0}\right) Y_M(a,z_1)Y_M(b,z_2)- z_{0}^{-1}\delta\left(\frac{-z_2+z_1}{z_0}\right)Y_M(b,z_2)Y_M(a,z_1)\\
&= z_{2}^{-1}\delta\left(\frac{z_1-z_0}{z_2}\right)\left(\frac{z_1-z_0}{z_2}\right)^{-r/T} Y_M(Y(a,z_0)b,z_2).
\end{aligned}
\end{equation}
A g-twisted weak V-module M is called admissible if M has a subspace decomposition:
\begin{equation*}M=\bigoplus_{n\in \frac{1}{T}\mathbb{N}}M(n),\end{equation*} such that 
$a_mM(n)\subseteq M(\mathrm{wt} a-m-1+n)$ for all 
$a\in V$ homogeneous, 
$m\in \frac{1}{T}\mathbb{Z}$, and 
$n\in \frac{1}{T}\mathbb{N}$.
An admissible g-twisted V-module M is called an (ordinary) g -twisted V -module if there exists 
$\lambda \in \mathbb{C}$, called the conformal weight, such that 
$M(n)=M_{\lambda+n}$ is an eigenspace of L(0) of eigenvalue 
$\lambda+n$, and M(n) is finite-dimensional, for all 
$n\in \frac{1}{T}\mathbb{N}$.
In particular, if 
$g=\mathrm{Id}_V$ and T = 1, then Definition 2.1 recovers the usual definitions of weak V-modules, admissible V-modules and ordinary V-modules.
2.2. The g-twisted Zhu’s algebra
$A_{g}(V)$ and it bimodule
$A_{g}(M)$
The g-twisted Zhu’s algebra 
$A_g(V)$ was constructed by Dong, Li and Mason in [Reference Dong, Li and Mason12], as a g-twisted generalization of the usual Zhu’s algebra A(V) in [Reference Zhu41], which controls the bottom level M(0) of a g-twisted admissible V-module.
2.2.1. Definition of
$A_g(V)$
By definition, for any 
$a\in V^r$ with 
$0\leq r\leq T-1$, and 
$b\in V$, let
\begin{equation}
a\circ_{g} b:= \mathrm{Res}_z Y(a,z)b\frac{(1+z)^{\mathrm{wt} a-1+\delta(r)+\frac{r}{T}}}{z^{1+\delta(r)}},\quad \mathrm{where}\quad \delta(r)=\begin{cases}1&\mathrm{if}\ r=0\\ 0&\mathrm{if}\ r \gt 0 \end{cases}.
\end{equation} Let 
$O_g(V):=\mathrm{span}\{a\circ_g b: a\in V^r,\ 0\leq r\leq T-1,\ b\in V \}$, and 
$A_g(V):=V/O_g(V)$. Define
\begin{equation}
a\ast_g b:=\begin{cases}\mathrm{Res}_z Y(a,z)b\frac{(1+z)^{\mathrm{wt} a}}{z}=\sum_{j\geq 0}\binom{\mathrm{wt} a}{j} a_{j-1}b &\mathrm{if}\ a\in V^0\\ 0&\mathrm{if}\ a\in V^r,\ r \gt 0. \end{cases}
\end{equation} By Theorem 2.4 in [Reference Dong, Li and Mason12], 
$A_g(V)$ is an associative algebra with respect to the product (2.4), with unit element 
$[\mathbf{1}]=\mathbf{1}+O_g(V)$. By Lemma 2.2 in [Reference Dong, Li and Mason12], we have
\begin{equation}
a\ast_g b-b\ast_g a=\begin{cases}\mathrm{Res}_z Y(a,z)b(1+z)^{\mathrm{wt} a-1}=\sum_{j\geq 0}\binom{\mathrm{wt} a-1}{j} a_jb &\mathrm{if}\ a\in V^0\\ 0&\mathrm{if}\ a\in V^r,\ r \gt 0. \end{cases}
\end{equation}2.2.2. Definition of
$A_g(M)$
Let M be a (untwisted) admissible V-module. The 
$A_g(V)$-bimodule 
$A_g(M)$ was first introduced in [Reference Qifen Jiang and Jiao37] as a g-twisted generalization of the A(V)-bimodule A(M) in [Reference Frenkel and Zhu18]. One can use 
$A_g(M)$ and 
$A_g(V)$ to calculate the fusion rules among one untwisted V-module M 1 and two g-twisted V-modules M 2 and M 3, see [Reference Gao, Liu and Zhu20].
Similar to (2.3), for any 
$a\in V^r$ with 
$0\leq r\leq T-1$, and 
$v\in M$, we let
\begin{equation}
a\circ_g v:=\mathrm{Res}_z Y_M(a,z)v\frac{(1+z)^{\mathrm{wt} a-1+\delta(r)+\frac{r}{T}}}{z^{1+\delta(r)}}.
\end{equation} Let 
$O_g(M)=\mathrm{span}\{a\circ_gv: a\in V^r,\ 0\leq r\leq T-1,\ v\in M \}$, and 
$A_g(M)=M/O_g(M)$. Define:
\begin{align}
a\ast_g v:&=\begin{cases}\mathrm{Res}_z Y_M(a,z)v\frac{(1+z)^{\mathrm{wt} a}}{z}=\sum_{j\geq 0}\binom{\mathrm{wt} a}{j} a_{j-1}v &\mathrm{if}\ a\in V^0\\ 0&\mathrm{if}\ a\in V^r,\ r \gt 0, \end{cases}
\end{align}
\begin{align}
v\ast_g a:&=\begin{cases}\mathrm{Res}_z Y_M(a,z)v\frac{(1+z)^{\mathrm{wt} a-1}}{z}=\sum_{j\geq 0}\binom{\mathrm{wt} a-1}{j} a_{j-1}v &\mathrm{if}\ a\in V^0\\ 0&\mathrm{if}\ a\in V^r,\ r \gt 0. \end{cases}
\end{align} Then 
$A_g(M)$ is a bimodule over 
$A_g(V)$ with respect to the left and right actions (2.7) and (2.8), see [Reference Qifen Jiang and Jiao37] Theorem 3.4 or [Reference Gao, Liu and Zhu20] Lemma 6.1. The following formula follows immediately from (2.7) and (2.8):
\begin{equation}
a\ast_g v-v\ast_g a=\begin{cases}\mathrm{Res}_z Y_M(a,z)v (1+z)^{\mathrm{wt} a-1}=\sum_{j\geq 0}\binom{\mathrm{wt} -1}{j} a_jv &\mathrm{if}\ a\in V^0\\ 0&\mathrm{if}\ a\in V^r,\ r \gt 0. \end{cases}
\end{equation} Moreover, using the 
$L(-1)$-derivative property of YM, one can show
\begin{equation}
\mathrm{Res}_z Y_M(a,z)v\frac{(1+z)^{\mathrm{wt} a-1+\delta(r)+\frac{r}{T}+n}}{z^{1+\delta(r)+m}}\in O_g(M),\quad m\geq n\geq 0.
\end{equation}2.2.3. Level filtration on
$A_g(V)$ and
$A_g(M)$
For the general theory of filtered rings and modules, we refer to [Reference McConnell and Robson35]. It was observed by Zhu in [Reference Zhu41] that A(V) has a canonical filtration obtained by the level decomposition of V:
\begin{equation*}A(V)=\bigcup_{p=0}^\infty F_pA(V),\quad \mathrm{where}\quad F_pA(V)=\left(\oplus_{n=0}^p V_n+O(V)\right)/O(V).\end{equation*} We can similarly define the level filtration on 
$A_g(V)$ and 
$A_g(M)$ as follows:
\begin{align}
A_g(V)&=\bigcup_{p=0}^\infty F_pA_g(V),\quad \mathrm{where}\quad F_pA_g(V)=\left(\oplus_{n=0}^p V_n+O_g(V)\right)/O_g(V).
\end{align}
\begin{align}
A_g(M)&=\bigcup_{p=0}^\infty F_pA(M),\quad \mathrm{where}\quad F_pA_g(M)=\left(\oplus_{n=0}^p M(n)+O_g(M)\right)/O_g(M).
\end{align}Lemma 2.2.
Let V be a VOA, and M be an admissible untwisted V-module. Then
(1)
$A_g(V)$ is a filtered associated algebra with respect to the filtration (2.11), and the associated graded algebra
\begin{equation*}\mathrm{gr} A_g(V)=\bigoplus_{p=0}^\infty F_{p}A_g(V)/F_{p-1}A_g(V)\quad \mathrm{with}\quad F_{-1}A_g(V)=0\end{equation*}is a commutative associative algebra with respect to the product:
(2.13)\begin{align}
& \left([a]+F_{p-1}A_g(V)\right)\ast_g \left([b]+F_{q-1}A_g(V)\right) \nonumber\\
&\quad =\begin{cases}[a_{-1}b]+F_{p+q-1}A_g(V) &\mathrm{if}\ a\in V^0,\\
0+F_{p+q-1}A_g(V)&\mathrm{if}\ a\in V^r,\ r \gt 0,
\end{cases}
\end{align}for any
$a\in \oplus_{n=0}^p V_n$ and
$b\in \oplus_{n=0}^q V_n$, and
$p,q\geq 0$, with identity element
$[\mathbf{1}]\in F_{0}A_g(V)$.(2)
$A_g(M)$ is a filtered
$A_g(V)$-bimodule with respect to (2.11) and (2.12), and the associated graded space
\begin{equation*}\mathrm{gr} A_g(M)=\bigoplus_{p=0}^\infty F_pA_g(M)/F_{p-1}A_g(M)\quad \mathrm{with}\quad F_{-1}A_g(M)=0\end{equation*}is a graded
$\mathrm{gr} A_g(V)$-module with respect to the module action:
(2.14)\begin{align}
&\left([a]+F_{p-1}A_g(V)\right)\ast_g \left([v]+F_{q-1}A_g(M)\right) \nonumber\\
&\quad =\begin{cases}[a_{-1}v]+F_{p+q-1}A_g(M) &\mathrm{if}\ a\in V^0,\\
0+F_{p+q-1}A_g(M)&\mathrm{if}\ a\in V^r,\ r \gt 0,
\end{cases}
\end{align}for any
$a\in \oplus_{n=0}^p V_n$ and
$v\in \oplus_{n=0}^q M(n)$, and
$p,q\geq 0$.
Proof. By (2.4), it is clear that 
$F_pA_g(V)\ast_g F_qA_g(V)\subseteq F_{p+q}A_g(V)$ for any 
$p,q\geq 0$, since we have 
$[a]\ast_g[b]=\sum_{j\geq 0}\binom{\mathrm{wt} a}{j} [a_{j-1} b]$ or 0, and 
$\mathrm{wt} (a_{j-1}b)=\mathrm{wt} a-j+\mathrm{wt} b\leq p+q$ for 
$a\in \oplus_{n=0}^p V_n$ and 
$b\in \oplus_{n=0}^q V_n$. Thus, 
$A_g(V)$ is a filtered algebra, and 
$\mathrm{gr} A_g(V)$ is a graded algebra with respect to the product (2.13). Assume 
$a\in V^0$. By (2.5), we have
\begin{align*}
&\left([a]+F_{p-1}A_g(V)\right)\ast_g \left([b]+F_{q-1}A_g(V)\right)-\left([b]+F_{p-1}A_g(V)\right)\ast_g \left([a]+F_{q-1}A_g(V)\right)\\
&=\sum_{j\geq 0}[a_{j}b]+F_{p+q-1}A_g(V)=0,
\end{align*} since 
$\mathrm{wt} a_{j}b=\mathrm{wt} a-j-1+\mathrm{wt} b \lt p+q$ and so 
$[a_jb]\in F_{p+q-1}A_g(V)$ for all 
$j\geq 0$. If 
$a\in V^r$ with r > 0, clearly 
$[a]+F_{p-1}A_g(V)$ commutes with any other elements in 
$\mathrm{gr} A_g(V)$. Thus, 
$\mathrm{gr} A_g(V)$ is a commutative associative algebra.
Similarly, by (2.7) and (2.8), we have 
$F_pA_g(V)\ast_g F_qA_g(M)\subseteq F_{p+q}A_g(M)$ and 
$F_pA_g(M)\ast_g F_qA_g(V)\subseteq F_{p+q}A_g(M)$. Thus, 
$A_g(M)$ is a filtered 
$A_g(V)$-bimodule, and 
$\mathrm{gr} A_g(M)$ is a 
$\mathrm{gr} A_g(V)$-bimodule. By (2.9), the left and right 
$\mathrm{gr} A_g(V)$-module actions on 
$\mathrm{gr} A_g(M)$ coincide. Hence 
$\mathrm{gr} A_g(M)$ is a graded 
$\mathrm{gr} A_g(V)$-module with respect to (2.14).
2.3. The cofinite conditions of a VOA
The C 2-cofinite condition of V was introduced by Zhu in [Reference Zhu41] to guarantee the convergence of the n-point trace functions. By definition, 
$C_2(V):=\mathrm{span}\{a_{-2}b: a,b\in V\}$, and V is called C 2-cofinite if 
$\dim V/C_2(V) \lt \infty$. Zhu also proved in [Reference Zhu41] that
\begin{equation*}R_2(V)=V/C_2(V)=\bigoplus_{p=0}^\infty V_p/(C_2(V)\cap V_p)\end{equation*}is a unital graded commutative associative algebra with respect to the product
\begin{equation}
(a+C_2(V))\cdot (b+C_2(V))=a_{-1}b+C_2(V),\quad a,b\in V,
\end{equation} with identity element 
$\mathbf{1}+C_2(V)$.
The notion of a strongly generated VOA was introduced by Kac [Reference Kac26]:
Definition 2.3.
Let V be a VOA, and 
$U\subseteq V$ be a subset. V is said to be strongly generated by U if V is spanned by elements of the form:
\begin{equation*}a^{1}_{-n_{1}}\dots a^{r}_{-n_{r}}u,\end{equation*} where 
$a^{1},\dots, a^{r},u\in U$ and 
$n_{i}\geq 1$ for all i. If V is strongly generated by a finite dimensional subspace, then V is called strongly finitely generated.
In the study of the strong generation property and Poincaré–Birkhoff–Witt PBW-basis of VOAs, Li introduced a similar condition in [Reference Karel and Li27, Reference Li30], called C 1-cofiniteness. By definition,
\begin{equation}
C_1(V)=\mathrm{span}\{a_{-1}b: a, b\in V_+\}+\mathrm{span}\{L(-1)c: c\in V\},
\end{equation} and V is called C 1 -cofinite if 
$\dim V/C_1(V) \lt \infty$. It is clear that 
$C_2(V)\subseteq C_1(V)$. Hence any C 2-cofinite VOA is also C 1-cofinite. The following theorem that relates the C 1-cofinite condition with the strong generation property of a VOA was proved by Li, see [Reference Karel and Li27, Reference Li30, Reference Li32]:
Theorem 2.4.
Let V be a VOA, and 
$U\subseteq V_{+}$ be a graded subspace. The following conditions are equivalent:
(1) V is strongly generated by U.
(2)
$V_{+}=U+C_{1}(V)$ as vector spaces.(3)
$(U+C_{2}(V))/C_{2}(V)$ generates
$R_2(V)$ as commutative algebra.
In particular, V is strongly finitely generated if and only if V is C 1-cofinite, if and only if 
$R_2(V)$ is a finitely generated commutative algebra.
The C 1-cofiniteness condition for V-modules was introduced by Huang in [Reference Huang23]. By definition, an admissible V-module M is called C 1-cofinite if 
$\dim M/C_1(M) \lt \infty$, where
\begin{equation}
C_1(M):=\mathrm{span}\{a_{-1}v: a\in V_+,v\in M\}.
\end{equation}There is a similar subspace of M introduced by Li in [Reference Li30]:
\begin{equation}
B(M):=\mathrm{span}\{a_{-1}v: a\in V_+,v\in M\}+\mathrm{span}\{b_0u:b\in \oplus_{n\geq 2}V_n, u\in M \}.
\end{equation} We need to adjust the definition of B(M) a little bit to make it compatible with 
$A_g(M)$:
\begin{equation}
\widetilde{C}^g_1(M):=\mathrm{span}\{a_{-1}v: a\in V_+,v\in M\}+\mathrm{span}\{b_0u:b\in \oplus_{n\geq 2}V_n\cap V^0, u\in M \},
\end{equation} where 
$V^0\subset V$ is the fixed point sub-VOA (2.1) with respect to g. Then 
$\widetilde{C}_1^g(M)=B(M)$ if 
$g=\mathrm{Id}_V$. Observe that the space spanned by 
$b_0u$ in (2.19) is non-zero, since 
$\omega \in \oplus_{n\geq 2}V_n\cap V^0$ and there exists 
$u\in M\backslash\{0\}$ such that 
$\omega_0u=L(-1)u\neq 0$, see [Reference Li29].
Definition 2.5.
Let M be an admissible untwisted V-module. We say that M is weakly C 1 -cofinite if 
$\dim M/B(M) \lt \infty$; we say that M is weakly C g1 -cofinite if 
$\dim M/ \widetilde{C}^g_1(M) \lt \infty$.
Since 
$C_1(M)\subseteq \widetilde{C}^g_1(M)\subseteq B(M)$, the following lemma is evident:
Lemma 2.6.
Let M be an admissible V-module. If M is C 1-cofinite, then it must be weakly 
$C_1^g$-cofinite. If M is weakly 
$C_1^g$-cofinite, then it must be weakly C 1-cofinite.
However, the converse of these statements is not true.
Example 2.7. Let 
$V=\bar{V}(c,0)$ be the universal Virasoro VOA with central charge c > 0. It is well-known that the Verma module 
$M=M(c,h)$ over the Virasoro Lie algebra of central charge c and highest weight h > 0 is an admissible module over V, see [Reference Frenkel and Zhu18, Reference Lepowsky and Li28]. Recall that
\begin{equation*}M(c,h)=\mathrm{span}\{L(-n_1)L(-n_2)\cdots L(-n_n)v_{c,h}: n_1\geq n_2\geq \dots\geq n_k\geq 1\}.\end{equation*} Then 
$M(c,h)=\mathbb{C} v_{c,h}+B(M(c,h))$ in view of (2.18). Hence 
$M(c,h)$ is weakly C 1-cofinite. However, 
$M(c,h)=\mathrm{span}\{L(-1)^kv_{c,h}:k\geq 0\}+C_1(M(c,h))$, and 
$L(-1)^kv_{c,h}\neq 0$ for all 
$k\geq 0$ in a Verma module. Thus, 
$M(c,h)$ is not C 1-cofinite.
Finally, we recall the following fact about the Noetherianity of a filtered ring, see Theorem 6.9 in [Reference McConnell and Robson35].
Proposition 2.8.
Let R be a filtered ring such that the associated graded ring 
$\mathrm{gr}R$ is left (respectively right) Noetherian, then R is left (respectively right) Noetherian.
3. Noetherianity of twisted Zhu’s algebra and its bimodule
We prove our main theorem of this paper in this section.
3.1. Noetherianity of
$A_g(V)$ for C 1-cofinite VOA V
In the study of Zhu’s algebra of the parafermion VOAs [Reference algebra3], Arakawa, Lam and Yamada introduced an epimorphism of commutative associative algebras:
\begin{equation}
\begin{aligned}
\phi: R_2(V)&\rightarrow \mathrm{gr} A(V)=\bigoplus_{p=0}^{\infty}F_pA(V)/F_{p-1}A(V),\\
a+C_{2}(V)&\mapsto [a]+F_{p-1}A(V),\quad a\in \oplus_{n=0}^p V_n,
\end{aligned}
\end{equation} where 
$\mathrm{gr} A(V)$ is the graded algebra in 2.2 with 
$g=\mathrm{Id}_V$. This map was also used to prove the Schur’s Lemma for irreducible modules of VOAs over an arbitrary field in [Reference Yang and Liu40].
Theorem 3.1.
Let V be a VOA, and 
$g\in \mathrm{Aut}(V)$ be a finite order automorphism. If V is strongly finitely generated, or equivalently, C 1-cofinite, then 
$A_g(V)$ is left and right Noetherian.
Proof. First, we generalize the epimorphism (3.1) to the following g-twisted case:
\begin{equation}
\begin{aligned}
\varphi: R_2(V)&\rightarrow \mathrm{gr} A_g(V)=\bigoplus_{p=0}^{\infty}F_pA_g(V)/F_{p-1}A_g(V),\\
a+C_{2}(V)&\mapsto [a]+F_{p-1}A_g(V),\quad a\in \oplus_{n=0}^p V_n.
\end{aligned}
\end{equation} For any 
$a\in V_p$ and 
$b\in V_q$ with 
$p,q\geq 0$, we have 
$[a_{-2}b]\in F_{p+q+1}A_g(V)$. To show φ is well-defined, we need to show 
$[a_{-2}b]\equiv 0\pmod{F_{p+q}A_g(V)}$. We may also assume 
$a\in V^r$ for some 
$0\leq r\leq T-1$. If r = 0, by (2.3), we have 
$[a\circ_g b]=\sum_{j\geq 0}\binom{\mathrm{wt} a}{j} [a_{j-2}b]=[0]$ in 
$ A_g(V).$ Hence
\begin{equation*}[a_{-2}b]=-\sum_{j\geq 1}\binom{\mathrm{wt} a}{j} [a_{j-2}b]\in F_{p+q}A_g(V)\end{equation*} since 
$\mathrm{wt} (a_{j-2}b)=p-j+1+q\leq p+q$. If r > 0, by Lemma 2.2 in [Reference Dong, Li and Mason12], we have
\begin{equation*}\mathrm{Res}_z Y(a,z)b\frac{(1+z)^{\mathrm{wt} a-1+\frac{r}{T}}}{z^2}=\sum_{j\geq 0}\binom{\mathrm{wt} a-1+\frac{r}{T}}{j} a_{j-2}b\in O_g(V).\end{equation*} Hence 
$[a_{-2}b]=-\sum_{j\geq 1}\binom{\mathrm{wt} a-1+\frac{r}{T}}{j} [a_{j-2}b]\in F_{p+q}A_g(V)$, and so φ is well-defined.
Clearly, φ is surjective and grading-preserving. Next, we show that φ is a homomorphism of commutative algebras. Let 
$a\in V^0\cap V_p$ and 
$b\in V_q$, by (2.15) and (2.13), we have
\begin{align*}
\varphi((a+C_2(V))\cdot (b+C_2(V)))&=\varphi(a_{-1}b+C_2(V))=[a_{-1}b]+F_{p+q-1}A_g(V)\\
&=\left([a]+F_{p-1}A_g(V)\right)\ast_g\left([b]+F_{q-1}A_g(V)\right)\\
&=\varphi(a+C_2(V))\ast_g \varphi(b+C_2(V)).
\end{align*} Now let 
$a\in V^r\cap V_p$ and 
$b\in V_q$, for some 
$1\leq r\leq T-1$. By (2.3), we have
\begin{equation*}a_{-1}b\equiv -\sum_{j\geq 1}\binom{\mathrm{wt} a-1+\frac{r}{T}}{j} a_{j-1}b\pmod{O_g(V)}.\end{equation*} Then 
$[a_{-1}b]+F_{p+q-1}A_g(V)=-\sum_{j\geq 1}\binom{\mathrm{wt} a-1+\frac{r}{T}}{j}[a_{j-1}b]+F_{p+q-1}A_g(V)=0+F_{p+q-1}A_g(V)$ since 
$\mathrm{wt} (a_{j-1}b)=p+q-j\leq p+q-1$ for any 
$j\geq 1$. Thus,
\begin{align*}
\varphi(a+C_2(V)\cdot (b+C_2(V)))=[a_{-1}b]+F_{p+q-1}A_g(V)=0=\varphi(a+C_2(V))\ast_g \varphi(b+C_2(V)),
\end{align*} in view of (2.13). Hence φ in (3.2) is an grading-preserving epimorphism of commutative algebras. Since V is strongly finitely generated, there exists a subspace 
$U=\mathrm{span}\{a^{1},\dots ,a^{m}\}\subset V$ of homogeneous elements 
$a^1\in V_{p_1},\dots, a^m\in V_{p_m}$ that strongly generates V. By Theorem 2.4, R(V) is generated by 
$\{a^{1}+C_{2}(V),\dots, a^{m}+C_{2}(V)\}$ as a commutative algebra. Since φ is an epimorphism, 
$\mathrm{gr} A_g(V)$ is generated by 
$\{[a^1]+F_{p_1-1}A_g(V),\dots , [a^m]+F_{p_m-1}A_g(V) \}$ as a commutative algebra. In particular, 
$\mathrm{gr} A_g(V)$ is Noetherian since it is quotient ring of the polynomial ring 
$\mathbb{C}[T_1,\dots, T_m]$. Then by Proposition 2.8, 
$A_g(V)$ is also left Noetherian.
Remark 3.2. If 
$g=\mathrm{Id}_V$, we have 
$A_g(V)=A(V)$. Note that the conclusion in Theorem 3.1 does not depend on the choice of g. Thus, A(V) is Noetherian if V is C 1-cofinite.
3.2. Noetherianity of
$A_g(M)$ for (weakly) C 1-cofinite V-module M
Let M be an admissible untwisted V-module. It was proved by Li that if 
$M=W+\widetilde{C}_1(M)$, then A(M) is generated by 
$(W+O(M))/O(M)$ as an A(V)-bimodule, see [Reference Li30] Proposition 3.16. We have a similar result about 
$A_g(M)$, combined with Huang’s C 1-cofinite condition (2.17).
Theorem 3.3.
Let M be an admissible untwisted V-module.
(1) Let
$M=U+C_1(M)$ and
$U=\mathrm{span}\{u^i:i\in I\}$. Then
(3.3)\begin{equation}
A_g(M)=\sum_{i\in I} A_g(V)\ast_g [u^i]=\sum_{i\in I} [u^i]\ast_g A_g(V)
\end{equation}as a left or right
$A_g(V)$-module. In particular,
$A_g(M)$ is Noetherian as a left or right
$A_g(V)$-module if V is C 1-cofinite and M is C 1-cofinite.(2) Let
$M=W+\widetilde{C}^g_1(M)$ and
$W=\mathrm{span}\{w^j:j\in J\}$. Then
(3.4)\begin{equation}
A_g(M)=\sum_{j\in J} A_g(V)\ast_g [w^j]\ast_g A_g(V)
\end{equation}as an
$A_g(V)$-bimodule. In particular,
$A_g(M)$ is Noetherian as an
$A_g(V)$-bimodule if V is C 1-cofinite and M is weakly C g1-cofinite.
Proof. (1) Denote the right submodule 
$\sum_{i\in I} [u^i]\ast_g A_g(V)$ of 
$A_g(M)$ by N. We use induction on degree n of M(n) to show 
$[M(n)]\subseteq N$ in 
$A_g(M)$. Since 
$\deg (a_{-1}v)=\mathrm{wt} a+\deg v\geq 1$ for any 
$a\in V_+$, we have 
$C_1(M)\subseteq \oplus_{m\geq 1}M(m)$. So 
$M(0)\subseteq U$ and 
$[M(0)]\subseteq N$. Suppose the conclusion holds for smaller n. Let 
$x\in M(n)$. We may assume
\begin{equation*}x=u+\sum_{k=1}^s a^k_{-1}v^k,\quad u\in U,\ a^k\in V_+\cap V^r,\ 0\leq r\leq T-1,\ v^k\in M,\end{equation*} with 
$\mathrm{wt} a^k+\deg v^k=n$ for all k. Since 
$[u]\in N$, we need to show 
$[a^k_{-1}v^k]\in N$ for all 
$1\leq k\leq s$.
Fix a 
$1\leq k\leq s$. If r = 0, by (2.8), we have
\begin{equation}
[a^k_{-1}v^k]=[v^k]\ast_g [a^k]-\sum_{j\geq 1}\binom{\mathrm{wt} a-1}{j}[a^k_{j-1}v^k].
\end{equation} Note that 
$\deg v^k \lt n$ since 
$\mathrm{wt} a^k\geq 1$. By the induction hypothesis, we have 
$[v^k]\in N$ which is a right 
$A_g(V)$-module. Hence 
$[v^k]\ast_g [a^k]\in N$. Moreover, since 
$\deg (a^k_{j-1}v^k)=\mathrm{wt} a^k-j+\deg v^k \lt n$ for any 
$j\geq 1$, we have 
$[a^k_{j-1}v^k]\in N$ by the induction hypothesis. Thus 
$[a^k_{-1}v^k]\in N$ in view of (3.5). If r > 0, by (2.6), we have the following equation in 
$A_g(M)$:
\begin{equation*}[a^k\circ_g v^k]=[a^k_{-1}v^k]+\sum_{j\geq 1}\binom{\mathrm{wt} a^k-1+\frac{r}{T}}{j} [a^k_{j-1}v^k]=0.\end{equation*} Since 
$\mathrm{wt} (a^k_{j-1}v^k) \lt n$ for any 
$j\geq 1$, we have 
$[a^k_{-1}v^k]=-\sum_{j\geq 1} \binom{\mathrm{wt} a^k-1+\frac{r}{T}}{j} [a^k_{j-1}v^k]\in N$ by the induction hypothesis. This proves 
$[M(n)]\subseteq N$ and finishes the induction step. Using a similar argument, we can show 
$ A_g(M)=\sum_{i\in I} A_g(V)\ast_g [u^i]$. Assume V is C 1-cofinite and M is C 1-cofinite. By Theorem 3.1, 
$A_g(V)$ is a left (respectively right) Noetherian algebra. By (3.3), 
$A_g(M)$ is a finitely generated left (respectively right) 
$A_g(V)$-module. Thus, 
$A_g(M)$ is left (respectively right) Noetherian as a left (respectively right) 
$A_g(V)$-module.
The proof of (2) is similar to the proof of (1) and the proof of Proposition 3.16 in [Reference Li30], we briefly sketch it. Again, we may denote 
$\sum_{j\in J} A_g(V)\ast_g [w^j]\ast_g A_g(V)$ by N ʹ and use induction on the degree n to show that 
$[M(n)]\subseteq N'$. Assume the conclusion holds for smaller n, for 
$y\in M(n)=W\cap M(n)+\widetilde{C}^g_1(M)\cap M(n)$, we may express it as
\begin{equation*}y=w+\sum_{k=1}^s a^k_{-1}v^k+\sum_{l=1}^t b^l_{0}u^l,\quad w\in W,\ a^k\in V_+\cap V^r,\ b^l\in \oplus_{p\geq 2}V_p\cap V^0,\ v^k,u^l\in M,\end{equation*} with 
$\deg(a^k_{-1}v^k)=\deg (b^l_{0}u^l)=n$ for all 
$k,l$. By adopting a similar argument as above, we can show 
$[a^k_{-1}v^k]\in N'$ for all k. Moreover, using the facts that
\begin{equation*}
b\ast_g u-u\ast_g b\equiv \mathrm{Res}_z Y_M(b,z)u (1+z)^{\mathrm{wt} b-1} \pmod{O_g(M)},
\end{equation*} for 
$b\in V^0$ and 
$u\in M$, and 
$\deg (b^l_ju^l) \lt n$ for any 
$j\geq 1$, we have
\begin{equation}
[b^l_0u^l]=-\sum_{j\geq 1} \binom{\mathrm{wt} b^l-1}{j} [b^l_j u^l]+[b^l]\ast_g [u^l]-[u^l]\ast_g [b^l]\in N'
\end{equation} for all l by the induction hypothesis. Thus 
$[y]=[w]+\sum_{k=1}^s [a^k_{-1}v^k]+\sum_{l=1}^t [b^l_{0}u^l]\in N'$.
Using Theorem 3.3, we can generalize Corollary 3.17 in [Reference Li30] and Theorem 3.1 in [Reference Huang23] about the finiteness of fusion rules under C 1-cofinite condition to the g-twisted case:
Corollary 3.4.
Let M 1 be an untwisted ordinary V-module, and 
$M^2,M^3$ be g-twisted ordinary V-modules. If the M 1 is weakly C g1-cofinite, then the fusion rule 
$N{\binom{M^3}{M^1\;M^2}}$ is finite.
Proof. Since M 1 is C 1-cofinite implies M 1 is weakly C 1-cofinite, it suffices to prove the finiteness of 
$N{\binom{M^3}{M^1\;M^2}}$ when M 1 is weakly C 1-cofinite. The following estimate for the fusion rule was proved by Gao, the author and Zhu, see [Reference Gao, Liu and Zhu20] Theorem 6.5:
\begin{equation}
N{\binom{M^3}{M^1\;M^2}}\leq \dim (M^3(0)^\ast\otimes_{A_g(V)}A_g(M)\otimes_{A_g(V)}M^2(0))^\ast .\end{equation} Let 
$M=W+\widetilde{C}_1(M)$, where 
$W=\mathrm{span}\{w^1,\dots, w^n\}$. Then by (3.4), we have
\begin{align*}
&M^3(0)^{\ast} \otimes_{A_g(V)}A_g(M)\otimes_{A_{g}(V)}M^2(0) \\
&\quad =\sum_{j=1}^n \ M^3 (0)^{\ast} \otimes_{A_{g}(V)} A_g(V){\ast}_{g} [w^j] {\ast}_{g} A_g(V) \otimes_{A_{g}(V)} M^2(0)\\
&\quad =\sum_{j=1}^n M^3(0)^{\ast} \otimes_{\mathbb{C}} \mathbb{C} [w^j]\otimes_{\mathbb{C}} M^2(0),
\end{align*} which is finite-dimensional since 
$M^2(0)$ and 
$M^3(0)$ are both finite-dimensional by Definition 2.1. Hence 
$N{\binom{M^3}{M^1\;M^2}}$ is finite, in view of (3.7).
4. Example of a non-C 1-cofinite CFT-type VOA whose Zhu’s algebra is non-Noetherian
For the classical non-C 2-cofinite CFT-type VOAs, like the vacuum module VOA 
$V_{\hat{\mathfrak{g}}}(\ell,0)$, the Heisenberg VOA 
$M_{\hat{\mathfrak{h}}}(\ell,0)$, and the universal Virasoro VOA 
$\bar{V}(c,0)$ [Reference Frenkel and Zhu18, Reference Lepowsky and Li28], it is well-known that they are C 1-cofinite [Reference Dong, Li and Mason14]. Although one can construct a non-C 1-cofinite VOA by taking infinite direct sum of a C 1-cofinite CFT-type VOA, such examples are not of CFT-type.
In this section, we give a natural example of non-C 1-cofinite CFT-type VOA based on the idea of the Borel-type sub-VOAs of a lattice VOA in [Reference Liu34]. We will also determine its Zhu’s algebra and show that it is not Noetherian.
4.1. The non-C 1-cofinite VOA VM
We refer to [Reference Dong and Lepowsky8, Reference Frenkel, Lepowsky and Meurman17] for the general construction of lattice VOAs. Let L be a positive definite even lattice. It was observed by the author in [Reference Liu34] that for any additive submonoid 
$M\leq L$ with identity element 0, the subspace 
$V_M:=\bigoplus_{\gamma\in M} M_{\hat{\mathfrak{h}}}(1,\gamma)$ is a CFT-type sub-VOA of VL, where 
$\mathfrak{h}=\mathbb{C}\otimes_{\mathbb{Z}} L$. See also [Reference Dong and Lepowsky8]. This follows from the fact that each 
$M_{\hat{\mathfrak{h}}}(1,\gamma)$ is a simple current module over the Heisenberg sub-VOA 
$M_{\hat{\mathfrak{h}}}(1,0)\leq V_L$.
Now let 
$L=A_2=\mathbb{Z}\alpha\oplus \mathbb{Z}\beta$ be the type A 2 root lattice, where 
$(\alpha|\alpha)=2, (\beta|\beta)=2,$ and 
$ (\alpha|\beta)=-1$. Choose a two-cocycle 
$\epsilon: A_2\times A_2\rightarrow \{\pm 1\}$, where 
$\epsilon(\alpha,\alpha)=1,\epsilon(\beta,\beta)=1,\epsilon(\alpha,\beta)=1 $ and 
$\epsilon(\beta,\alpha)=-1$. Let 
$M\leq A_2$ be the following submonoid:
\begin{equation}
M:=\{m\alpha+n\beta: m\geq n\geq 1\}\cup \{0\}.
\end{equation}M is represented in Figure 1 by the red dots.
The submonoid M of root lattice A2.
Consider the sub-VOA of the lattice VOA 
$V_{A_2}$ associated with M:
\begin{equation}
V_M=M_{\hat{\mathfrak{h}}}(1,0)\oplus\bigoplus_{m\geq n\geq 1} M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta).
\end{equation} Then VM is of CFT-type. In the rest of this section, we fix the VOA VM as in (4.2). We will show that VM is not C 1-cofinite, and 
$A(V_M)$ is not Noetherian.
Lemma 4.1.
For any 
$m\geq 1$, we have 
$\mathrm{e}^{m\alpha+\beta}\notin C_1(V_M)$.
Proof. In view of (4.2) and (2.16), we can express 
$C_1(V_M)$ as follows:
\begin{align*}
C_1(V_M)&=\mathrm{span}\{u_{-1}v: u\in M_{\hat{\mathfrak{h}}}(1,\gamma)\cap (V_M)_+, v\in M_{\hat{\mathfrak{h}}}(1,\gamma')\cap(V_M)_+, \gamma,\gamma'\in M \}\\
&\ +\mathrm{span}\{L(-1)w:w\in M_{\hat{\mathfrak{h}}}(1,\theta)\cap(V_M)_+, \theta\in M \}.
\end{align*} Suppose 
$\mathrm{e}^{m\alpha+\beta}\in C_1(V_M)$ for some 
$m\geq 1$. Note that 
$u_{-1}v\in M_{\hat{\mathfrak{h}}}(1,\gamma+\gamma')$ for 
$u\in M_{\hat{\mathfrak{h}}}(1,\gamma)$ and 
$v\in M_{\hat{\mathfrak{h}}}(1,\gamma')$, and 
$L(-1) M_{\hat{\mathfrak{h}}}(1,\theta)\subseteq M_{\hat{\mathfrak{h}}}(1,\theta)$. Moreover, if 
$\gamma,\gamma'$ are non-zero elements in M (4.1), then 
$\gamma+\gamma'\neq m\alpha+\beta$. Since the Heisenberg modules 
$M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)$ are in direct sum in (4.2), and 
$C_1(V_M)$ is a graded subspace of VM, it follows that 
$\mathrm{e}^{m\alpha+\beta}$ must be contained in 
$W_1+W_2+W_3\subset C_1(V_M)$, where
\begin{align*}
W_1&= \mathrm{span}\{u_{-1}v: u\in M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\cap (V_M)_+, v\in M_{\hat{\mathfrak{h}}}(1,0)\cap (V_M)_+ \},\\
W_2&=\mathrm{span}\{u'_{-1}v': u'\in M_{\hat{\mathfrak{h}}}(1,0)\cap (V_M)_+,\ v'\in M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\cap (V_M)_+ \},\\
W_3&=\mathrm{span}\{L(-1)w: w\in M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\cap (V_M)_+\}.
\end{align*} Note that 
$W_1,W_2\subset M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)=\bigoplus_{k=0}^\infty M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)_{(m^2-m+1)+k}$, where 
$m^2-m+1=\mathrm{wt} (\mathrm{e}^{m\alpha+\beta})$. For homogeneous elements 
$u\in M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\cap (V_M)_+$ and 
$ v\in M_{\hat{\mathfrak{h}}}(1,0)\cap (V_M)_+ $, since 
$\mathrm{wt} v \gt 0$, we must have 
$u_{-1}v\in \sum_{k=1}^\infty M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)_{(m^2-m+1)+k}$ as 
$\mathrm{wt} (u_{-1}v)=\mathrm{wt} u+\mathrm{wt} v \gt \mathrm{wt} u\geq m^2-m+1$. This shows 
$W_1,W_2\subseteq \sum_{k=1}^\infty M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)_{(m^2-m+1)+k}$. On the other hand, since 
$\mathrm{wt} (L(-1)w)=\mathrm{wt} w+1$, it is clear that 
$W_3\subseteq \sum_{k=1}^\infty M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)_{(m^2-m+1)+k}$. Then we have
\begin{equation*}\mathrm{e}^{m\alpha+\beta}\in M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)_{m^2-m+1}\cap \sum_{k=1}^\infty M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)_{(m^2-m+1)+k}=0,\end{equation*} which is a contradiction. Thus, 
$\mathrm{e}^{m\alpha+\beta}\notin C_1(V_M)$ for any 
$m\geq 1$.
Theorem 4.2.
$V_M/C_1(V_M)$ has a basis 
$\{\mathbf{1}+C_1(V_M),\mathrm{e}^{m\alpha+\beta}+C_1(V_M):m\geq 1\}$. In particular, the CFT-type VOA VM is not C 1-cofinite.
Proof. Since 
$(m\alpha+n\beta|m\alpha+n\beta)/2=m^2-mn+n^2\geq 1$ for all 
$m\geq n\geq 1$, we have 
$M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)\subseteq (V_M)_+$ for any such a pair of 
$m,n$. Also note that 
$a_{-n}b\in C_1(V_M)$ for any 
$a,b\in (V_M)_+$ and 
$n\geq 1$ since 
$C_1(V)\supset C_2(V)\supset C_3(V)\supset \cdots $, see [Reference Li30].
First, we show that 
$\mathrm{e}^{m\alpha+n\beta}\in C_1(V_M)$ for any 
$m\geq n\geq 2$. Indeed, for any 
$m\geq n\geq 1$, since 
$(\alpha+\beta|m\alpha+n\beta)=m+n\geq 2$, by the definition of lattice vertex operators, we have
\begin{align*}
\mathrm{e}^{\alpha+\beta}_{-m-n-1}\mathrm{e}^{m\alpha+n\beta}&=\mathrm{Res}_z E^-(-\alpha-\beta,z)E^+(-\alpha-\beta,z)\mathrm{e}_{\alpha+\beta}z^{\alpha+\beta}\mathrm{e}^{m\alpha+n\beta}\\
&=(-1)^m \mathrm{e}^{(m+1)\alpha+(n+1)\beta}\in C_1(V_M).
\end{align*} Hence 
$\mathrm{e}^{(m+1)\alpha+(n+1)\beta}\in C_1(V_M)$ for any 
$(m+1)\geq (n+1)\geq 2$. This proves 
$\mathrm{e}^{m\alpha+n\beta}\in C_1(V_M)$ for any 
$m\geq n\geq 2$. Since 
$h(-n)C_1(V_M)\subseteq C_1(V_M)$ for any 
$h\in \mathfrak{h}$ and 
$n\geq 1$, we have 
$M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)\subseteq C_1(V_M)$ for any 
$m\geq n\geq 2$. Then by the decomposition (4.2), we have
\begin{align*}
V_M/C_1(V_M)&=\left(M_{\hat{\mathfrak{h}}}(1,0)+\sum_{m\geq 1} M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\right)+C_1(V_M)\\
&= \mathrm{span}\{\mathbf{1}+C_1(V_M), \mathrm{e}^{m\alpha+\beta}+C_1(V_M): m\geq 1 \}.
\end{align*} It remains to show 
$\{\mathbf{1}+C_1(V_M), \mathrm{e}^{m\alpha+\beta}+C_1(V_M): m\geq 1 \}$ are linearly independent.
Note that 
$\mathbf{1}+C_1(V_M)\neq 0$ in view of (2.16). By Lemma 4.1, 
$\mathrm{e}^{m\alpha+\beta}+C_1(V_M)\neq 0$ for any 
$m\geq 1.$ Since 
$L(0)(u_{-1}v)=u_{-1}L(0)v+(L(0)u)_{-1}v$ and 
$L(0)L(-1)w=L(-1)L(0)w$, we have 
$L(0)C_1(V_M)\subseteq C_1(V_M)$. Hence 
$L(0): V_M/C_1(V_M)\rightarrow V_M/C_1(V_M), a+C_1(V_M)\mapsto L(0)a+C_1(V_M)$ is a well-defined linear map. Since 
$L(0)\mathrm{e}^{m\alpha+\beta}=(m^2-m+1) \mathrm{e}^{m\alpha+\beta}$, it follows that 
$\mathbf{1}+C_1(V_M), \mathrm{e}^{\alpha+\beta}+C_1(V_M),\mathrm{e}^{2\alpha+\beta}+C_1(V_M),\mathrm{e}^{3\alpha+\beta}+C_1(V_M),\ldots $ are eigenvectors of L(0) of distinct eigenvalues. Thus, 
$\{\mathbf{1}+C_1(V_M),e^{m\alpha+\beta}+C_1(V_M):m\geq 1\}$ is a basis of 
$V_M/C_1(V_M)$.
Remark 4.3. From the proofs of Lemma 4.1 and Theorem 4.2, we see that the essential reason why VM is not C 1-cofinite is that the chain of lattice points 
$\{\alpha+\beta, 2\alpha+\beta,3\alpha+\beta,\dots \}$, which is the first horizontal row of red dots in Figure 1, cannot be generated by finitely many points in the submonoid M. Using this idea, one can construct many examples of non-C 1-cofinite CFT-type VOAs inside a lattice VOA.
4.2. Non-Noetherianity of the Zhu’s algebra of VM
We determine the (untwisted) Zhu’s algebra of VM (4.2) based on a similar method as in [Reference Liu34]. In particular, we will see that 
$A(V_M)$ is not Noetherian. Hence our example VM in this section verifies Theorem 3.1 when 
$g=\mathrm{Id}_V$.
4.2.1. A spanning set of
$O(V_M)$
Lemma 4.4.
For any 
$m\geq n\geq 2$, we have 
$\mathrm{e}^{m\alpha+n\beta}\in O(V_M)$.
Proof. Similar to the proof of Theorem 4.2, for any 
$m\geq n\geq 1$, since 
$(\alpha+\beta|m\alpha+n\beta)=m+n\geq 2$, we have the following formula for any 
$k\geq 0$:
\begin{align*}
\mathrm{e}^{\alpha+\beta}_{-k-1}\mathrm{e}^{m\alpha+n\beta}&=\mathrm{Res}_z (-1)^mE^-(-\alpha-\beta,z) z^{(\alpha+\beta|m\alpha+n\beta)-k-1} \mathrm{e}^{(m+1)\alpha+(n+1)\beta}\\
&=\begin{cases}(-1)^m \mathrm{e}^{(m+1)\alpha+(n+1)\beta}&\mathrm{if}\ k=m+n,\\
0&\mathrm{if}\ k \lt m+n. \end{cases}
\end{align*} Then by (2.10) with M = V and 
$g=\mathrm{Id}_V$, noting that 
$\mathrm{wt} (\mathrm{e}^{\alpha+\beta})=2$, we have
\begin{align*}
&\mathrm{Res}_z Y(\mathrm{e}^{\alpha+\beta},z) \mathrm{e}^{m\alpha+n\beta}\frac{(1+z)^2}{z^{m+n+1}}\\
&\quad =\mathrm{e}^{\alpha+\beta}_{-m-n-1}\mathrm{e}^{m\alpha+n\beta}+2\mathrm{e}^{\alpha+\beta}_{-m-n}\mathrm{e}^{m\alpha+n\beta}+\mathrm{e}^{\alpha+\beta}_{-m-n+1}\mathrm{e}^{m\alpha+n\beta}\\
&\quad =(-1)^m \mathrm{e}^{(m+1)\alpha+(n+1)\beta}\equiv 0\pmod{O(V_M)}.
\end{align*} Thus, 
$\mathrm{e}^{(m+1)\alpha+(n+1)\beta}\in O(V_M)$ for any 
$(m+1)\geq (n+1)\geq 2$.
Let O be the subspace of VM spanned by the following elements:
\begin{equation}
\begin{cases}
&h(-n-2)u+h(-n-1)u, \qquad u\in V_M,\ \mathrm{and}\ n\geq 0,\\
&m\alpha(-1)v+\beta(-1)v+(m^2-m+1)v,\qquad v\in M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta),\ m\geq 1,\\
&M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta),\qquad m\geq n\geq 2.
\end{cases}\end{equation}Lemma 4.5.
We have 
$O\subseteq O(V_M)$ as subspace of VM.
Proof. Clearly, 
$h(-n-2)u+h(-n-1)u\in O(V_M)$ for any 
$u\in V_M$. Let 
$m\geq 1$. Note that 
$\mathrm{e}^{m\alpha+\beta}_{-2}\mathbf{1}= \mathrm{Res}_z E^-(-\alpha-\beta,z) z^{-2}\mathrm{e}^{m\alpha+\beta}=m\alpha(-1)\mathrm{e}^{m\alpha+\beta}+\beta(-1)\mathrm{e}^{m\alpha+\beta}$. Since 
$\mathrm{wt} (\mathrm{e}^{m\alpha+\beta})=m^2-m+1$, we have
\begin{align*}
\mathrm{e}^{m\alpha+\beta}\circ \mathbf{1}&=\mathrm{Res}_z Y(\mathrm{e}^{m\alpha+\beta},z)\mathbf{1}\frac{(1+z)^{m^2-m+1}}{z^2}= \mathrm{e}^{m\alpha+\beta}_{-2}\mathbf{1}+ (m^2-m+1)\mathrm{e}^{m\alpha+\beta}_{-1}\mathbf{1}\\
&=m\alpha(-1)\mathrm{e}^{m\alpha+\beta}+\beta(-1)\mathrm{e}^{m\alpha+\beta}+(m^2-m+1)\mathrm{e}^{m\alpha+\beta}\equiv 0\pmod{O(V_M)}.
\end{align*} Since 
$\alpha(-1)$ and 
$\beta(-1)$ commute with 
$h(-n)$, for any 
$h\in \mathfrak{h}$ and 
$n\geq 1$, we have 
$m\alpha(-1)v+\beta(-1)v+(m^2-m+1)v\in O(V_M)$, for any 
$v=h^1(-n_1)\cdots h^r(-n_r)\mathrm{e}^{m\alpha+\beta}\in M_{\hat{\mathfrak{h}}}(1,0)$.
Finally, for 
$m\geq m\geq 2$, let 
$w=h^1(-n_1-1)\cdots h^r(-n_r-1)\mathrm{e}^{m\alpha+\beta}$ be a spanning element of 
$M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)$, where 
$n_1\geq \dots \geq n_r\geq 0$. Since 
$h(-n-1)v\equiv (-1)^n v{\ast} (h(-1)\mathbf{1})\pmod{O(V_M)}$ for any 
$h\in \mathfrak{h}$ and 
$v\in V_M$ by (2.8), we have
\begin{equation*}
w\equiv (-1)^{n_1+\dots +n_r} \mathrm{e}^{m\alpha+\beta}{\ast} (h^1(-1)\mathbf{1}){\ast} \dots {\ast} (h^r(-1)\mathbf{1})\pmod{O(V_M)}.
\end{equation*} Moreover, by Theorem 2.1.1 in [Reference Zhu41], 
$O(V_M){\ast} V_M\subseteq O(V_M)$. It follows from Lemma 4.4 that 
$w\in O(V_M)$. Hence 
$M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)\subseteq O(V_M)$ for any 
$m\geq m\geq 2$.
Conversely, we want to show 
$O(V_M)\subseteq O$. By (4.2), it suffices to show 
$M_{\hat{\mathfrak{h}}}(1,\gamma)\circ M_{\hat{\mathfrak{h}}}(1,\gamma')\subseteq O,$ for any 
$\gamma,\gamma'\in M$. If 
$\gamma=m\alpha+n\beta$ and 
$\gamma'=m'\alpha+n'\beta$, where 
$m\geq n\geq 1$ and 
$m'\geq n'\geq 1$, then 
$\gamma+\gamma'=(m+m')\alpha+(n+n')\beta$, with 
$m+m'\geq n+n'\geq 2$. By (4.3), we have
\begin{equation*}M_{\hat{\mathfrak{h}}}(1,\gamma)\circ M_{\hat{\mathfrak{h}}}(1,\gamma')\subseteq M_{\hat{\mathfrak{h}}}(1,\gamma+\gamma')\subset O.\end{equation*} Moreover, if 
$m\geq n\geq 2$, we also have 
$M_{\hat{\mathfrak{h}}}(1,0)\circ M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)\subset O$ and 
$M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)\circ M_{\hat{\mathfrak{h}}}(1,0)\subset O$. Hence we only need to show
\begin{equation}
M_{\hat{\mathfrak{h}}}(1,0)\circ M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\subset O\quad \mathrm{and}\quad
M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)\circ M_{\hat{\mathfrak{h}}}(1,0)\subset O,
\end{equation} for any 
$m\geq 1$. The proof of (4.4) is a slight modification of the induction process in Section 3.2 in [Reference Liu34], we omit the details. In conclusion, we have the following:
Proposition 4.6.
Let O be the subspace of VM spanned by elements in (4.3). Then 
$O=O(V_M)$.
4.2.2. Structure of
$A(V_M)$
Consider the associative algebra
\begin{equation}
A_M=\mathbb{C} \lt x,y,z_1,z_2,\ldots \gt /R,
\end{equation} where 
$\mathbb{C} \lt x,y,z_1,z_2,\ldots \gt $ is the tensor algebra on infinitely many generators 
$x,y,z_1,z_2,\dots $, and R is the two-sided ideal generated by the following elements:
\begin{equation}
\begin{aligned}
&xy-yx,\quad z_m(mx+y)+(m^2-m+1)z_m,\ m\geq 1,\quad z_iz_j,\ i,j\geq 1,\\
& xz_m-z_mx-(2m-1)z_m,\quad yz_m-z_my-(2-m)z_m,\ m\geq 1.
\end{aligned}
\end{equation}It is clear that AM has the following subspace decomposition:
\begin{equation}
A_M=\mathbb{C}[x,y]\oplus \left(\bigoplus_{m=1}^\infty z_m\mathbb{C}[y]\right),
\end{equation} where 
$z_m\mathbb{C}[y]$ is a vector space with basis 
$\{z_m,z_my,z_my^2,\dots\}$, and we use the same symbols 
$x,y,z_m$ to denote their equivalent classes in the quotient space.
Theorem 4.7.
Define an algebra homomorphism 
$F: \mathbb{C} \lt x,y,z_1,z_2,\ldots \gt \rightarrow A(V_M)$ by letting
\begin{equation}
F(x):=[\alpha(-1)\mathbf{1}],\ F(y):=[\beta(-1)\mathbf{1}],\ F(z_m)=[\mathrm{e}^{m\alpha+\beta}],\ m\geq 1.
\end{equation} Then F factors through AM and induces an isomorphism 
$F:A_M\rightarrow A(V_M)$.
Proof. We first show that 
$F(R)=0$. Indeed, by (2.5), (4.3) and Lemma 4.5, we have
\begin{align*}
&F(xy-yx)=[\alpha(-1)\mathbf{1}]{\ast}[\beta(-1)\mathbf{1}]-[\beta(-1)\mathbf{1}]\ast[\alpha(-1)\mathbf{1}]\\
& =[\alpha(0)\beta(-1)\mathbf{1}]=0;\\
&F(z_m(mx+y)+(m^2-m+1)z_m)\\
&= [\mathrm{e}^{m\alpha+\beta}]\ast [m\alpha(-1)\mathbf{1}]+[\mathrm{e}^{m\alpha+\beta}]\ast [\beta(-1)\mathbf{1}]+(m^2-m+1)[\mathrm{e}^{m\alpha+\beta}] \\
&=[m\alpha(-1)\mathrm{e}^{m\alpha+\beta}+\beta(-1)\mathrm{e}^{m\alpha+\beta}+(m^2-m+1)\mathrm{e}^{m\alpha+\beta}]=0;\\
&F(xz_m-z_mx-(2m-1)z_m)=[\alpha(-1)\mathbf{1}]\ast [\mathrm{e}^{m\alpha+\beta}]\\
&\quad -[\mathrm{e}^{m\alpha+\beta}]\ast [\alpha(-1)\mathbf{1}]-(2m-1)[\mathrm{e}^{m\alpha+\beta}]\\
&=[\alpha(0)\mathrm{e}^{m\alpha+\beta}-(2m-1)\mathrm{e}^{m\alpha+\beta}]=0;\\
& F(yz_m-z_my-(2-m)z_m)= [\beta(-1)\mathbf{1}]\ast [\mathrm{e}^{m\alpha+\beta}]\\
&\quad -[\mathrm{e}^{m\alpha+\beta}]\ast [\beta(-1)\mathbf{1}]-(2-m)[\mathrm{e}^{m\alpha+\beta}]\\
&=[\beta(0)\mathrm{e}^{m\alpha+\beta}-(2-m)\mathrm{e}^{m\alpha+\beta}]=0.
\end{align*} Moreover, since 
$\mathrm{e}^{i\alpha+\beta}\ast \mathrm{e}^{j\alpha+\beta}\in M_{\hat{\mathfrak{h}}}(1,(i+j)\alpha+2\beta)\subset O(V_M)$ in view of Proposition 4.5, it follows that 
$F(z_iz_j)=[\mathrm{e}^{i\alpha+\beta}\ast \mathrm{e}^{i\alpha+\beta}]=0,$ for any 
$i,j\geq 1$. This shows F factors though AM. To show F is an isomorphism, we construct an inverse map of F. Similar to the proof of Theorem 4.11 in [Reference Liu34], we first define a linear map
\begin{equation}
\bar{(\cdot)}: \mathfrak{h}=\mathbb{C}\alpha \oplus \mathbb{C} \beta\rightarrow A_M, \quad h=\lambda\alpha+\mu \beta\mapsto \bar{h}=\lambda x+\mu y,\quad \lambda,\mu\in\mathbb{C}.
\end{equation} Then we define a linear map 
$G: V_M=M_{\hat{\mathfrak{h}}}(1,0)\oplus\bigoplus_{m\geq n\geq 1} M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta) \rightarrow A_M$ as follows:
\begin{align}
h^1(-n_1-1)\cdots h^r(-n_r-1)\mathbf{1}&\mapsto (-1)^{n_1+\cdots +n_r} \overline{h^r}\cdot \overline{h^{r-1}}\cdot \dots \cdot \overline{h^1},
\end{align}
\begin{align}
h^1(-n_1-1)\cdots h^r(-n_r-1)\mathrm{e}^{m\alpha+\beta}&\mapsto (-1)^{n_1+\cdots +n_r} z_m\cdot \overline{h^r}\cdot \overline{h^{r-1}}\cdot \cdots \cdot \overline{h^1},\quad m\geq 1,
\end{align}
\begin{align}
M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta)&\mapsto 0,\quad m\geq n\geq 2,
\end{align} where 
$n_1\geq \dots\geq n_r\geq 0$, and 
$\overline{h^i}$ is the image of hi in AM under (4.9). We claim that 
$G(O)=0$.
Indeed, for any 
$u\in V_M$, 
$h\in \mathfrak{h}$, and 
$n\geq 0$, it is clear from (4.10)–(4.12) that
\begin{equation*}G(h(-n-2)u+h(-n-1)u)=(-1)^{n+1}G(u)\cdot \overline{h}+(-1)^n G(u)\cdot \overline{h}=0.\end{equation*} Let 
$v=h^1(-n_1-1)\cdots h^r(-n_r-1)\mathrm{e}^{m\alpha+\beta}$ be a spanning element of 
$ M_{\hat{\mathfrak{h}}}(1,m\alpha+\beta)$ with 
$m\geq 1$, then by (4.9), (4.11) and (4.6), we have
\begin{align*}
&G(m\alpha(-1)v+\beta(-1)v+(m^2-m+1)v)\\
&=m (-1)^{n_1+\cdots +n_r} z_m\cdot \overline{h^r}\cdot \cdots \cdot \overline{h^1}\cdot \overline{\alpha} + (-1)^{n_1+\cdots +n_r} z_m\cdot \overline{h^r}\cdot \cdots \cdot \overline{h^1}\cdot \overline{\beta}\\
&\quad + (m^2-m+1)z_m\cdot \overline{h^r}\cdot \cdots \cdot \overline{h^1}\\
&=(-1)^{n_1+\cdots+n_r}\left( z_m(mx+y)+(m^2-m+1)z_m \right)\cdot \overline{h^r}\cdot \cdots \cdot \overline{h^1}\\
&=0.
\end{align*} Finally, by (4.12), we have 
$G(M_{\hat{\mathfrak{h}}}(1,m\alpha+n\beta))=0$ for any 
$m\geq n\geq 2$. Thus, we have 
$G(O(V_M))=0$ by (4.3) and Proposition 4.6, and G induces a well-defined linear map 
$G:A(V_M)=V_M/O(V_M)\rightarrow A_M$, such that
\begin{equation}
G([\alpha(-1)\mathbf{1}])=x,\quad G([\beta(-1)\mathbf{1}])=y,\quad G([\mathrm{e}^{m\alpha+\beta}])=z_m,\ m\geq 1,
\end{equation} in view of (4.10)–(4.12). By (4.8) and (4.13), it is clear that G is an inverse of 
$F:A_M\rightarrow A(V_M)$. Hence 
$A_M\cong A(V_M)$ as associative algebras.
Corollary 4.8.
The untwisted Zhu’s algebra 
$A(V_M)$ is not Noetherian.
Proof. By Theorem 4.7 and (4.7), we have an isomorphism
\begin{equation*}A(V_M)\cong A_M= \mathbb{C}[x,y]\oplus \left(\bigoplus_{m=1}^\infty z_m\mathbb{C}[y]\right)=\mathbb{C}[x,y]\oplus J,\end{equation*} where 
$J=\bigoplus_{m=1}^\infty z_m\mathbb{C}[y]$. By (4.6), it is clear that J is a two-sided ideal of AM. Suppose J can be generated by finitely many elements 
$w_1,\dots,w_k\in J$. There must exist an index N > 0 s.t. 
$w_1,\dots ,w_k\in \bigoplus_{m=1}^N z_m \mathbb{C}[y]$. But it follows from (4.6) that
\begin{equation*}A_M\cdot \left(\bigoplus_{m=1}^N z_m \mathbb{C}[y]\right)\cdot A_M\subseteq \bigoplus_{m=1}^N z_m \mathbb{C}[y],\end{equation*} since 
$z_iz_j=0$ for all 
$i,j\geq 1$. Then we have 
$J\subseteq \bigoplus_{m=1}^N z_m \mathbb{C}[y]$, which is a contradiction. Therefore, 
$A(V_M)$ has a two-sided ideal J that is not finitely generated. This shows 
$A(V_M)$ is neither left nor right Noetherian.
5. Finiteness of g-twisted higher Zhu’s algebra
In this section, using a higher order analog of the epimorphism (3.2), we prove that the g-twisted higher Zhu’s algebra 
$A_{g,n}(V)$ constructed by Dong, Li and Mason in [Reference Dong, Li and Mason10] and its bimodule 
$A_{g,n}(M)$ constructed by Jiang and Jiao in [Reference Qifen Jiang and Jiao37] are finite-dimensional if V is C 2-cofinite, which generalizes Miyamoto’s result on finiteness of 
$A_{n}(V)$ and Buhl’s result on finiteness of 
$A_n(M)$ under the C 2-cofinite condition in [Reference Miyamoto36] to the g-twisted case.
5.1. Shifted level filtration on
$A_{g,n}(V)$
First, we recall the definition of 
$A_{g,n}(V)$ in [Reference Dong, Li and Mason10]. Fix a rational number 
$n=l+\frac{i}{T}\in \frac{1}{T}\mathbb{Z}$, where 
$l\in \mathbb{N}$ and 
$0\leq i\leq T-1$ are uniquely determined by n.
For 
$a\in V^r$ with 
$0\leq r\leq T-1$, and 
$b\in V$, define
\begin{equation}
a\circ_{g,n} b:=\mathrm{Res}_z Y(a,z)b\frac{(1+z)^{\mathrm{wt} a-1+\delta_{i}(r)+l+r/T}}{z^{2l+\delta_i(r)+\delta_i(T-r)}},\quad \mathrm{where}\quad
\delta_i(r)=\begin{cases}
1&\mathrm{if}\ r\leq i\\
0&\mathrm{if}\ r \gt i
\end{cases},
\end{equation} and set 
$\delta_i(T)=1$. Let 
$O_{g,n}(V)$ be the subspace of V spanned by all 
$a\circ _{g,n}b$ and 
$L(-1)c+L(0)c$, and let 
$A_{g,n}(V):=V/O_{g,n}(V)$. Define
\begin{equation}
a\ast_{g,n}b:=\begin{cases}\sum_{m=0}^l (-1)^m\binom{m+l}{l} \mathrm{Res}_z Y(a,z)b (1+z)^{\mathrm{wt} a+l}/z^{l+m+1} &\mathrm{if}\ a\in V^0,\\
0&\mathrm{if}\ a\in V^r,\ r \gt 0.
\end{cases}
\end{equation} By Theorem 2.4 in [Reference Dong, Li and Mason10], 
$A_{g,n}(V)$ is an associative algebra with respect to (5.2). Again, we denote the equivalent class of an element 
$a\in V$ in 
$A_{g,n}(V)$ by 
$[a]$.
For the rest of this paper, we fix the rational number 
$n=l+\frac{i}{T}$. The usual level filtration (2.11) cannot give us a desirable higher order analog of the epimorphism (3.2). So we introduce a new level filtration on 
$A_{g,n}(V)$ as follows: For 
$p\geq 2l$, let
\begin{equation}
F_pA_{g,n}(V):=\mathrm{span}\{[a]: a\in V\ \mathrm{homogeneous},\ \mathrm{wt} a+2l\leq p \}.
\end{equation} For 
$p \lt 2l$, let 
$F_{p}A_{g,n}(V):=0$. Clearly, we have
\begin{equation}
F_{2l}A_{g,n}(V)\subseteq F_{2l+1}A_{g,n}(V)\subseteq \cdots \quad \mathrm{and}\quad A_{g,n}(V)=\bigcup_{p=2l}^\infty F_{p}A_{g,n}(V).
\end{equation}Lemma 5.1.
$A_{g,n}(V)$ is a filtered algebra with respect to the filtration (5.3). The product on the associated graded algebra 
$\mathrm{gr} A_{g,n}(V)=\bigoplus_{p=2l}^\infty F_{p}A_{g,n}(V)/F_{p-1}A_{g,n}(V)$ is given by
\begin{align}
&\left([a]+F_{p-1}A_{g,n}(V)\right)\ast _{g,n} \left([b]+F_{q-1}A_{g,n}(V)\right) \nonumber\\
&\quad =\begin{cases}(-1)^l\binom{2l}{l}[a_{-2l-1}b]+F_{p+q-1}A_{g,n}(V)&\mathrm{if}\ r=0\\
0&\mathrm{if}\ r \gt 0
\end{cases},
\end{align} where 
$a\in V^r,b\in V$ are homogeneous, with 
$\mathrm{wt} a+2l\leq p$ and 
$\mathrm{wt} b+2l\leq q$, and 
$p,q\geq 2l$.
Proof. Let 
$a\in V^0$ and 
$b\in V$ be homogeneous elements such that 
$\mathrm{wt} a+2l\leq p$ and 
$\mathrm{wt} b+2l\leq q$. By the definitions of product on 
$A_{g,n}(V)$ (5.2), we have
\begin{equation}
[a\ast_{g,n}b]=\sum_{m=0}^l \sum_{j\geq 0} (-1)^{m} \binom{m+l}{l} \binom{\mathrm{wt} a+l}{j} [a_{j-l-m-1}b]\in F_{p+q}A_{g,n}(V)
\end{equation} since 
$\mathrm{wt} (a_{j-l-m-1}b)+2l=\mathrm{wt} a-j+l+m+\mathrm{wt} b+2l\leq (\mathrm{wt} a+2l)+(\mathrm{wt} b+2l)\leq p+q$, for any 
$j\geq 0$ and 
$0\leq m\leq l$. Hence 
$F_{p}A_{g,n}(V)\ast_{g,n} F_qA_{g,n}(V)\subseteq F_{p+q}A_{g,n}(V)$, and so 
$A_{g,n}(V)$ is a filtered algebra. Moreover, by (5.3) and the equality about weight above, we have 
$[a_{j-l-m-1}b]\in F_{p+q-1}A_{g,n}(V)$ unless j = 0 and m = l. It follows from (5.6) that
\begin{equation*}[a\ast_{g,n}b]+F_{p+q-1}A_{g,n}(V)=(-1)^l\binom{2l}{l} [a_{-2l-1}b]+F_{p+q-1}A_{g,n}(V).\end{equation*}This proves (5.5).
Remark 5.2. By Lemma 2.2 in [Reference Dong, Li and Mason10], for any 
$a\in V^0$ and 
$b\in V$, we have
\begin{equation}
a\ast_{g,n}b-b\ast_{g,n}a\equiv \mathrm{Res}_z Y(a,z)b (1+z)^{\mathrm{wt} a-1}=\sum_{j\geq 0} \binom{\mathrm{wt} a-1}{j} a_jb \pmod{O_{g,n}(V)}.
\end{equation} Since 
$\mathrm{wt} (a_jb)+2l\leq (\mathrm{wt} a+2l)+(\mathrm{wt} b+2l)-1$, it follows that 
$\mathrm{gr} A_{g,n}(V)$ is a commutative graded algebra with respect to the product (5.5). However, unlike 
$\mathrm{gr} A_g(V)$ in the previous sections, if 
$l\geq 1$, the element 
$[\mathbf{1}]+F_{2l-1}A_{g,n}(V)$ is not the unit element of 
$\mathrm{gr} A_{g,n}(V)$.
5.2. Finiteness of
$A_{g,n}(V)$
Recall that 
$n=l+\frac{i}{T}$, where 
$l\in \mathbb{N}$. Consider the following higher level generalization of Zhu’s C 2-algebra 
$R_2(V)$:
\begin{equation}
R_{2l+2}(V):=V/C_{2l+2}(V),\quad \mathrm{where}\quad C_{2l+2}(V)=\mathrm{span}\{a_{-2l-2}b: a,b\in V\}.
\end{equation}Lemma 5.3.
$R_{2l+2}(V)$ is a graded associative algebra with respect to the product
\begin{equation}
(a+C_{2l+2}(V))\cdot (b+C_{2l+2}(V))=(-1)^l\binom{2l}{l}a_{-2l-1}b+C_{2l+2}(V),
\end{equation}and the grading
\begin{equation}
R_{2l+2}(V)=\bigoplus_{p=2l}^\infty R_{2l+2}(V)_p,\quad R_{2l+2}(V)_p=\mathrm{span}\{a+C_{2l+2}(V):\mathrm{wt} a+2l= p\}.
\end{equation} It is commutative if and only if 
$\sum_{j=1}^{2l} (-1)^j (b_{j-2l-1}a)_{-1-j}\mathbf{1}\in C_{2l+2}(V)$, for any 
$a,b\in V$.
Proof. For 
$a,b,c\in V$, by the Jacobi identity of VOA, we have
\begin{align*}
&(a_{-2l-1}b)_{-2l-1}c-a_{-2l-1}(b_{-2l-1}c)\\
&=\sum_{j\geq 1}\binom{-2l-1}{j}(-1)^j a_{-2l-1-j}b_{-2l-1+j}c-\sum_{j\geq 0}\binom{-2l-1}{j} (-1)^{-2l-1+j} b_{-4l-2-j}a_{j}c\\
&\equiv 0\pmod{C_{2l+2}(V)}.
\end{align*} Hence the product (5.9) is associative since the coefficient 
$(-1)^l \binom{2l}{l}$ does not depend on a and b. By (5.8), 
$C_{2l+2}(V)$ is spanned by homogeneous elements, hence 
$R_{2l+2}(V)$ is a graded algebra. Since 
$R_{2l+2}(V)_p=(V_{p-2l}+C_{2l+2}(V))/C_{2l+2}(V)$ for any 
$p\geq 2l$ and 
$V=\bigoplus_{p=2l}^\infty V_{p-2l}$, we have 
$ R_{2l+2}(V)=\bigoplus_{p=2l}^\infty R_{2l+2}(V)_p$. It is clear that 
$R_{2l+2}(V)_{p}\cdot R_{2l+2}(V)_q\subseteq R_{2l+2}(V)_{p+q}$ for any 
$p,q\geq 2l$ since 
$\mathrm{wt} (a_{-2l-1}b)+2l=(\mathrm{wt} a+2l)+(\mathrm{wt} b+2l)=p+q$ if 
$\mathrm{wt} a+2l=p$ and 
$\mathrm{wt} b+2l=q$.
Finally, by the skew-symmetry of the vertex operator, we have
\begin{align*}
a_{-2l-1}b&=\mathrm{Res}_z Y(a,z)bz^{-2l-1}=\mathrm{Res}_z \mathrm{e}^{zL(-1)}Y(b,-z)a z^{-2l-1}\\
&=\mathrm{Res}_z \sum_{j\geq 0} \frac{L(-1)^j}{j!} z^j \sum_{n\in \mathbb{Z}} (-1)^{n+1} z^{-n-1-2l-1} b_n a\\
&=b_{-2l-1}a+\sum_{j\geq 1} \frac{L(-1)^j}{j!} (-1)^j b_{j-2l-1}a\\
&\equiv b_{-2l-1}a+\sum_{j=1}^{2l} (-1)^j (b_{j-2l-1}a)_{-1-j}\mathbf{1} \pmod{C_{2l+2}(V)}.
\end{align*} Thus, 
$R_{2l+2}(V)$ is commutative if and only if the obstruction term 
$\sum_{j=1}^{2l} (-1)^j (b_{j-2l-1}a)_{-1-j}\mathbf{1}$ is in 
$C_{2l+2}(V)$.
Remark 5.4. If l = 0, then 
$R_{2l+2}(V)=R_2(V)$ in view of (5.9) and (2.15). The obstruction term for the commutativity in Lemma 5.3 does not exist in this case. Hence 
$R_2(V)$ is commutative.
We wish to find a g-twisted higher order analog of the epimorphism φ in (3.2). However, it turns out that φ is not always generalizable without any extra assumptions.
Theorem 5.5.
Let V be a VOA, 
$g\in \mathrm{Aut}(V)$ be of order T, and 
$n=l+\frac{i}{T}\in \frac{1}{T}\mathbb{Z}$, where 
$l\in \mathbb{N}$ and 
$0\leq i\leq T-1$. Then there is a surjective linear map:
\begin{equation}
\begin{aligned}
\varphi_n: R_{2l+2}(V)&\rightarrow \mathrm{gr} A_{g,n}(V)=\bigoplus_{p=2l}^{\infty}F_pA_{g,n}(V)/F_{p-1}A_{g,n}(V),\\
a+C_{2l+2}(V)&\mapsto [a]+F_{p-1}A_{g,n}(V),\quad a+C_{2l+2}(V)\in R_{2l+2}(V)_p.
\end{aligned}
\end{equation} If, furthermore, 
$i \lt \lfloor T/2\rfloor$, then φn is an epimorphism of associative algebras.
Proof. Similar to Theorem 3.1, we first show φn is well-defined. Let 
$a\in V_{p-2l}\cap V^r$ with 
$0\leq r\leq T-1$ and 
$b\in V_{q-2l}$, for some 
$p,q\geq 2l$. Then 
$a_{-2l-2}b\in V_{p+q+1-2l}$, and 
$\varphi_n(a_{-2l-2}b+C_{2l+2}(V))=[a_{-2l-2}b]+F_{p+q}A_{g,n}(V)$.
We need to show 
$[a_{-2l-2}b]\equiv 0\pmod{F_{p+q}A_{g,n}(V)}$.
Indeed, for by Lemma 2.2 in [Reference Dong, Li and Mason10], we have
\begin{equation}
\mathrm{Res}_z Y(a,z)b\frac{(1+z)^{\mathrm{wt} a-1+\delta_{i}(r)+l+r/T}}{z^{2l+\delta_i(r)+\delta_i(T-r)+m}}\in O_{g,n}(V),
\end{equation} for any 
$m\geq 0$. Since 
$\delta_i(r)$ and 
$\delta_i(T-r)$ are either 0 or 1 in view of (5.2), we may choose 
$m\geq 0$ in such a way that 
$2l+\delta_i(r)+\delta_i(T-r)+m=2l+2$. Then by (5.12),
\begin{equation*}[a_{-2l-2}b]=-\sum_{j\geq 1} \binom{\mathrm{wt} a-1+\delta_i(r)+l+r/T}{j} [a_{j-2l-2}b]\in F_{p+q}A_{g,n}(V)\end{equation*} since 
$\mathrm{wt} (a_{j-2l-2}b)+2l=(p-2l)-j+2l+1+(q-2l)+2l\leq p+q$ for any 
$j\geq 1$, which means 
$[a_{j-2l-2}b]\in F_{p+q}A_{g,n}(V)$ by (5.3). This proves the well-definedness of φn.
Clearly, φn is surjective. For any 
$a+C_{2l+2}(V)\in R_{2l+2}(V)_{p}$, with 
$\mathrm{wt} a+2l=p$, we have 
$[a]\in F_{p}A_{g,n}(V)$ by (5.3). Hence 
$\varphi_{n}(R_{2l+2}(V)_{p})\subseteq F_pA_{g,n}(V)/F_{p-1}A_{g,n}(V)$ for any 
$p\geq 2l$.
Finally, we show φn is a homomorphism when 
$i \lt \lfloor T/2\rfloor$. Again, we let 
$a\in V_{p-2l}\cap V^r$ with 
$0\leq r\leq T-1$ and 
$b\in V_{q-2l}$. If r = 0, since 
$\mathrm{wt} (a_{-2l-1}b)+2l=p+q$, then by (5.5) and (5.9),
\begin{align*}
&\varphi_n ((a+C_{2l+2}(V))\cdot (b+C_{2l+2}(V)))=\varphi_n ((-1)^l\binom{2l}{l}a_{-2l-1}b+C_{2l+2}(V))\\
&=(-1)^l \binom{2l}{l} [a_{-2l-1}b]+F_{p+q-1}A_{g,n}(V)= \left([a]+F_{p-1}A_{g,n}(V)\right)\ast _{g,n} \left([b]+F_{q-1}A_{g,n}(V)\right)\\
&= \varphi_n(a+C_{2l+2}(V))\ast_{g,n} \varphi_n(b+C_{2l+2}(V)).
\end{align*} Now consider the case when r > 0. Since 
$i \lt \lfloor T/2\rfloor$, the inequalities 
$r\leq i$ and 
$T-r\leq i$ cannot be satisfied simultaneously. Thus 
$\delta_i(r)+\delta_{i}(T-r)=1$ for any r > 0. By (5.1), we have
\begin{equation*}[a_{-2l-1}b]=-\sum_{j\geq 1}\binom{\mathrm{wt} a-1+\delta_i(r)+l+r/T}{j} [a_{j-2l-1}b]\in F_{p+q-1}A_{g,n}(V),\end{equation*} since 
$\mathrm{wt} (a_{j-2l-1}b)+2l=(p-2l)-j+2l+(q-2l)+2l\leq p+q-1$ for any 
$j\geq 1$. Then
\begin{align*}
&\varphi_n ((a+C_{2l+2}(V))\cdot (b+C_{2l+2}(V))) =(-1)^l \binom{2l}{l} [a_{-2l-1}b]+F_{p+q-1}A_{g,n}(V)\\
&=0= \varphi_n(a+C_{2l+2}(V))\ast_{g,n} \varphi_n(b+C_{2l+2}(V)),
\end{align*}in view of (5.5).
In the proof of the modular invariance of C 2-cofinite VOAs, Miyamoto proved that 
$A_n(V)$ are finite-dimensional for all 
$n\geq 0$ if V is C 2-cofinite, see Theorem 2.5 in [Reference Miyamoto36]. The following Corollary of Theorem 5.5 generalizes Miyamoto’s result to the g-twisted case.
Corollary 5.6.
Let 
$n=l+\frac{i}{T}\in \frac{1}{T}\mathbb{Z}$, where 
$l\in \mathbb{N}$ and 
$0\leq i\leq T-1$.
(1) If V is C 2-cofinite, then
$ A_{g,n}(V)$ is a finite-dimensional associative algebra.(2) If
$i \lt \lfloor T/2\rfloor$, and
$R_{2l+2}(V)$ is a finitely generated associative algebra with respect to the product (5.9), then
$ A_{g,n}(V)$ is left and right Noetherian.
Proof. It was proved by Gaberdiel and Neitzke that V is Cu-cofinite for any 
$u\geq 2$ if V is C 2-cofinite, see Theorem 11 in [Reference Gaberdiel and Neitzke19]. Since the filtration (5.4) is exhaustive, by (5.11),
\begin{equation*}\dim A_{g,n}(V)=\dim \mathrm{gr} A_{g,n}(V) \leq \dim R_{2l+2}(V) \lt \infty\end{equation*} if V is C 2-cofinite. Now let 
$i \lt \lfloor T/2\rfloor$ and assume that 
$R_{2l+2}(V)$ is a finitely generated algebra. By Theorem 5.5, φn is an epimorphism of associative algebras, then 
$\mathrm{gr} A_{g,n}(V)$ is a finitely generated commutative algebra in view of Remark 5.2, which is necessarily Noetherian. Hence 
$A_{g,n}(V)$ is left and right Noetherian by Proposition 2.8.
Example 5.7. Let 
$V=M_{\hat{\mathfrak{h}}}(1,0)$ be the rank-one Heisenberg VOA, and let 
$g=\mathrm{Id}_V$. Then V is C 1-cofinite by Theorem 2.4, since 
$R_2(V)\cong \mathbb{C}[x]$ as a commutative algebra, see [Reference Dong, Li and Mason14]. In [Reference Addabbo and Barron2], Addabbo and Barron conjectured that for any 
$n\geq 1$, one has
\begin{equation}
A_n(V)\cong \mathrm{Mat}_{p(n)}(\mathbb{C}[x])\oplus A_{n-1}(V)
\end{equation} as a direct product of associative algebras, where p(n) is the number of partitions of n. The isomorphism (5.13) was proved recently by Damiolini, Gibney and Krashen, see Corollary 7.3.1 [Reference Damiolini, Gibney and Krashen5]. In particular, since 
$ \mathrm{Mat}_{p(n)}(\mathbb{C}[x])$ is a finitely generated module over a Noetherian ring 
$\mathbb{C}[x]$, using induction on n, it is easy to show that 
$A_n(V)$ is Noetherian for all 
$n\geq 0$.
We believe the following statement that generalizes Theorem 3.1 is true:
Conjecture 5.8.
Let V be a VOA, 
$g\in \mathrm{Aut}(V)$ be of order T, and 
$n=l+\frac{i}{T}\in \frac{1}{T}\mathbb{Z}$, where 
$l\in \mathbb{N}$ and 
$i \lt \lfloor T/2\rfloor$. Then 
$A_{g,n}(V)$ is left and right Noetherian if V is C 1-cofinite.
5.3. Finiteness of
$A_{g,n}(M)$
Buhl extended Gaberdiel and Neitzke’s theorem to the case of V-modules. He proved that 
$A_{n}(M)$ are finite-dimensional for all 
$n\geq 0$ if V is C 2-cofinite and M is C 2-cofinite, see [Reference Buhl4] Corollary 5.5. As another Corollary of Theorem 5.5, we generalize Buhl’s theorem to the twisted case.
Abe, Buhl and Dong proved that if V is C 2-cofinite then an irreducible V-module M is C 2-cofnite, see Proposition 5.2 in [Reference Abe, Buhl and Dong1]. In fact, it is easy to show that M is also C 1-cofinite by adopting a similar proof. Moreover, Buhl proved that M is Cn-cofinite for all 
$n\geq 2$, if M is C 2-cofinite, see Corollary 5.3 in [Reference Buhl4]. Hence we have the following:
Lemma 5.9.
Let V be C 2-cofinite, and let M be an irreducible admissible V-module. Then M is Cn-cofinite, for any 
$n\geq 1$.
We can generalize 
$\widetilde{C}^g_1(M)$ (2.19) to the higher order case. For 
$l\geq 0$, define
\begin{equation}
\widetilde{C}^g_{2l+1}(M):=\mathrm{span}\{a_{-2l-1}v:a\in V_+, v\in M\}+\mathrm{span}\{b_0u: b\in \oplus_{p\geq 2}V_p\cap V^0, u\in M\}.
\end{equation} We say that M is weakly 
$C^g_{2l+1}$ -cofinite if 
$\dim M/\widetilde{C}^g_{2l+1}(M) \lt \infty$.
Corollary 5.10.
Let 
$n=l+\frac{i}{T}\in \frac{1}{T}\mathbb{Z}$, where 
$l\in \mathbb{N}$ and 
$0\leq i\leq T-1$.
(1) If
$M=Y+\widetilde{C}^g_{2l+1}(M)$,
$Y=\mathrm{span}\{y^p\in:p\in \Lambda\}$ and
$i \lt \lfloor T/2\rfloor$, then
(5.15)\begin{equation}
A_{g,n}(M)=\sum_{p\in \Lambda} A_{g,n}(V)\ast_{g,n} [y^p]\ast_{g,n} A_{g,n}(V).
\end{equation}In particular, if M is weakly
$C_{2l+1}^g$-cofinite, then
$A_{g,n}(M)$ is a finitely generated
$A_{g,n}(V)$-bimodule.(2) If
$M=U+C_{2l+1}(M)$,
$U=\mathrm{span}\{u^\alpha:\alpha\in I\}$ and
$i \lt \lfloor T/2\rfloor$, then
(5.16)\begin{equation}
A_{g,n}(M)=\sum_{\alpha\in I} A_{g,n}(V)\ast _{g,n}[u^\alpha]=\sum_{\alpha\in I} [u^\alpha]\ast_{g,n} A_{g,n}(V).
\end{equation}In particular, if M is
$C_{2l+1}$-cofinite, then
$A_{g,n}(M)$ is finitely generated as a left or right
$A_{g,n}(V)$-module.(3) If
$M=W+C_{2l+2}(M)$ and
$W=\mathrm{span}\{w^j:j\in J\}$, then
(5.17)\begin{equation}
A_{g,n}(M)=\sum_{j\in J} \mathbb{C} [w^j].
\end{equation}In particular, if V is C 2-cofinite, then
$A_{g,n}(M)$ is finite-dimensional for all
$n\geq 0$.
Proof. The proof is similar to the proof of Theorem 3.3. We write out the details for (5.15) and omit the rests. Denote 
$\sum_{p\in \Lambda} A_{g,n}(V)\ast_{g,n}[y^p]\ast_{g,n}A_{g,n}(V)$ by N. We use induction on the degree m of 
$M=\bigoplus_{m=0}^\infty M(m)$ to show 
$[M(m)]\subseteq N$. Since 
$\deg (a_{-2l-1}v) \gt 0$ and 
$\deg (b_0u) \gt 0$ for any 
$a\in V_+$, 
$b\in \oplus_{p\in 2}V_p$, and 
$u,v\in M$, we have 
$\widetilde{C}^g_{2l+1}(M)\cap M(0)=0$ in view of (5.14). Hence 
$[M(0)]\subseteq N$. Suppose the conclusion holds for smaller m. For 
$x\in M(m)$, we may assume
\begin{equation*}x=u+\sum_{k=1}^s a^k_{-2l-1}v^k+\sum_{q=1}^t b^q_0 u^q ,\quad u\in U,\ a^k\in V_+\cap V^r,\ b\in \oplus_{p\geq 2}V_p\cap V^r,\ v^k,u^q\in M,\end{equation*} where 
$0\leq r\leq T-1$, 
$\mathrm{wt} a^k+2l+\deg v^k=m$, and 
$\mathrm{wt} b^q-1+\deg u^q=m$ for all 
$k,q$. We need to show that 
$[a^k_{-2l-1}v^k]\in N$ and 
$[b^q_0 u^q]\in N$ for all 
$k,q$.
Recall the following formulas for the definition of 
$A_{g,n}(M)=M/O_{g,n}(M)$ in [Reference Qifen Jiang and Jiao37]:
\begin{equation}
a\circ_{g,n} v=\mathrm{Res}_z Y_M(a,z)v\frac{(1+z)^{\mathrm{wt} a+l-1+\delta_i(r)+\frac{r}{T}}}{z^{2l+\delta_i(r)+\delta_i(T-r)}},
\end{equation} where δi is defined by (5.1). For 
$a\in V^r$ and 
$v\in M$, one has
\begin{equation}
a\ast_{g,n}v=\mathrm{Res}_z \sum_{m=0}^l (-1)^m\binom{m+l}{l} \mathrm{Res}_z Y(a,z)v \frac{(1+z)^{\mathrm{wt} a+l}}{z^{l+m+1}}
\end{equation} if 
$a\in V^0$, and 
$a\ast_{g,n}v=0$ if 
$a\in V^r$ with r > 0.
For each 
$1\leq k\leq s$, if 
$a^k\in V^0$, by (5.19) and induction hypothesis, we have
\begin{align*}
[a^k_{-2k-1}v^k]=-\sum_{j,m\geq 0, j-m \gt -l}(-1)^m \binom{m+l}{l} \binom{\mathrm{wt} a^k+l}{j} [a^k_{j-l-m-1} v^k]\in N,
\end{align*} since 
$\deg (a^k_{j-l-m-2} v^k) \lt \mathrm{wt} a^k+2l+\deg v^k=m$ when 
$j-m \gt -l$. On the other hand, if 
$a^k\in V^r$ with r > 0, we have 
$\delta_i(r)+\delta_i(T-r)=1$ since 
$i \lt \lfloor T/2\rfloor$. By (5.18), we have
\begin{equation*}[0]=[a^k\circ_{g,n}v^k]=[a^k_{-2l-1}v^k]+\sum_{j\geq 1}\binom{\mathrm{wt} a^k+l-1+\delta_i(r)+r/T}{j} [a^k_{j-2l-1}v^k]\end{equation*} in 
$A_{g,n}(M)$. Since 
$\deg (a^k_{j-2l-1}v^k)=\mathrm{wt} a^k-j+2l+\deg v^k \lt m$, then 
$[a^k_{j-2l-1}v^k]\in N$ for all 
$j\geq 1$ by the induction hypothesis. Hence 
$[a^k_{-2l-1}v^k]\in N$. Moreover, we have
\begin{equation}
b\ast_{g,n}u-u\ast_{g,n}b\equiv \mathrm{Res}_{z} Y(b,z)u(1+z)^{\mathrm{wt} b-1}\pmod{O_{g,n}(M)}
\end{equation} for any 
$b\in V^0$ and 
$u\in M$, see Lemma 3.1 [Reference Qifen Jiang and Jiao37]. Then by (5.20) and (3.6), we have
\begin{equation*}[b^q_0 u^q]=-\sum_{j\geq 1} \binom{\mathrm{wt} b^q-1}{j} [b^q_j u^q]+[b^q]\ast_{g,n} [u^q]-[u^q]\ast_{g,n} [b^q]\in N,\end{equation*} since 
$\deg (b^q_j u^q)=\mathrm{wt} b^q-j-1+\deg u^q \lt m$ for any 
$j\geq 1$. Thus 
$[x]=[u]+\sum_{k=1}^s [a^k_{-2l-1}v^k]+\sum_{q=1}^t[b^q_0u^q]\in N$. This shows (5.15).
Finally, assume V is C 2-cofinite. By Lemma 5.9, M is 
$C_{2l+2}$-cofinite, for any 
$l\geq 0$. Then by (5.17), 
$A_{g,n}(M)$ is finite-dimensional.
Acknowledgements
I’m deeply grateful to professors Angela Gibney and Danny Karshen for their encouragement and many valuable comments and suggestions for this paper.
