#  Representations of Toroidal and Full toroidal Lie algebras over polynomial
algebras

Santanu Tantubay  S. Tantubay: Department of Mathematics, The Institute of
Mathematical Sciences, A CI of Homi Bhabha National Institute, IV Cross Road,
CIT Campus Taramani Chennai 600 113 Tamil Nadu, India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Priyanshu Chakraborty  Priyanshu
Chakraborty:School of Mathematical Sciences, Ministry of Education Key
Laboratory of Mathematics and Engineering Applications and Shanghai Key
Laboratory of PMMP, East China Normal University, No. 500 Dongchuan Rd.,
Shanghai 200241, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Toroidal Lie algebras are $n$ variable generalizations of affine Kac-Moody Lie
algebras. Full toroidal Lie algebra is the semidirect product of derived Lie
algebra of toroidal Lie algebra and Witt algebra, also it can be thought of
$n$-variable generalization of Affine-Virasoro algebras. Let
$\tilde{\mathfrak{h}}$ be a Cartan subalgebra of a toroidal Lie algebra as
well as full toroidal Lie algebra without containing the zero-degree central
elements. In this paper, we classify the module structure on
$U(\tilde{\mathfrak{h}})$ for all toroidal Lie algebras as well as full
toroidal Lie algebras which are free $U(\tilde{\mathfrak{h}})$-modules of rank
1. These modules exist only for type $A_{l}(l\geq 1)$, $C_{l}(l\geq 2)$
toroidal Lie algebras and the same is true for full toroidal Lie algebras.
Also, we determined the irreducibility condition for these classes of modules
for both the Lie algebras.

###### Key words and phrases:

Kac-Moody Algebras, Toroidal Lie algebras, Full toroidal Lie algebras

###### 2020 Mathematics Subject Classification:

17B65; 17B66, 17B68

⋆ Corresponding author

##  1\. Introduction

In order to understand the module category of any algebras, one of the initial
steps is to classify its irreducible objects. Lie algebra is a fundamental
subject in mathematics. Lie algebra theory also plays an important role in
physics. The classification of simple modules over finite dimensional simple
complex Lie algebra seems wild; for $\mathfrak{sl}_{2}$ there is a weak
version of such classification (see [4], [22] ). Some other classes of simple
modules for simple finite-dimensional Lie algebras are well studied. For
example, simple weight modules with finite-dimensional weight spaces are
classified for simple finite-dimensional Lie algebras, there are two types of
such modules: parabolically induced modules and cuspidal module. Parabolically
induced modules include simple finite-dimensional modules ([10], [16]) and
more generally highest-weight modules ([16], [23], [3]). Simple cuspidal
modules were classified by O. Matheieu (see [21]). Other well-studied classes
of simple modules for finite-dimensional simple Lie algebras are Whittaker
modules ([25]) and Gelfan-Zetlin modules ([15]).
Affine Kac-Moody Lie algebras are natural generalizations of finite-
dimensional simple Lie algebras. The structure theory and representation
theory of affine Kac-Moody algebra has been used in different areas of
mathematics ([26], [28], [29]) as well as in physics ([5]). Integrable weight
representation theory is well-studied for affine Kac-Moody Lie algebras (See
[27]). Simple integrable modules with finite-dimensional weight spaces have
been classified in [11] and [12]. Verma-type modules for this Lie algebras
were first studied by Jakobsen-Kac ([24]) and then by Futorny ([17], [18]).
The classification of all non-zero level simple weight modules with finite-
dimensional weight spaces over Kac-Moody Lie algebras was obtained in [19] and
for level zero case it was announced in [9]. There are some other classes of
weight modules for affine Kac-Moody Lie algebras with infinite-dimensional
weight spaces(see [13], [1], [20], [30]) .
Non-weight simple modules for any Lie algebras are less understood. A class of
non-weight modules for finite-dimensional simple Lie algebra of type $A_{l}$
is studied by J. Nilsson in [31], which are known as Cartan-free modules. In
[32], the author showed that Cartan-free modules exist only for type
$A_{l}(l\geq 1)$ and type $C_{l}(l\geq 2)$. For affine Kac-Moody Lie algebras
rank one Cartan free modules are classified in [7]. For Witt algebra such
classification was done in [37]. For other classes of Lie algebras,
classification of Cartan free modules one can see [6], [39], [40],. This paper
aims to study non-weight modules for two classes of Lie algebras namely
toroidal Lie algebras and full toroidal Lie algebras. For integrable weight
representations of toroidal Lie algebra and full toroidal Lie algebra, one can
see [33], [34] and reference therein.
This paper is organized as follows: In Section 2, we recall the definitions of
Kac-Moody Lie algebra, toroidal Lie algebra, and full toroidal Lie algebra.
Then we defined a full subcategory of the category of modules corresponding to
a finite-dimensional commutative subalgebra of a Lie algebra. In Section 3, we
recall some auxiliary lemmas related to finite-dimensional simple Lie
algebras. In Section 4, we classified all Cartan-free modules of rank one over
toroidal Lie algebras and we determined the condition of simplicity for these
modules. In Section 5, we classified all Cartan-free modules of rank one over
full toroidal Lie algebras and we determined the condition of simplicity for
these modules.

##  2\. Notations and Preliminaries

Throughout this paper, $\mathbb{Z}$, $\mathbb{C}$, and $\mathbb{C}^{*}$ denote
the sets of integers, complex numbers, and nonzero complex numbers,
respectively. For a Lie algebra $\mathfrak{a}$, we denote the universal
enveloping algebra of $\mathfrak{a}$ as $\mathcal{U}(\mathfrak{a})$. All the
vector spaces, algebras, and tensor products are over $\mathbb{C}$, unless it
is specified.

In this section, we shall introduce the notions of affine Kac-Moody algebras,
toroidal Lie algebra and full toroidal Lie algebra.

### 2.1.

Suppose $\mathfrak{g}=\mathfrak{g}(X_{l})$ is a finite-dimensional simple Lie
algebra with a fixed Cartan subalgebra $\mathfrak{h}$ of type $X_{l}$, where
$X$ is one of the types $A,B,\cdots G$ . Let $\\{x_{i},y_{i},h_{i}:1\leq i\leq
l\\}$ be a set of Chevalley generators for $\mathfrak{g}$, where
$l=\hbox{dim}(\mathfrak{h})$. For type $\mathfrak{g}=\mathfrak{g}(A_{l})$ or
type $\mathfrak{g}=\mathfrak{g}(C_{l})$, we take a basis $\\{H_{i}:1\leq i\leq
l\\}$ of $\mathfrak{h}$ by

$[H_{i},x_{j}]=\delta_{ij}x_{i},\;[H_{i},y_{j}]=-\delta_{ij}y_{j},\;\forall\;1\leq
i,j\leq l.$

###  2.2. Affine Kac-Moody algebras.

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra. Let us fix a
Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$. Suppose that $\Delta$ is
the set of roots of $\mathfrak{g}$ and $\Delta_{+}$ is the set of positive
roots of $\mathfrak{g}$. Then
$\mathfrak{g}=\mathfrak{h}\oplus\displaystyle{\bigoplus_{\alpha\in\Delta}}\mathfrak{g}_{\alpha}$
is the root space decomposition of $\mathfrak{g}$ with respect to
$\mathfrak{h}$. Assume that $(-,-)$ is the Killing form on $\mathfrak{g}$.

Let $\mathbb{C}[t^{\pm 1}]$ be the Laurent polynomial ring over $\mathbb{C}$.
Then the algebra $\widehat{\mathfrak{g}}$ is a central extension of loop
algebra $\mathfrak{g}\otimes\mathbb{C}[t^{\pm 1}]$, i.e.
$\widehat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t^{\pm
1}]\oplus\mathbb{C}{\bf{k}}$ with the Lie brackets

$[x(m),y(n)]=[x,y](m+n)+m(x,y)\delta_{m+n,0}{\bf{k}},\quad[\widehat{\mathfrak{g}},{\bf{k}}]=0,$
(2.1)

for all $x,y\in\mathfrak{g},m,n\in\mathbb{Z}$, where $x(m)=x\otimes t^{m}$.

Moreover, we can define the affine Kac-Moody algebra
$\widetilde{\mathfrak{g}}:=\widehat{\mathfrak{g}}\oplus\mathbb{C}d=\mathfrak{g}\otimes\mathbb{C}[t^{\pm
1}]\oplus\mathbb{C}{\bf{k}}\oplus\mathbb{C}d$, with above Lie brackets and

$[d,x(m)]=mx(m).$

Let $\widetilde{\mathfrak{h}}=\mathfrak{h}\oplus\mathbb{C}d$ be a Cartan
subalgebra of Kac-Moody Lie algebra $\widetilde{\mathfrak{g}}$.

###  2.3. Toroidal Lie algebras

Let $A=\mathbb{C}[t_{1}^{\pm 1},\cdots,t_{n}^{\pm 1}]$ be the Laurent
polynomial ring in $n$ commuting variables $t_{1},\cdots,t_{n}$ over
$\mathbb{C}$. Assume $\Omega_{A}$ is the free $A$-module with basis
${K_{1},\dots,K_{n}}$ and $dA$ is the subspace spanned by the elements
$\sum_{i=1}^{n}m_{i}t^{m}K_{i},\;m\in\mathbb{Z}^{n}$. Then
$L(\mathfrak{g})=\mathfrak{g}\otimes A$ with the brackets $[x\otimes
t^{r},y\otimes t^{s}]=[x,y]\otimes t^{r+s}$ is known as the loop algebra of
$\mathfrak{g}$ by $A$. Define $\mathcal{K}_{A}=\Omega_{A}/dA$, then it is well
known that (See [35], [36])
$\widetilde{L(\mathfrak{g})}=L(\mathfrak{g})\oplus\mathcal{K}_{A}$ is the
universal central extension of $L(\mathfrak{g})$ with the Lie brackets

$[x(a),y(b)]=[x,y](a+b)+(x,y)\sum_{i=1}^{n}a_{1}t^{a+b}K_{i},\;[L(\mathfrak{g}),\mathcal{K}_{A}]=0$
(2.2)

Let $D=\oplus_{i=1}^{n}\mathbb{C}d_{i}$ be a $n$ dimensional vectorspace. We
define the toroidal Lie algebra
$\widetilde{\mathfrak{g}}_{n}=\widetilde{L(\mathfrak{g})}\oplus D$ with the
Lie brackets (2.2) and $[d_{i},x(a)]=a_{i}x(a),\;[d_{i},d_{j}]=0$, where
$x(a)\in\widetilde{L(\mathfrak{g})}$ and $1\leq i,j\leq n$. Then it is easy to
see that $\widetilde{\mathfrak{g}}_{1}=\widetilde{\mathfrak{g}}$. We define a
Cartan subalgebra of toroidal Lie algebra by
$\widetilde{\mathfrak{h}}_{n}=\mathfrak{h}\oplus D$.

###  2.4. Full toroidal Lie algebras ([34])

Let $Der(A)$ be the space of all derivations on $A$. It has a basis consisting
of elements of the form $\\{t^{r}d_{i}:r\in\mathbb{Z}^{n},\;1\leq i\leq n\\}$.
Now define $D(u,r)=\sum_{i=1}^{n}u_{i}t^{r}d_{i}$, where $u\in\mathbb{C}^{n}$
and $r\in\mathbb{Z}^{n}$. We see that $Der(A)$ is a Lie algebra, with the Lie
brackets $[D(u,r),D(v,s)]=D(w,r+s)$, where $w=(u,s)v-(v,r)u$,
$u,v\in\mathbb{C}^{n},\;r,s\in\mathbb{Z}^{n}$. This Lie algebra is known as
Witt algebra. Every derivation $d$ of $A$ can be naturally extended to a
derivation on the tensor product $\mathfrak{g}\otimes A$ by $d(x\otimes
f)=x\otimes df$, where $x\in\mathfrak{g}$, $f\in A$ and $d$ has a unique
natural extension to $\mathcal{K}_{A}$ by

$t^{r}d_{i}(t^{s}K_{j})=s_{i}t^{(r+s)}K_{j}+\delta_{ij}\sum_{p=1}^{n}r_{p}t^{r+s}K_{p}.$

It is well known that $Der(A)$ admits two non-trivial $2$-cocyles with values
in $\mathcal{K}_{A}$ (See [2]):

$\phi_{1}(t^{r}d_{i},t^{s}d_{j})=-s_{i}r_{j}\sum_{p=1}^{n}r_{p}t^{r+s}K_{p},$
$\phi_{2}(t^{r}d_{i},t^{s}d_{j})=r_{i}s_{j}\sum_{p=1}^{n}r_{p}t^{r+s}K_{p}.$

Let $\phi$ be an arbitrary Linear combination of $\phi_{1}$ and $\phi_{2}$.
Then there is a corresponding Lie algebra

$\mathcal{L}=\mathfrak{g}\otimes A\oplus\mathcal{K}_{A}\oplus Der(A)$

with the Lie brackets 2.2 and the following:

$[t^{r}d_{i},t^{s}K_{j}]=s_{i}t^{r+s}K_{j}+\delta_{ij}\sum_{p=1}^{n}r_{p}t^{r+s}K_{p}$
$[D(u,r),D(v,s)]=D(w,r+s)+\phi(D(u,r),D(w,s))$ $[D(u,r),x(s)]=(u,s)x(r+s).$

This lie algebra is known as full toroidal Lie algebra.

### 2.5.

Let $\mathfrak{a}$ be any Lie algebra over $\mathbb{C}$ and $\mathfrak{b}$ be
any finite-dimensional abelian subalgebra of $\mathfrak{a}$. Denote
$\mathcal{M}(\mathfrak{a},\mathfrak{b})$ by the full subcategory of
$U(\mathfrak{a})$-modules consisting of objects whose restriction to
$U(\mathfrak{b})$ is a free module of rank $1$, i.e.,

$\mathcal{M}(\mathfrak{a},\mathfrak{b})=\\{M\in
U(\mathfrak{a})-\hbox{Mod}|\hbox{Res}_{U(\mathfrak{b})}^{U(\mathfrak{a})}M\cong_{U(\mathfrak{b})}U(\mathfrak{b})\\}.$

##  3\. Some Auxiliary Lemmas

In this section, we will recall several known results which are essential for
our paper.

######  Lemma 3.1.

([32], Cor.11) Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra
with a Cartan subalgebra $\mathfrak{h}$. Then
$\mathcal{M}(\mathfrak{g},\mathfrak{h})\neq\varnothing$ if and only if
$\mathfrak{g}$ is of type $A_{l}(l\geq 1)$ or $C_{l}(l\geq 2)$.

For any finite-dimensional abelian Lie algebra $Z$, set
$\tilde{\mathbf{g}}=\mathfrak{g}\oplus Z$ be the trivial central extension.
Set $\tilde{\mathfrak{h}}=\mathfrak{h}\oplus Z$, this will be again abelian
finite-dimensional Lie algebra.

######  Lemma 3.2.

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra and ${Z}$ be a
finite-dimensional Lie algebra.

  1. (1)

Suppose $\mathfrak{g}$ is of type $A_{l}(l\geq 1)$. For any
$\mathbf{a}=(a_{1},\cdots,a_{l})\in(\mathbb{C}^{*})^{l}$,
$b\in\mathcal{U}(Z)$, $\subseteq\overline{1,l+1}$, the vector space
$M(\mathbf{a},b,S)=\mathcal{U}(\tilde{\mathfrak{h}})$ becomes a
$\tilde{\mathfrak{g}}$-module under the following action: for all
$g\in\mathcal{U}(\tilde{\mathfrak{h}})$,

$H_{i}.\tilde{g}=H_{i}\tilde{g},\;i\in\mathbf{l};\hskip
28.45274ptz.\tilde{g}=z\tilde{g},\;\forall z\in Z,$

$x_{i}.\tilde{g}=a_{i}\\{\delta_{i\in S}+\delta_{i\notin
S}(H_{i}-H_{i-1}-b-1)\\}\\{\delta_{i+1\in
S}(H_{i+1}-H_{i}-b)+\delta_{i+1\notin S}\\}w_{i}(\tilde{g}),$

$y_{i}.\tilde{g}=a_{i}^{-1}\\{\delta_{i\in S}(H_{i}-H_{i-1}-b)+\delta_{i\notin
S}\\}\\{\delta_{i+1\in S}+\delta_{i+1\notin
S}(H_{i+1}-H_{i}-b-1)\\}w_{i}^{-1}(\tilde{g}),$

where $H_{l+1}=H_{0}:=0$ and $w_{i}$ is the automorphism of
$\mathcal{U}(\tilde{\mathfrak{h}})$ which sends $H_{i}$ to $H_{i}-1$ and fixes
other $H_{j},\;j\neq i$ and $Z$. Moreover,

$\mathcal{M}(\tilde{\mathfrak{g}},\tilde{\mathfrak{h}})=\\{M(a,b,S):a\in(\mathbb{C}^{*})^{l},\;b\in\mathcal{U}(Z),S\subseteq\overline{1,l+1}\\}.$

  2. (2)

$\mathcal{M}(\tilde{\mathfrak{g}},\tilde{\mathfrak{h}})=$ if and only if
$\mathcal{M}(\mathfrak{g},\mathfrak{h})=\varnothing$.

Now suppose $\mathfrak{g}$ is of type $C_{l}(l\geq 2)$. Now for any
$\mathbf{a}\in(\mathbb{C}^{*})^{l},\;S\subseteq\overline{1,l}$, let us recall
the $\tilde{\mathfrak{g}}$-module
$M(\mathbf{a},S)=\mathcal{U}(\tilde{\mathfrak{h}})$ from [8] with the
following actions:

$H_{i}.\tilde{g}=H_{i}\tilde{g},\;i\in\mathbf{l};\hskip
28.45274ptz.\tilde{g}=z\tilde{g},\;\forall z\in Z,$
$x_{k}.\tilde{g}=a_{k}\\{\delta_{k\in S}+\delta_{k\notin
S}(H_{k}-H_{k-1}-b-\frac{1}{2})\\}\\{\delta_{k+1\in
S}((1+\delta_{k,l-1})H_{k+1}-H_{k}+\frac{1}{2})+\delta_{k+1\notin
S}\\}w_{k}(\tilde{g}),$

$y_{k}.\tilde{g}=a_{k}^{-1}\\{\delta_{k\in
S}(H_{k}-H_{k-1}-\frac{1}{2})+\delta_{k\notin S}\\}\\{\delta_{k+1\in
S}+\delta_{k+1\notin
S}((1+\delta_{k,l-1})H_{k+1}-H_{k}-\frac{1}{2})\\}w_{k}^{-1}(\tilde{g}),$

$x_{l}.\tilde{g}=a_{l}\\{\delta_{l\in S}-\delta_{l\notin
S}(H_{l}-\frac{1}{2}H_{l-1}-\frac{3}{4})(H_{l}-\frac{1}{2})H_{l-1}-\frac{1}{4}\\}w_{l}(\tilde{g}),$

$y_{l}.\tilde{g}=a_{l}^{-1}\\{-\delta_{l\in
S}(H_{l}-\frac{1}{2}H_{l-1}+\frac{3}{4})(H_{l}-\frac{1}{2}H_{l-1}+\frac{1}{4})+\delta_{l\notin
S}\\}w_{l}^{-1}(\tilde{g}),$

where $\tilde{g}\in\mathcal{U}(\tilde{\mathfrak{h}}),\;1\leq k\leq l-1$,
$H_{0}:=0$ and $w_{i}$ is the automorphism of
$\mathcal{U}(\tilde{\mathfrak{h}})$ which sends $H_{i}$ to $H_{i}-1$ and fixes
other $H_{j},\;j\neq i$ and $Z$.

######  Lemma 3.3.

([7], Lemma 4) Let $\mathfrak{g}=\mathfrak{sp}_{2l}(l\geq 2)$ with a Cartan
subalgebra $\mathfrak{h}$. For a finite-dimensional abelian Lie algebra $Z$,
we have
$\mathcal{M}(\tilde{\mathfrak{g}},\tilde{\mathfrak{h}})=\\{M(\mathbf{a},S):\mathbf{a}\in(\mathbb{C}^{*})^{l},\;S\subseteq\overline{1,l}\\}$.

######  Lemma 3.4.

([7], [31], [32])

  1. (1)

$M(\mathbf{a},b,S)\in\mathcal{M}(\mathfrak{g}(A_{l}),\mathfrak{h})$ is simple
if and only if $(l+1)b\notin\mathbb{Z}_{+}$ or $S\neq\emptyset$ or
$S\neq\overline{1,l+1}$

  2. (2)

Any module in $\mathcal{M}(\mathfrak{g}(C_{l}),\mathfrak{h})$ is simple.

######  Lemma 3.5.

([8]), Lemma 2) Let $\mathbb{C}[T_{1},\dots,T_{m}]$ be a polynomial algebra in
$m$ indeterminates $T_{1},\dots,T_{m}$ over $\mathbb{C}$ and $w_{i}$ be the
automorphism of $\mathbb{C}[T_{1},\dots,T_{m}]$ which sends $T_{i}$ to
$T_{i}-1$ and fixes $T_{j}$ for $j\neq i$. For any
$g\in\mathbb{C}[T_{1},\dots,T_{m}],\;k\in\mathbb{Z}\setminus\\{0\\}$ and
$k^{\prime}\in\mathbb{Z}_{+}$ we have

$\hbox{deg}_{T_{i}}(w_{i}^{k}-\hbox{Id})(g)=\hbox{deg}_{T_{i}}(g)-1,$

and

$\hbox{deg}_{T_{i}}(w_{i}-\hbox{Id})^{k^{\prime}}(g)=\begin{cases}\hbox{deg}_{T_{i}}(g)-k^{\prime}&\hbox{if}\;k^{\prime}\leq
deg_{T_{i}}(g)\\\ -1&\hbox{if}\;k^{\prime}>deg_{T_{i}}(g),\end{cases}$

where $\hbox{deg}_{T_{i}}(0)=-1$

##  4\. Cartan Free modules over $\widetilde{\mathfrak{g}}_{n}$

In this section, we will study the categories
$\mathcal{M}(\widetilde{\mathfrak{g}}_{n},\widetilde{\mathfrak{h}}_{n})$ for
toroidal Lie algebras $\widetilde{\mathfrak{g}}_{n}$.

We see that $\widetilde{\mathfrak{h}}_{n}$ is an abelian Lie algebra, so
$\mathcal{U}(\widetilde{\mathfrak{h}}_{n})=\mathbb{C}[H_{1},\dots,H_{l},d_{1},\dots,d_{n}]$.
Now define some automorphisms of
$\mathbb{C}[H_{1},\dots,H_{l},d_{1},\dots,d_{n}]$ as follows: for any
$\tilde{g}(H_{1},\dots,H_{l},d_{1},\dots,d_{n})\in\mathcal{U}(\widetilde{\mathfrak{h}}_{n}),\;1\leq
i\leq l,\;1\leq j\leq n$,

$\sigma_{i}:\mathcal{U}(\widetilde{\mathfrak{h}}_{n})\rightarrow\mathcal{U}(\widetilde{\mathfrak{h}}_{n}),\;\tilde{g}(H_{1},\dots,H_{l},d_{1},\dots,d_{n})\mapsto\tilde{g}(H_{1},\dots,H_{i}-1,\dots,H_{l},d_{1},\dots,d_{n});$
$\tau_{j}:\mathcal{U}(\widetilde{\mathfrak{h}}_{n})\rightarrow\mathcal{U}(\widetilde{\mathfrak{h}}_{n}),\;\tilde{g}(H_{1},\dots,H_{l},d_{1},\dots,d_{n})\mapsto\tilde{g}(H_{1},\dots,H_{l},d_{1},\dots,d_{j}-1,\dots,d_{n}).$

Note that $\\{\sigma_{i},\tau_{j}:1\leq i\leq l,\;1\leq j\leq n\\}$ is a
commuting family of automorphisms of
$\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$.
Now we define
$\mathcal{P}_{H}:=\mathbb{C}[H_{1},\dots,H_{l}],\;\mathcal{P}_{D}:=\mathbb{C}[d_{1},\dots,d_{n}]$
and
$\mathcal{P}_{i}=\mathbb{C}[H_{1},\dots,H_{i-1},H_{i+1},\dots
H_{l},d_{1},\dots,d_{n}],\;\mathcal{P}^{j}=\mathbb{C}[H_{1},\dots,H_{l},d_{1},\dots,d_{j-1},d_{j+1},\dots
d_{n}],$ for $1\leq i\leq l,\;1\leq j\leq n$. Let
$\mathbf{a}=(a_{1})\in\mathbb{Z}^{n}$ and for any
$\mathbf{\lambda}=(\lambda_{1},\dots,\lambda_{n})\in(\mathbb{C}^{*})^{n}$ we
define
$\mathbf{\lambda}^{\mathbf{a}}=\lambda_{1}^{a_{1}}\dots\lambda_{n}^{a_{n}}\in\mathbb{C}^{*}$
and
$\tau^{\mathbf{a}}=\tau_{1}^{a_{1}}\dots\tau_{n}^{a_{n}}\in\hbox{Aut}(\mathcal{U}(\widetilde{\mathfrak{h}}_{n}))$.

For any $M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h})$, we have
$M_{\mathfrak{g}}=\mathcal{U}(\mathfrak{h})\subset\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$.
Let $\mathbf{\lambda}\in(\mathbb{C}^{*})^{n}$. For any
$\tilde{g}\in\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$ and
$a\in\mathbb{Z}^{n}$, we define

$\begin{cases}t^{a}K_{j}.\tilde{g}=0,\;d_{j}.\tilde{g}=d_{j}\tilde{g}\\\
h(a).\tilde{g}=\mathbf{\lambda}^{a}h\tau^{a}(\tilde{g})\\\
x_{i}(a).\tilde{g}=\mathbf{\lambda}^{a}\sigma_{i}\tau^{a}(\tilde{g})(x_{i}.1)\\\
y_{i}(a).\tilde{g}=\mathbf{\lambda}^{a}\sigma_{i}^{-1}\tau^{a}(\tilde{g})(y_{i}.1)\end{cases}$
(4.1)

where $i\in\overline{1,l}$ and $j\in\overline{1,n}$ and $x_{i}.1,y_{i}.1$ are
determined by the $\mathfrak{g}$-module structure on $M_{\mathfrak{g}}$. It is
easy to see that (4.1) defines a
$\mathcal{U}(\widetilde{\mathfrak{g}}_{n})$-module structure on
$\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$. We denote the corresponding
module as
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$.
It is easy to see that
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})\in\mathcal{M}(\widetilde{\mathfrak{g}}_{n},\widetilde{\mathfrak{h}}_{n})$.
We see that as vector space
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})\cong
M_{\mathfrak{g}}\otimes\mathcal{P}_{D}$. We can also define a
$\widetilde{\mathfrak{g}}_{n}$-module structure on
$M_{\mathfrak{g}}\otimes\mathcal{P}_{D}$ by similar actions as in 4.1. In this
case we will have
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})\cong
M_{\mathfrak{g}}\otimes\mathcal{P}_{D}$ as
$\widetilde{\mathfrak{g}}_{n}$-module.

######  Proposition 4.1.

The $\widetilde{\mathfrak{g}}_{n}$-module
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$
is simple if and only if the corresponding $\mathfrak{g}$-module
$M_{\mathfrak{g}}$ is simple.

###### Proof.

Suppose $M_{\mathfrak{g}}$ is not simple as $\mathfrak{g}$-module and $N$ be a
non-zero proper submodule of $M_{\mathfrak{g}}$. Then by (4.1) we see that
$N\otimes\mathcal{P}_{D}$ is a nonzero proper
$\widetilde{\mathfrak{g}}_{n}$-module of
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$.
Conversely assume $M_{\mathfrak{g}}$ is a simple $\mathfrak{g}$-module. Let
$W$ be a nonzero submodule of $M_{\mathfrak{g}}\otimes\mathcal{P}_{D}$.
Claim: There exists $w\in W$ such that $\hbox{deg}_{d_{j}}(w)=0$ for all
$1\leq j\leq n$.
Let $w_{1}\in W$ be a non-zero element with a minimal degree in $d_{1}$. Now
for any $h\in\mathfrak{h}$, we see that
$\hbox{deg}_{d_{1}}(h(e_{1})-\lambda_{1}h)(w_{1})<\hbox{deg}_{d_{1}}(w_{1})$,
hence $\hbox{deg}_{d_{1}}(w_{1})=0$. Now let $w_{2}\in W$ be a nonzero element
from the set $\\{w\in W:\hbox{deg}_{d_{1}}(w)=0\\}$ with minimal degree in
$d_{2}$. Now again for any $h\in\mathfrak{h}$, we have
$\hbox{deg}_{d_{2}}(h(e_{2})-\lambda_{2}h)(w_{2})<\hbox{deg}_{d_{2}}(w_{2})$,
which forces $\hbox{deg}_{d_{2}}(w_{2})=0$. Continuing this process we find
$w\in W$ such that $\hbox{deg}_{d_{j}}(w)=0$ for all $1\leq j\leq n$.
Hence $w\in M_{\mathfrak{g}}$, which implies
$\mathcal{U}(\mathfrak{g})w=M_{\mathfrak{g}}$. Note that
$M_{\mathfrak{g}}\otimes\mathcal{P}_{D}\subset W$, we obtain that
$W=M_{\mathfrak{g}}\otimes\mathcal{P}_{D}$. So the
$\widetilde{\mathfrak{g}}_{n}$-module
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$
is simple. ∎

From this Proposition 4.1 and Lemma 3.4, we have the following Corollary.

######  Corollary 4.2.

  1. (1)

The $\widetilde{\mathfrak{g}}_{n}(A_{l})$-module
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}(A_{l})}(\mathbf{\lambda},M(\mathbf{a},b,S))$
is simple if and only if $(l+1)b\notin\mathbb{Z}_{+}$ or $1\leq|S|\leq l.$

  2. (2)

$\widetilde{\mathfrak{g}}_{n}(C_{l})$-module
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}(C_{l})}(\mathbf{\lambda},M_{\mathfrak{g}(C_{l})}$
is simple for any $\mathbf{\lambda}\in(\mathbb{C}^{*})^{n}$ and
$M_{\mathfrak{g}(C_{l})}\in\mathcal{M}(\mathfrak{g}(C_{l}),\mathfrak{h})$.

######  Remark 4.3.

If
$M\in\mathcal{M}(\widetilde{\mathfrak{g}}_{n},\widetilde{\mathfrak{h}}_{n})$,
then
$\hbox{Res}_{\tilde{\mathfrak{g}}}^{\widetilde{\mathfrak{g}}_{n}}(M)\in\mathcal{M}(\tilde{\mathfrak{g}},\tilde{\mathfrak{h}})$,
where $\tilde{\mathfrak{g}}=\mathfrak{g}\oplus D$ and
$\tilde{\mathfrak{h}}=\mathfrak{h}\oplus D$. Then from Lemma 3.2, we have
$\mathcal{M}(\widetilde{\mathfrak{g}}_{n},\widetilde{\mathfrak{h}}_{n})\neq\varnothing$
only for type $A_{l}(l\geq 1)$ and $C_{l}(l\geq 2)$.

######  Theorem 4.4.

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra of type $A_{l}(l\geq
1)$ or of type $C_{l}(l\geq 2)$, then

$\mathcal{M}(\widetilde{\mathfrak{g}}_{n},\widetilde{\mathfrak{h}}_{n})=\\{\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}}):\mathbf{\lambda}\in(\mathbb{C}^{*})^{n},M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h})\\}.$

To prove this theorem it is enough to prove that for any
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}\in\mathcal{M}(\widetilde{\mathfrak{g}}_{n},\widetilde{\mathfrak{h}}_{n})$
there exists a pair
$(\mathbf{\lambda},M_{\mathfrak{g}})\in(\mathbb{C}^{*})^{n}\times\mathcal{M}(\mathfrak{g},\mathfrak{h})$,
such that
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}=\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$.
We have the following easy observations.

######  Lemma 4.5.

Suppose
$\tilde{g}\in\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}},\;a\in\mathbb{Z}^{n}$,
$i\in\overline{1,l},\;j\in\overline{1,n}$. Then we have

$\begin{cases}t^{a}K_{j}.(\tilde{g})=\tau^{a}(\tilde{g})(t^{a}K_{j}.1)\\\
h(a).(\tilde{g})=\tau^{a}(\tilde{g})(h(a).1)\\\
x_{i}(a).(\tilde{g})=(\sigma_{i}\tau^{a}(\tilde{g}))(x_{i}(a).1)\\\
y_{i}(a).(\tilde{g})=(\sigma_{i}^{-1}\tau^{a}(\tilde{g}))(y_{i}(a).1)\end{cases}$

Lemma 4.5 shows that to prove
$\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}=\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$
for some $\mathbf{\lambda}\in(\mathbb{C}^{*})^{n}$ and
$M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h})$ it is enough to
prove the following two assertions:
Assertion (A): $t^{a}K_{j}.1=0$ for $j\in\overline{1,n}$.
Assertion (B): $X(a).1=\mathbf{\lambda}^{a}(X.1)$ for some
$\mathbf{\lambda}\in(\mathbb{C}^{*})^{n}$ and $X.1$ is determined by some
$\mathcal{U}(\mathfrak{g})$-module
$M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h})$, where
$X=x_{i},y_{i},h_{i}$ for all $i\in\overline{1,l}$.
We will prove these two assertions for type $A_{l}(l\geq 1)$ and the same
method is valid for type $C_{l}(l\geq 2)$.

######  Lemma 4.6.

For $\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}$, we have

  1. (1)

$x_{i}(e_{j}).1\neq 0,\;y_{i}(e_{j}).1\neq 0$ for all $i\in\overline{1,l}$ and
$j\in\overline{1,n}$.

  2. (2)

There exists $a_{j}\in\mathbb{C}$ such that $K_{j}.1=a_{j}$.

###### Proof.

The proof is similar to Lemma 12 of [7]. Just note that we need to run the
same Lemma $n$ times. ∎

######  Lemma 4.7.

  1. (1)

In (3.2), $b\in\mathbb{C}$, hence
$x_{i}.1,y_{i}.1\in\mathcal{U}(\mathfrak{h})$.

  2. (2)

$K_{j}.1=0$, for all $j\in\overline{1,n}$.

  3. (3)

There exists $\lambda_{i}^{j}\in\mathbb{C}^{*}$ such that

$x_{i}(-e_{j}).1=\frac{1}{\lambda_{i}^{j}}(x_{i}.1),\;y_{i}(e_{j}).1=\lambda_{i}^{j}(f_{i}.1).$

###### Proof.

The proof is similar to Lemma 13 of [7]. ∎

######  Lemma 4.8.

  1. (1)

$h_{i}(\pm e_{j}).1=(\lambda_{i}^{j})^{\pm 1}h_{i}.1.$

  2. (2)

$\lambda_{1}^{j}=\lambda_{2}^{j}=\cdots=\lambda_{l}^{j}$ for all
$j\in\overline{1,n}$.

  3. (3)

For any $a\in\mathbb{Z}^{n}$ and $x\in\mathfrak{g}$,we have
$x(a)=\mathbf{\lambda}^{a}(x.1)$ for some
$\mathbf{\lambda}\in(\mathbb{C}^{*})^{n}$.

###### Proof.

From Lemma 14 of [7], we can prove the first two parts of the proposition
together with $x(re_{j})=(\lambda^{j})^{r}(x.1)$, where
$\lambda^{j}=\lambda_{i}^{j}\in\mathbb{C}^{*}$, $r\in\mathbb{Z}$ and
$x\in\mathfrak{g}$.
Now we have
$x_{i}(r_{j}e_{j}+r_{k}e_{k})=[H_{i}(r_{j}e_{j}),x_{i}(r_{k}e_{k})]$, where
$1\leq i\leq l$ and $1\leq j\neq k\leq n$, therefore

$\displaystyle x_{i}(r_{j}e_{j}+r_{k}e_{k}).1$
$\displaystyle=H_{i}(r_{j}e_{j}).x_{i}(r_{k}e_{k}).1-x_{i}(r_{k}e_{k}).H_{i}(r_{j}e_{j}).1$
$\displaystyle=\lambda_{k}^{r_{k}}H_{i}(r_{j}e_{j})(x_{i}.1)-\lambda_{j}^{r_{j}}x_{i}(r_{k}e_{k})(H_{i}.1)$
$\displaystyle=\lambda_{k}^{r_{k}}(\tau_{j}^{r_{j}}(x_{i}.1))H_{i}(r_{j}e_{j}).1-\lambda_{j}^{r_{j}}(\sigma_{i}\tau_{k}^{r_{k}}(H_{i}.1))(x_{i}(r_{k}e_{k}).1)$
$\displaystyle=\lambda_{i}^{r_{j}}\lambda_{k}^{r_{k}}x_{i}.1$

Similarly from the Lie brackets
$y_{i}(r_{j}e_{j}+r_{k}e_{k})=[y_{i}(r_{j}e_{j}),H_{i}(r_{k}e_{k})]$, where
$1\leq i\leq l$ and $1\leq j\neq k\leq n$, we prove
$y_{i}(r_{j}e_{j}+r_{k}e_{k})=\lambda_{i}^{r_{j}}\lambda_{k}^{r_{k}}y_{i}.1$

Proceeding inductively we can prove $x_{i}(r).1=\lambda^{r}x_{i}.1$ and
$y_{i}(r).1=\lambda^{r}y_{i}.1$ for any $r\in\mathbb{Z}^{n}$.
Now

$\displaystyle H_{i}(r).1$ $\displaystyle=[x_{i}(0),y_{i}(r)].1$
$\displaystyle=x_{i}(0)y_{i}(r).1-y_{i}(r)x_{i}(0).1$
$\displaystyle=\lambda^{r}x_{i}.y_{i}.1-\sigma_{i}^{-1}\tau^{r}(x_{i}.1)y_{i}(r).1$
$\displaystyle=\lambda^{r}\\{x_{i}.y_{i}.1-\sigma_{i}^{-1}(x_{i}.1)y_{i}.1\\}$
$\displaystyle=\lambda^{r}\\{x_{i}.y_{i}.1-y_{i}.x_{i}.1\\}$
$\displaystyle=\lambda^{r}H_{i}.1,$

hence we have part 3 of our Lemma. ∎

######  Lemma 4.9.

$t^{a}K_{j}.1=0$ for $j\in\overline{1,n}$ and $0\neq a\in\mathbb{Z}^{n}$.

###### Proof.

We have
$t^{a}K_{j}.1=[x_{i}(e_{j}),y_{i}(a-e_{j})].1-H_{i}(a).1=x_{i}(e_{j}).y_{i}(a-e_{j}).1-y_{i}(a-e_{j}).x_{i}(e_{j}).1-\lambda^{a}H_{i}.1=\lambda^{(a-e_{j})}x_{i}(e_{j})(y_{i}.1)-\lambda_{j}y_{i}(a-e_{j}).(x_{i}.1)-\lambda^{a}H_{i}=\lambda^{(a-e_{j})}\sigma_{i}\tau_{j}(y_{i}.1)(x_{i}(e_{j}).1)-\lambda_{j}\sigma_{i}^{-1}\tau^{(a-e_{j})}(x_{i}.1)(y_{i}(a-e_{j}).1)-\lambda^{a}H_{i}=\lambda^{a}\sigma_{i}(y_{i}.1)(x_{i}.1)-\lambda^{a}\sigma_{i}^{-1}(x_{i}.1)(y_{i}.1)-\lambda^{a}H_{i}=0$.
∎

The proof of Theorem 4.4 will follow from, Lemma 4.5, Lemma 4.6, Lemma 4.7,
Lemma 4.8 and Lemma 4.9.

##  5\. Cartan free modules over full toroidal Lie algebras

### 5.1.

In this section, we will consider the categories
$\mathcal{M}(\mathcal{L},\widetilde{\mathfrak{h}}_{n})$. First, we construct a
class of modules in this category. Suppose
$M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h}),\;\Lambda\in(\mathbb{C}^{*})^{n}$
and $a\in\mathbb{C}$. For any $g\in\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$
and $r\in\mathbb{Z}^{n}$, we define the following action:

$\begin{cases}t^{r}K_{j}.\tilde{g}=0,\\\
h(a).\tilde{g}=\mathbf{\lambda}^{a}h\tau^{a}(\tilde{g})\\\
x_{i}(a).\tilde{g}=\mathbf{\lambda}^{a}\sigma_{i}\tau^{a}(\tilde{g})(x_{i}.1)\\\
y_{i}(a).\tilde{g}=\mathbf{\lambda}^{a}\sigma_{i}^{-1}\tau^{a}(\tilde{g})(y_{i}.1)\\\
t^{r}d_{i}.\tilde{g}=\lambda^{r}\tau^{r}(\tilde{g})(d_{i}-r_{i}(a+1)).\end{cases}$
(5.1)

We see that with the above actions $\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$
becomes $\mathcal{L}$-module where $\widetilde{\mathfrak{h}}_{n}$ acts freely
on $\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$. We denote this module by
$\widehat{M}_{\mathcal{L}}(\Lambda,a,M_{\mathfrak{g}})$.

######  Proposition 5.1.

The $\mathcal{L}$-module
$\widehat{M}_{\mathcal{L}}(\Lambda,a,M_{\mathfrak{g}})$ is simple if and only
if the corresponding $\mathfrak{g}$-module $M_{\mathfrak{g}}$ is simple.

###### Proof.

The proof is same as Proposition 4.1. ∎

######  Corollary 5.2.

  1. (1)

The $\mathcal{L}(A_{l})$-module
$\widehat{M}_{\tau(A_{l})}(\mathbf{\lambda},a,M(\mathbf{b},c,S))$ is simple if
and only if $(l+1)c\notin\mathbb{Z}_{+}$ or $1\leq|S|\leq l.$

  2. (2)

$\mathcal{L}(C_{l})$-module
$\widehat{M}_{\tau(C_{l})}(\mathbf{\lambda},a,M_{\mathfrak{g}(C_{l})})$ is
simple for any $\mathbf{\lambda}\in(\mathbb{C}^{*})^{n}$ and
$M_{\mathfrak{g}(C_{l})}\in\mathcal{M}(\mathfrak{g}(C_{l}),\mathfrak{h})$.

Suppose $M\in\mathcal{M}(\mathcal{L},\widetilde{\mathfrak{h}}_{n})$. Note that
in order to understand the module structure on $M$, we need first to
understand the Cartan free modules of extension of Witt algebras by finite-
dimensional abelian Lie algebras.
Let $\mathcal{W}_{n}=Der(A)$ be the Witt algebra. In [38], the authors defined
a class of $\mathcal{W}_{n}$ modules as follows. For $a\in\mathbb{C}$ and
$\Lambda_{n}=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in(\mathbb{C}^{*})^{n}$,
denote $\Omega(\Lambda_{n},a)=\mathbb{C}[d_{1},d_{2},\cdots,d_{n}]$ the
polynomial algebra over $\mathbb{C}$ in commuting indeterminates
$d_{1},d_{2},\cdots d_{n}$. The action of $\mathcal{W}_{n}$ on
$\Omega(\Lambda_{n},a)$ is defined by

$t^{k}d_{i}.f(d_{1},\cdots,d_{n})=\Lambda_{n}^{k}(d_{i}-k_{i}(a+1))f(d_{1}-k_{1},\cdots,d_{n}-k_{n}),$
(5.2)

where
$k=(k_{1},k_{2},\cdots,k_{n})\in\mathbb{Z}^{n},f\in\mathbb{C}[d_{1},d_{2},\cdots,d_{n}],\Lambda_{n}^{k}=\lambda_{1}^{k_{1}}\lambda_{2}^{k_{2}}\cdots\lambda_{n}^{k_{n}},i=1,2,\cdots,n$.
Now we recall a result from [37].

######  Theorem 5.3.

(Theorem 9, [37]) Let $M\in\mathcal{M}(\mathcal{W}_{n},D)$, where $D$ is the
$\mathbb{C}$-linear span of $\\{d_{1},d_{2},\dots d_{n}\\}$. Then
$M\cong\Omega(\Lambda_{n},a)$ for some $\Lambda_{n}\in(\mathbb{C}^{*})^{n}$
and some $a\in\mathbb{C}$.

Let $\widetilde{\mathcal{W}_{n}}=\mathcal{W}_{n}\oplus Z$ and
$\widetilde{D}=D\oplus Z$, where $Z$ is any finite-dimensional abelian Lie
algebra. Now our aim is to understand the category
$\mathcal{M}(\widetilde{\mathcal{W}_{n}},\widetilde{D})$. Now we will follow
the idea of Lemma 13 from [8].
Let $M\in\mathcal{M}(\widetilde{\mathcal{W}_{n}},\widetilde{D})$. By
assumption $[\mathcal{W}_{n},Z]=0$, so it is easy to see that
$\widetilde{\mathcal{W}_{n}}$ module structure on $M$ is determined by the
actions of $t^{r}d_{i}$ on $1$, where $r\in\mathbb{Z}^{n}$ and $1\leq i\leq
n$.
Define a new Lie algebra
$\widehat{\mathcal{W}_{n}}:=\mathcal{U}(Z)\otimes_{\mathbb{C}}\mathcal{W}_{n}\subset\mathcal{U}(\widetilde{\mathcal{W}_{n}})$
over the PID $\mathcal{U}(Z)$. Then $M=\mathcal{U}(\widetilde{D})$ becomes a
$\widehat{\mathcal{W}_{n}}$-module over $\mathcal{U}(Z)$ where the action of
$\widehat{\mathcal{W}_{n}}$ on $\mathcal{U}(\widetilde{D})$ inherits from the
action of $\widetilde{\mathcal{W}_{n}}$ on $M$.
Now let us extend the base field. Suppose $\mathbb{C}(Z)$ be the field of
fraction of $\mathcal{U}(Z)$ and $\overline{\mathbb{C}(Z)}$ be the algebraic
closed extension field of $\mathbb{C}(Z)$. Let

$G=\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\widehat{\mathcal{W}_{n}}\cong\overline{\mathbb{C}(Z)}\otimes_{\mathbb{C}}\mathcal{W}_{n}$

with the Cartan subalgebra
$\mathfrak{D}\cong\overline{\mathbb{C}(Z)}\otimes_{\mathbb{C}}D$. Note that
$\mathcal{U}(\widetilde{D})\cong\mathcal{U}(Z)\otimes_{\mathbb{C}}\mathcal{U}(D)$,
we have vector space isomorphism over $\overline{\mathbb{C}(Z)}$ as follows:

$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})\cong\overline{\mathbb{C}(Z)}\otimes_{\mathbb{C}}\mathcal{U}(D)\cong\overline{\mathbb{C}(Z)}[\mathfrak{D}]=\mathcal{U}(\mathfrak{D}),$

where the last term is the universal enveloping algebra of $\mathfrak{D}$ over
$\overline{\mathbb{C}(Z)}$. So we can identify $\mathcal{U}(\mathfrak{D})$
with with
$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})$.
Now it is natural to extend the module action of $\widehat{\mathcal{W}_{n}}$
on $\mathcal{U}(\widetilde{D})$ to the module action of $G$ on
$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})$
by

$\phi_{1}\otimes\phi_{2}\otimes
x\circ\psi_{1}\otimes\psi_{2}g=\phi_{1}\psi_{1}\otimes(\phi_{2}\otimes
x.\psi_{2}g),$

where
$\phi_{1},\psi_{1}\in\overline{\mathbb{C}(Z)},\;\phi_{2},\psi_{2}\in\mathcal{U}(Z),\;x\in\mathcal{W}_{n},\;g\in\mathcal{U}(D)$.
We see that the action of $\mathfrak{D}$ on
$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})$
is just multiplication, so the $G$-module
$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})$
is inside $\mathcal{M}(G,\mathfrak{D})$, where all the modules are over
$\overline{\mathbb{C}(Z)}$.
Now the method used in [37] to prove Theorem 9 is valid for the $G$-module
$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})$.
So
$\overline{\mathbb{C}(Z)}\otimes_{\mathcal{U}(Z)}\mathcal{U}(\widetilde{D})\cong\Omega(\Lambda_{n},a)$
for some $\Lambda_{n}\in((\overline{\mathbb{C}(Z)})^{*})^{n}$ and some
$a\in\overline{\mathbb{C}(Z)}$.
Note that the action of $\mathcal{W}_{n}\subset G$ on
$\mathcal{U}(\widetilde{D})$ coincide with the action of $\mathcal{W}_{n}$ on
the $\widetilde{\mathcal{W}_{n}}$-module $M=\mathcal{U}(\widetilde{D})$, we
must have $t^{r}d_{i}\circ 1\in\mathcal{U}(\widetilde{D})$ for
$r\in\mathbb{Z}^{n},\;1\leq i\leq n$, which forces that
$a,\lambda_{1},\lambda_{2},\dots,\lambda_{n}\in\mathcal{U}(Z)$. Again from the
action 5.2, we see that
$\lambda_{1}^{-1},\lambda_{2}^{-1}\dots,\lambda_{n}^{-1}\in\mathcal{U}(Z)$. So
$\Lambda_{n}\in(\mathbb{C}^{*})^{n}$ and $a\in\mathcal{U}(Z)$ and we have the
following Proposition.

######  Proposition 5.4.

Let $M\in\mathcal{M}(\widetilde{\mathcal{W}_{n}},\widetilde{D})$. Then
$M\cong\Omega(\Lambda_{n},a)$ for some $\Lambda_{n}\in(\mathbb{C}^{*})^{n}$
and $a\in\mathcal{U}(Z)$.

Suppose $M\in\mathcal{M}(\mathcal{L},\widetilde{\mathfrak{h}}_{n})$. From the
Lie brackets of Full toroidal Lie algebras, we see that for any
$\tilde{g}\in\mathcal{U}(\widetilde{\mathfrak{h}}_{n})$:

$\begin{cases}t^{a}K_{j}.(\tilde{g})=\tau^{a}(\tilde{g})(t^{a}K_{j}.1),\\\
h(a).(\tilde{g})=\tau^{a}(\tilde{g})(h(a).1),\\\
x_{i}(a).(\tilde{g})=(\sigma_{i}\tau^{a}(\tilde{g}))(x_{i}(a).1),\\\
y_{i}(a).(\tilde{g})=(\sigma_{i}^{-1}\tau^{a}(\tilde{g}))(y_{i}(a).1),\\\
D(u,r).(\tilde{g})=\tau^{r}(\tilde{g})D(u,r).1\end{cases}$

where $1\leq i\leq l$, $j\in\overline{1,n}$, $u\in\mathbb{C}^{n}$ and
$r\in\mathbb{Z}^{n}$.

Note that
${Res}^{\mathcal{L}}_{\widetilde{\mathfrak{g}_{n}}}M\in\mathcal{M}(\mathcal{L},\widetilde{\mathfrak{h}}_{n})$,
there fore we have $t^{a}K_{j}.1=0$, where $1\leq j\leq n$. Now from Theorem
4.4, we have
${Res}^{\mathcal{L}}_{\widetilde{\mathfrak{g}_{n}}}M\cong\widetilde{M}_{\widetilde{\mathfrak{g}}_{n}}(\mathbf{\lambda},M_{\mathfrak{g}})$,
for some
$\mathbf{\lambda}\in(\mathbb{C}^{*})^{n},M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h})$.

Also
$Res^{\mathcal{L}}_{\widetilde{\mathcal{W}_{n}}}M\in\mathcal{M}(\widetilde{\mathcal{W}_{n}},\widetilde{D})$,
where $\widetilde{\mathcal{W}_{n}}=\mathcal{W}_{n}\oplus\mathfrak{h}_{n}$ and
$\widetilde{D}=D\oplus\mathfrak{h}_{n}.$ So from Proposition 5.4, we see that
$Res^{\mathcal{L}}_{\widetilde{\mathcal{W}_{n}}}M\cong\Omega(\gamma,a)$ for
some $\gamma\in(\mathbb{C}^{*})^{n}$ and $a\in\mathcal{U}(Z)$.
Claim: $a\in\mathbb{C}$.
We know that $[t^{r}d_{j},x_{i}]=0$ for all $r\in\mathbb{Z}^{n}$, $1\leq j\leq
n$ and $1\leq i\leq l$. Therefore we will have the following:

$\displaystyle 0$ $\displaystyle=t^{r}d_{j}.x_{i}.1-x_{i}.t^{r}d_{j}.1$
$\displaystyle=\tau^{r}(x_{i}.1)t^{r}d_{i}.1-x_{i}.t^{r}d_{j}.1$
$\displaystyle=\gamma^{r}(x_{i}.1)(d_{j}-r_{j}(a+1))-x_{i}.(t^{r}d_{j}.1)$
$\displaystyle=\gamma^{r}\\{(x_{i}.1)(d_{j}-r_{j}(a+1))-\sigma_{i}(d_{j}-r_{j}(a+1))(x_{i}.1)\\}$

From the above equation, we see that if we choose $r\in\mathbb{Z}^{n}$ such
that $r_{j}\neq 0$, we will have $a-\sigma_{i}(a)=0$ for all $1\leq i\leq l$.
So $a\in\cap_{i=1}^{l}\mathcal{P}_{i}=\mathcal{P}_{D}$. We also know that
$a\in\mathcal{U}(\mathfrak{h}_{n})=\mathcal{P}_{H}$, therefore
$a\in\mathbb{C}$. Now consider the relation
$[t^{r}d_{j},x_{i}(s)].1=s_{j}x_{i}(r+s).1$, which will give us the following:

$\displaystyle s_{j}\Lambda_{n}^{r+s}x_{i}.1$
$\displaystyle=t^{r}d_{j}.x_{i}(s).1-x_{i}(s).t^{r}d_{j}.1$
$\displaystyle=\Lambda_{n}^{s}t^{r}d_{j}.x_{i}.1-x_{i}(s).\gamma^{r}(d_{j}-s_{j}(a+1))$
$\displaystyle=\Lambda_{n}^{s}\tau^{r}\\{(x_{i}.1)\\}t^{r}d_{j}.1-\gamma^{r}\sigma_{i}\tau^{s}\\{(d_{j}-s_{j}(a+1))\\}x_{i}(s).1$
$\displaystyle=\Lambda_{n}^{s}(x_{i}.1)\gamma^{r}(d_{j}-r_{j}(a+1))-\Lambda_{n}^{s}\gamma^{r}(d_{j}-s_{j}-s_{j}(a+1))\
=s_{j}\Lambda_{n}^{s}\gamma^{r}(x_{i}.1).$

From the above equation, if we choose $s\in\mathbb{Z}^{n}$ such that
$s_{j}\neq 0$, then we see that $\Lambda_{n}^{r}=\gamma^{r}$ for all
$r\in\mathbb{Z}^{n}$. Therefore we will have $\Lambda_{n}=\gamma$.

######  Theorem 5.5.

Let $M\in\mathcal{M}(\mathcal{L},\widetilde{\mathfrak{h}}_{n})$, then
$M\cong\widehat{M}_{\mathcal{L}}(\Lambda,a,M_{\mathfrak{g}})$, for some
$\Lambda\in(\mathbb{C}^{*})^{n},\;a\in\mathbb{C},\;M_{\mathfrak{g}}\in\mathcal{M}(\mathfrak{g},\mathfrak{h})$.

###### Proof.

The proof follows from the above discussion. ∎

######  Theorem 5.6.

$\widehat{M}_{\mathcal{L}}(\Lambda,a,M_{\mathfrak{g}})\cong\widehat{M}_{\mathcal{L}}(\Lambda^{\prime},a^{\prime},M_{\mathfrak{g^{\prime}}})$
if and only if $M_{\mathfrak{g}}\cong$ $M_{\mathfrak{g^{\prime}}}$ as
$\mathfrak{g}$-module and $\Lambda=\Lambda^{\prime}$, $a=a^{\prime}.$

###### Proof.

It is clear to see the sufficient part. Let
$\phi:\widehat{M}_{\mathcal{L}}(\Lambda,a,M_{\mathfrak{g}})\to\widehat{M}_{\mathcal{L}}(\Lambda^{\prime},a^{\prime},M_{\mathfrak{g^{\prime}}})$
be the isomorphism with its inverse $\phi^{-1}$. Therefore we have the
following relations

$\displaystyle\phi(\tilde{g})=\tilde{g}\phi(1)$ (5.3)
$\displaystyle\phi^{-1}(\tilde{g})=\tilde{g}\phi^{-1}(1),$ (5.4)

for all $\tilde{g}\in U(\tilde{\mathfrak{h}}_{n}).$ In particular we have
$\phi^{-1}(\phi(1))=\phi(1)\phi^{-1}(1)=1$, which implies that $\phi(1)$ is a
non-zero scalar. Now consider $\psi=\phi|_{U(\mathfrak{h})}$ and note that
$\psi(U(\mathfrak{h})=U(\mathfrak{h}).$ Hence $\psi$ is an isomorphism between
$M_{\mathfrak{g}}$ and $M_{\mathfrak{g^{\prime}}}$.
Let $\Lambda=(\lambda_{1},\dots,\lambda_{n})$ and
$\Lambda^{\prime}=(\lambda_{1}^{\prime},\dots,\lambda_{n}^{\prime}).$ Now
consider the relation
$\phi(h(e_{i}).\tilde{g})=h(e_{i}).\phi(\tilde{g})=h(e_{i}).\tilde{g}\phi(1)$.
This implies that $\lambda_{i}=\lambda_{i}^{\prime}.$ Also consider the
relation
$\phi(t_{i}d_{i}.\tilde{g})=t_{i}d_{i}.\phi(\tilde{g})=t_{i}d_{i}.\tilde{g}\phi(1)$
and conclude that $a=a^{\prime}$ using the fact that
$\lambda_{i}=\lambda_{i}^{\prime}.$ ∎

## References

  * [1] V. Bekkert, G. Benkart, V. Futorny, I. Kashuba, New irreducible modules for Heisenberg and affine Lie algebras, J. Algebra 373 (2013) 284–298.
  * [2] S. Berman, Y. Billig, Irreducible representations for toroidal Lie algebras, J. Algebra 221 (1999) 188-231.
  * [3] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, A certain category of $\mathfrak{g}$-modules Funkcional. Anal. i Prilozen 10 (1976) 1-8.
  * [4] R. Block, The irreducible representations of the Lie algebra $\mathfrak{sl}(2)$ and of the Weyl algebra, Adv. Math. 139\. (1) (1981) 69-110.
  * [5] R. Blumenhagen, E. Plauschinn, Introduction to Conformal Field Theory: With Applications to String Theory, Lect. Notes Phys., vol. 779, Springer, Berlin Heidelberg, 2009.
  * [6] Q. Chen, J. Han, Non-weight modules over the affine-Virasoro algebra of type $A_{1}$, J. Math. Phys. 60, 071707 (2019).
  * [7] Y. A. Cai, H. Tan, K. Zhao, New representations of affine Kac-Moody algebras, J. Algebra 547 (2020), 95-115.
  * [8] Y. A. Cai, H. Tan, K. Zhao, Module structures on $U(\mathfrak{h})$ for Kac-Moody algebras, Sci. Sin. Math. 47 (11) (2017) 1491-1514, arXiv:1606.01891.
  * [9] I. Dimitrov, D. Grantcharov, Classification of simple weight modules over affine Lie algebras, arXiv: 0910.0688v1.
  * [10] E. Cartan, Les groups projectifs qui ne laissent invariante aucune multiplicite‘ planet, Bull. Soc. Math. France 41 (1913) 53-96.
  * [11] V. Chari, Integrable representations of affine Lie-algebras, Invent. Math. 85 (1986) 317–335.
  * [12] V. Chari, A. Pressley, Integrable representations of twisted affine Kac-Moody algebras, J. Algebra 113 (1988) 438–464.
  * [13] V. Chari, A. Pressley, Integrable modules for affine Lie algebras, Math.Ann. 277 (3) (1987) 543–562.
  * [14] P. Chakraborty, Irreducible modules for map Heisenberg-Virasoro Lie algebras, arXiv:2311.02635 .
  * [15] Yu. Drozd, S. Ovsienko, V. Futorny, On Gelfand–Zetlin modules, in: Proceedings of the Winter School on Geometry and Physics, Srni, 1990, in: Rend. Circ. Mat. Palermo (2), vol. 26, 1991, pp. 143–147.
  * [16] J. Dixmier, Enveloping Algebras, American Mathematical Society, 1977.
  * [17] V. Futorny, Irreducible non-dense $A(1)^{1}$ -modules, Pacific J. Math. 172 (1996) 83–99.
  * [18] V. Futorny, Representations of Affine Lie Algebras, Queens Papers in Pure and Appl. Math., vol. 106, Queens University, Kingston, ON, 1997.
  * [19] V. Futorny, A. Tsylke,Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras, J. Algebra 238 (2001) 426–441.
  * [20] V. Futorny, I. Kashuba, Structure of parabolically induced modules for affine Kac-Moody algebras, J. Algebra 500 (2018) 362–374.
  * [21] O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2) (2000) 537-592.
  * [22] V. Mazorchuk, Lectures on $\mathfrak{sl}_{2}(\mathbb{C})$-Modules, Imperial College Press, London, 2010.
  * [23] J. E. Humphreys, Representations of Semisimple Lie algebras in BGG category $\mathcal{O}$, American Mathematical Society, 2008.
  * [24] H.P. Jakobsen, V.G. Kac, A new class of unitarizable highest weight representations of infi- nite dimensional Lie algebras, in: Nonlinear Equations in Classical and Quantum Field Theory, Meudon/Paris, 1983/1984, in: Lecture Notes in Phys., vol. 226, Springer, Berlin, 1985, p. 120.
  * [25] B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (2) (1978) 101–184.
  * [26] V.G. Kac, Infinite-dimensional Lie algebras and Dedekind’s $\eta$-function, Funkt. Anal. Prilozh 8 (1) (1974) 77–78, English translation Funct. Anal. Appl. 8 (1974) 68–70.
  * [27] V.G. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, 1990.
  * [28] V.G. Kac, A. Raina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, World Sci, Singapore, 1987.
  * [29] J. Lepowsky, H. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Mathematics, vol. 227, Birkhauser Boston Inc., Boston, 2004.
  * [30] V. Mazorchuk, K. Zhao, Characterization of simple highest weight modules, Canad. Math. Bull. 56 (3) (2013) 606–614.
  * [31] J. Nilsson, Simple $\mathfrak{sl}_{n+1}$-module structures on $U(h)$, J. Algebra 424 (2015) 294-329.
  * [32] J. Nilsson, $\mathcal{U}(\mathfrak{h})$-free modules and coherent families, J. Pure Appl. Algebra 220(4) (2016) 1475-1488.
  * [33] S.E. Rao, Classification of irreducible integrable modules for toroidal Lie algebras with finite-dimensional weight spaces , J. Algebra 277 (2004), no. 1, 318–348.
  * [34] E. S. Rao, C. Jiang, Classification of irreducible integrable representations for the full toroidal Lie algebras, J. Pure Appl Algebra 2005 (2005) 71-85.
  * [35] E. S. Rao, R. V. Moody, T. Yokonuma, Toroidal Lie algebras and vertex representations, Geom, Dedicata 35 (1990), 283-307.
  * [36] C. Kassel, Kahler differentials and coverings extended over a commutative algebra, Journal of Pure and Applied Algebra 34 (1984), 265-275.
  * [37] H. Tan, K. Zhao, $\mathcal{W}_{n}^{+}$\- and $\mathcal{W}_{n}$-module structures on $\mathcal{U}(\mathfrak{h}_{n})$, J. Algebra 424 (2015) 357-375.
  * [38] H. Tan, K. Zhao, Irreducible modules over Witt algebras $\mathcal{W}_{n}$ and over $\mathfrak{sl}_{n+1}(\mathbb{C})$, Algebr. Represent. Theory 21 (2018), no. 4, 787–806.
  * [39] J. Zhang, Non-weight representations of Cartan type S Lie algebras, Comm. Algebra 2018, Vol. 46, NO. 10, 4243-4264.
  * [40] X. Zhu, Simple modules over Takif Lie algebra $\mathfrak{sl}_{2}$, arXiv:2211.07261v1.


# Ensemble Learning for Vietnamese Scene Text Spotting in Urban Environments

Hieu Nguyen1,2,$\dagger$, Cong-Hoang Ta1,2,$\dagger$, Phuong-Thuy Le-
Nguyen1,2,$\dagger$,
Minh-Triet Tran1,2, Trung-Nghia Le1,2,*
$\dagger$Equal contributors*Corresponding author. Email address:
<EMAIL_ADDRESS>0009-0001-7726-5301 0009-0009-7946-736X
0009-0005-0493-581X 0000-0003-3046-3041 0000-0002-7363-2610 1Faculty of
Information Technology, University of Science, VNU-HCM, Vietnam
2Vietnam National University, Ho Chi Minh City, Vietnam

###### Abstract

This paper presents a simple yet efficient ensemble learning framework for
Vietnamese scene text spotting. Leveraging the power of ensemble learning,
which combines multiple models to yield more accurate predictions, our
approach aims to significantly enhance the performance of scene text spotting
in challenging urban settings. Through experimental evaluations on the VinText
dataset, our proposed method achieves a significant improvement in accuracy
compared to existing methods with an impressive accuracy of 5%. These results
unequivocally demonstrate the efficacy of ensemble learning in the context of
Vietnamese scene text spotting in urban environments, highlighting its
potential for real-world applications, such as text detection and recognition
in urban signage, advertisements, and various text-rich urban scenes.

###### Index Terms:

Ensemble learning, scene text spotting, Vietnamese scene text

##  I INTRODUCTION

The detection and recognition of text in images, namely scene text spotting,
represent a highly formidable task in computer vision[1], demanding the
precise localization and identification of text sequences within real-world
contexts[2]. The implications of scene text spotting are far-reaching,
spanning crucial applications such as key entities recognition [3], autonomous
driving [4], intelligent navigation [5, 6], etc. Scene text spotting has been
recently drawing much attention from the community, resulting in rapid
investigations [7, 8].

Nonetheless, the intricacies of the Vietnamese language, with its rich set of
characters and diacritics, pose significant challenges for scene text
spotting. Certain Vietnamese characters, especially when accompanied by
diacritics, exhibit visual similarities that can lead to confusion, such as,
‘ă’ and ‘â’, ‘ô’ versus ‘ó’, ‘é’ and ‘ê’. Consequently, conventional
individual models may encounter limitations when it comes to accurately
detecting and recognizing Vietnamese scene text.

In addition, the realm of scene text spotting in Vietnamese urban environments
is rife with challenges for existing methods [9, 1] In bustling city contexts,
scene texts are frequently occluded, either partially or entirely, due to the
presence of diverse surrounding objects such as trees, towers, traffic signs,
electric poles, etc. The variability in lighting conditions, encompassing
weather fluctuations and low contrast scenarios caused by background
reflections, further exacerbates the difficulty of scene text spotting.

Figure 1: Scene text spotting in Vietnamese urban environments poses various
challenges, such as obscured by trees and perspective-shifted.

Existing approaches predominantly rely on individual models [10, 11, 2, 7],
each exhibiting varying degrees of success across different subsets of images.
However, when faced with the complexities of Vietnamese urban environments, an
intricate domain with large and diverse contexts, such single-model solutions
often yield limited results.

To overcome these limitations, we propose an ensemble learning framework, a
powerful technique that has shown remarkable efficacy in various computer
vision tasks[12, 13]. Our ensemble learning framework combines multiple state-
of-the-art methods, leveraging their individual strengths and mitigating their
weaknesses. Indeed, each method is tailored to address distinct challenges
related to text detection, recognition, and the intricacies of the Vietnamese
script. By aggregating their predictions, we can significantly enhance the
performance of Vietnamese scene text spotting in urban environments. Extensive
experiments on the VinText dataset [10] show that our proposed ensemble
learning framework outperforms individual models for Vietnamese scene text
spotting, demonstrating superior accuracy and robustness in challenging urban
environments by boosting the accuracy up to 5%. The experimental findings
underscore the potential of ensemble learning as a powerful tool for advancing
scene text spotting in dynamic urban environments.

Our paper contributes to the field of Vietnamese scene text spotting in urban
environments in the following ways:

  * •

We propose a novel ensemble learning framework specifically designed for
Vietnamese scene text spotting, addressing the complexities of urban
environments and the unique characteristics of the Vietnamese script.

  * •

We conduct thorough experiments using an extensive dataset encompassing
diverse urban scenes in Vietnam. Widely adopted in the research community,
this dataset enables a comprehensive evaluation of our proposed approach and
facilitates fair comparisons with existing methods. Extensive experimental
findings demonstrate the utility and superiority of our ensemble approach,
exhibiting higher accuracy and enhanced robustness in challenging urban
contexts.

Figure 2: Workflow of the proposed ensemble learning framework for Vietnamese
scene text spotting.

##  II RELATED WORK

###  II-A Scene Text Spotting

In recent decades, the growth of deep learning significantly contributes to
the advancement of scene text spotting. There are typically approaches: two-
stage and end-to-end approaches. The two-stage approach consists of two major
stages: scene text detection and scene text recognition. A plethora of
effective algorithms have been proposed for each stage, including DB/DB++ [14,
15], SAST [16], EAST [17], etc. for detection and SPIN [18], SRN [19], SVTR
[20], ABINet [21], etc. for recognization. Efforts have also been directed
towards integrating detection and recognition processes, with works like
TextBoxes by Liao et al. [22] combining a single-shot detector and a text
recognizer. Furthermore, Nguyen et al. [10] leveraged a dictionary to generate
a list of potential results and identify the most visually compatible outcome
with the text’s appearance to train and improve the recognition stage.
Additionally, He et al. contributed VinText dataset [10], a challenging
benchmark for Vietnamese scene text spotting. Despite these advancements,
existing techniques treat detection and recognition as independent tasks,
lacking seamless information exchange between them.

Meanwhile, end-to-end approach focuses on merging detection and recognition
into a unified system. Peng et al. [23] presented an end-to-end scene text
spotting method that approaches scene text spotting as a sequence prediction
task. Wang et al. proposed PGNet [24], which revolves around developing a
model that combines a detection unit and a recognition module, allowing for
shared CNN features and joint training. Huang et al. [1] introduced a novel
end-to-end scene text spotting framework called SwinTextSpotter. By
integrating both functionalities into a single algorithm, the resulting end-
to-end model becomes more compact and efficient, leading to improved speed and
performance recognition.

###  II-B Ensemble Learning

Ensemble learning represents a powerful methodology that amalgamates the
strengths and benefits of multiple approaches, culminating in a superior
model. As early as 1996, Rosen proposed [25] a technique called decorrelation
network training to enhance the accuracy of regression learning in ensemble
neural networks. Deng et al. [26] introduced linear and log-linear stacking
methods for ensemble learning, focusing on applications to speech class
posterior probabilities computed by convolutional, recurrent, and fully-
connected deep neural networks. Recently, Casado-García and Heras [27]
explored ensemble methods for object detection, addressing the challenges
associated with ensembling object detectors, such as the limitations of
existing ensemble approaches that depended on specific detection models or
frameworks. Leveraging the advantage of different methods, we introduce an
ensemble learning framework to integrate results from individual models of
both the two-stage and end-to-end approaches to improve the performance of
scene text spotting.

##  III PROPOSED METHOD

###  III-A Overview

We present an ensemble learning framework designed to effectively combine the
outputs of multiple scene text spotting methods. Fig. 2 illustrates the
overview of our method, which consists of three main components: data
converter, base models, and meta-model.

In the first stage of the workflow, we transform the raw images and labels of
the VinText dataset [10] into different formats suitable for training and
predicting of each base model. The data used to train the recognition model is
images with a single word, while the dataset includes images with many
different text lines which contain more than one word. Therefore, we crop the
original images to smaller images that contain only one word and also create
new labels for the new images.

After that, base models are trained on converted data and used for prediction
to obtain initial results. Subsequently, the results of all models are
combined with an ensemble technique and are essential data to create a meta-
model. The meta-model is then used for prediction and generates final result,
which is expected to be better than the initial ones.

###  III-B Text Box Combination:

Our posed Meta-model undergoes two processes: merging non-overlapping text
boxes and merging overlapping text boxes. Given an image and $n$ base models,
the input of our ensemble algorithm is a list of prediction results:
$D=[D_{1},\ldots,D_{n}]$ where each $D_{i}$, with $i\in\\{1,\ldots,n\\}$, is a
list of predicted text boxes $D_{i}=[B_{1},B_{2},\ldots]$. In general, each
$D_{i}$ comes from the predictions of a scene text spotting model using a
particular method $M_{i}$ for a given image.

####  III-B1 Merging non-overlapping text boxes

Firstly, we combine the non-overlapping text boxes belonging to different
$D_{i}$ results together to form the result $D_{s}$:

$D_{s}=D_{1}\cup D_{2}\cup\ldots\cup D_{n},$ (1)

where pair of non-overlapping text boxes ($B_{h},B_{k}$) are defined by:

$\begin{cases}\begin{aligned} \forall P_{o}\in B_{h}\ \implies P_{o}\notin
B_{k},\\\ \forall P_{o}\in B_{k}\ \implies P_{o}\notin B_{h},\\\
\end{aligned}\end{cases}$ (2)

where $B_{h},B_{k}$ are text boxes consisting of the coordinates of four
points at the four corners of the text box, starting from the bottom left
corner in counterclockwise order. $P_{o}$ is the coordinates of a point in
Oxy, which is one of four points of a text box (B).

####  III-B2 Merging overlapping text boxes

Subsequently, we combine the overlapping text boxes belonging to different
$D_{i}$ results based on the Intersection over Union (IoU). IoU is utilized to
determine whether to merge two text boxes into one or separate them into two
distinct text boxes.

Given two bounding boxes, denoted as $B_{h}$ and $B_{k}$, the IoU formula is
used to determine the overlapped region between them, which is represented by
the equation:

$IoU_{(B_{h},B_{k})}=\frac{S(B_{h}\cap B_{k})}{S(B_{h}\cup B_{k})},$ (3)

where $S(\cdot)$ is the area of the text box. From empirically experimental
findings, we notice that when two text boxes contain different words, IoU of
them is always less than 0.5. Therefore, we propose the following formula to
ensemble two overlapping text boxes:

$\begin{cases}\begin{aligned} B_{s}&=B_{h}\cap B_{k}&\text{if }IoU\geq 0.5,\\\
B_{n}&=B_{h}-(B_{h}\cap B_{k})&\text{if }IoU<0.5,\\\ B_{m}&=B_{k}-(B_{h}\cap
B_{k})&\text{if }IoU<0.5,\end{aligned}\end{cases}$ (4)

where $B_{n}$, $B_{m}$, and $B_{s}$ are new text boxes after merging two text
boxes $B_{h}$ and $B_{k}$.

##  IV EXPERIMENTS

###  IV-A Implementation Details

All experiments were performed on Google Colab with the following
configuration GPU T4 and 13GB of RAM. Detection methods were pre-trained on
the ICDAR 2015 [28] for $1,000$ epochs. The initialized learning rate was
$7\times 10^{-3}$ and followed a decay learning rate schedule with a decay
factor of $0.9$. After that, the models were fine-tuned for $30$ epochs on
VinText dataset [10], and weight decay regularization was applied with a value
of $10^{-4}$. The dataset underwent decoding in BGR format, and various data
augmentation techniques were applied, such as horizontal flipping, affine
transformations (rotations between $-10^{\circ}$ to $10^{\circ}$), and
resizing between $0.5$ and $3$ times the original size.

Recognition methods were pre-trained on MJSynth [29, 30] and SynthText [31]
datasets. After that, they underwent a total of 20 epochs of fine-tuning on
the VinText dataset [10]. We utilized the Adam optimizer [32] with a
$\beta_{1}$ value of 0.9 and $\beta_{2}$ value of $0.99$. Gradient clipping
with a norm threshold of $20.0$ was applied to prevent exploding gradients.
The learning rate followed a piecewise schedule, decaying at epoch 20 tow
$10^{-4}$ and $10^{-5}$ for subsequent steps. $L_{2}$ regularization with a
factor of 0 was employed.

###  IV-B Experimental Settings

The VinText dataset [10], a recently proposed Vietnamese text dataset, was
used to evaluate methods. This benchmark dataset has a total of 56K text
instances and 2,000 images, which consists of 1,200 training images, 500
testing images, and 300 unseen test images. Most text instances have usually
too many patterns and chaotic scenes with scene text instances of various
types, appearances, sizes, and orientations. We evaluated methods on unseen
test images.

In order to evaluate the performance and effectiveness of methods, we used
Character Accuracy ($CA$) metric for recognition task, Precision ($P$), Recall
($R$), and F-measure ($F1$) metrics for detection task.

###  IV-C Vietnamese Scene Text Spotting Benchmark

TABLE I: Result of detection methods on the VinText dataset [10]. $1^{st}$ and
$2^{nd}$ places are shown in blue and red, respectively.

Method | Backbone |  | Pre-trained
---
Model
| Fine-tuned on
---
VinText [10]
Precision | Recall | Hmean
SAST [16] | ResNet50_vd [33] | Total-Text [34] | $\checkmark$ | 87.82 | 53.40 | 66.40
| 86.09 | 59.98 | 70.70
ICDAR2015 [28] | $\checkmark$ | 89.53 | 70.56 | 78.92
| 87.45 | 56.43 | 68.59
DB++ [15] | ResNet50 [35] | ICDAR2015 [28] | $\checkmark$ | 92.17 | 71.17 | 80.32
| 89.05 | 65.18 | 75.26
DB [14] |  ResNet50_vd [33] | ICDAR2015 [28] | $\checkmark$ | 85.56 | 65.17 | 73.97
| 84.36 | 56.43 | 67.62
MobileNetV3 [36] | $\checkmark$ | 78.73 | 65.34 | 71.41
| 76.71 | 50.44 | 60.86
EAST [17] | ResNet50_vd [33] | ICDAR2015 [28] | $\checkmark$ | 67.00 | 71.86 | 69.35
| 62.59 | 54.73 | 58.40
MobileNetV3 [36] | $\checkmark$ | 68.13 | 69.50 | 68.80
| 63.80 | 51.73 | 57.13

####  IV-C1 Scene Text Detection

We conducted experiments on the Vintext dataset [10] using various detection
methods, including SAST [16], DB++ [15], DB [14], EAST [17], in different
settings.

Table I showcases the results with DB++ [15] outperforming other methods.
Specifically, DB++ with the ResNet50 [35] backbone achieved the highest
performance (80.32% in terms of Hmean) when pre-trained on the ICDAR2015
dataset [28] and fine-tuned on the VinText dataset. On the other hand, SAST
[16] with the ResNet50_vd backbone, pre-trained on the ICDAR2015 dataset,
demonstrated competitive results across all metrics. However, it is noteworthy
that there exists a trade-off between the accuracy and comprehensiveness of
these methods, as reflected by relatively lower recall values. In contrast,
EAST [17] exhibited a balanced performance in terms of precision and recall.
These findings shed light on the strengths and limitations of these methods,
facilitating informed decision-making when selecting an appropriate scene text
detection method for our ensemble learning framework.

Backbone evaluation:  Our investigation focuses on two scene text detection
methods, DB[14] and EAST [17], where we explore the influence of different
backbone architectures, ResNet50_vd [33] and MobileNetV3 [36]. The comparative
results in Table I highlight the superiority of ResNet50_vd over MobileNetV3
in terms of performance. These findings emphasize the critical importance of
selecting an appropriate backbone architecture for each model, as it
significantly influences the effectiveness of text detection and recognition
tasks.

Effectiveness of fine-tuning models:  Table I shows that the fine-tuned models
on the Vintext dataset [10] exhibit a substantial improvement in results
compared to their pre-training counterparts. These findings demonstrate the
potency of leveraging the Vintext dataset for fine-tuning to significantly
enhance the performance of Vietnamese scene text detection models.

####  IV-C2 Scene Text Recognition

TABLE II: Result of recognition methods on the VinText [10]. $1^{st}$ and
$2^{nd}$ places are shown in blue and red, respectively. We remark that
methods were pre-trained on MJSynth [29, 30] and SynthText [31] datasets in
default.

Method | Backbone |  | Pre-trained
---
Model
| Fine-tuned on
---
VinText [10]
Accuracy
SRN [19] | ResNet50_vd_fpn [33] | $\checkmark$ | $\checkmark$ | 70.73
$\checkmark$ |  | 25.81
| $\checkmark$ | 26.01
ABINet [21] | ResNet45 [37] |  | $\checkmark$ | 73.03
$\checkmark$ |  | 28.57
| $\checkmark$ | 36.89
SPIN [18] | ResNet32 [38] | $\checkmark$ | $\checkmark$ | 78.35
$\checkmark$ |  | 20.69
RobustScanner [39] | ResNet31[38] | $\checkmark$ | $\checkmark$ | 80.77
$\checkmark$ |  | 28.01
| $\checkmark$ | 42.18
SAR [40] | ResNet31[38] | $\checkmark$ | $\checkmark$ | 69.30
$\checkmark$ |  | 26.10
| $\checkmark$ | 49.21
SVTR [20] | SVTR_Tiny [41] | $\checkmark$ | $\checkmark$ | 66.04
$\checkmark$ |  | 28.04
| $\checkmark$ | 30.04
NRTR [42] | NRTR_MTB [42] | $\checkmark$ | $\checkmark$ | 30.30
$\checkmark$ |  | 4.44
| $\checkmark$ | 28.00
RFL [43] | ResNetRFL [43] | $\checkmark$ | $\checkmark$ | 52.18
$\checkmark$ |  | 26.77
| $\checkmark$ | 27.07

(a) DB++ [15] and SPIN [18]

(b) SAST [16] and ABINet [21]

(c) Our ensemble method

Figure 3: Visualization of results of our ensemble learning framework.

We also evaluated the performance of various recognition methods, such as SRN
[19], ABINet [21], SPIN [18], RobustScanner [39], SAR [40], SVTR [20], NRTR
[42], RFL [43], in different settings. Table II presents a comparative
analysis of these methods based on their average accuracy. RobustScanner[39],
SPIN[18], ABINet[21], and SRN[19] stand out with impressive results.
RobustScanner achieved the highest accuracy of 80.77%, followed by SPIN with
78.35%, ABINet with 73.03%, and SRN with 70.73%. These methods consistently
demonstrate strong performance. On the other hand, RFL[43] and NRTR[42]
exhibit relatively lower accuracies of 52.18% and 30.30%, respectively.

The experimental results also reveal compelling evidence of the importance of
pre-training models on MJSynth [29, 30] and SynthText [31] datasets. Training
from scratch on the VinText dataset[10] yields notably inferior performance
across all methods. In addition, the results of fine-tuned models exhibit a
remarkable improvement, which indicates the necessity of fine-tuning pre-
trained models on the VinText dataset[10].

###  IV-D Ensemble Learning Evaluation

TABLE III: End-to-end scene text spotting results on the VinText dataset[10]. | Method | Char_acc | F-measure
---|---|---|---
(1) | DB++ [15] and SPIN [18] | 54.08 | 60.19
(2) | SAST [16] and ABINet [21] | 52.59 | 58.59
(3) | DB [14] and SRN [19] | 50.51 | 56.87
(4) | PGNet (4) [24] | 53.73 | 60.03
| Ensemble learning (Ours): |  |
| (1), (2) | 58.75 | 65.21
| (1), (3) | 51.09 | 55.06
| (1), (4) | 56.24 | 64.37
| (1), (2), (3) | 49.83 | 57.33
| (1), (2), (3), (4) | 52.91 | 55.42

We carefully analyzed the results of detection and recognition methods in
Table I and Table II to select appropriate models for our ensemble learning
framework. The objective is to identify models that excel in distinct aspects
and can synergistically complement each other when combined.

In the evaluation of detection methods from Table I, we meticulously analyze
their performance on both the ICDAR2015 [28] and Total-Text datasets [34].
Notably, DB++[15], EAST[17], and SAST [16] stand out in different metrics.
DB++ [15] showcases high precision, while EAST [17] demonstrates good recall,
and SAST exhibits a strong F-measure specifically on the Total-Text dataset
[34].

Shifting our focus to the recognition methods from Table II, we assess their
average accuracy. SPIN [18], ABINet [21], and RobustScanner [39] stand out as
top performers, demonstrating remarkable accuracy in text recognition.
However, upon merging the RobustScanner [39] with the detection methods, we
observe challenges with overlapping text boxes, leading to a decline in
overall performance. As a result, we make the decision to exclude the
RobustScanner [39] from our final ensemble, striking a balance between
detection and recognition performance.

After the individual evaluations, we selected DB++ [15], EAST [17], SAST
[16])for the detection task, SPIN [18], ABINet [21], SRN [19] for the
recognition task. Additionally, PSNet [24] was chosen as an end-to-end model
capable of addressing both detection and recognition aspects. Subsequently, we
employed our ensemble algorithm to combine model pairs, harnessing the unique
strengths of each end-to-end model. The results of this comprehensive ensemble
are presented in Table III, showcasing a powerful and complementary fusion of
models for Vietnamese scene text spotting.

While potential of ensemble learning is vast, it is crucial to recognize that
mere combination of more models does not always guarantee improved results. In
fact, some models, when merged, may yield even worse performance compared to
their individual counterparts (see Table III). Our rigorous experimentation
has led us to discover that the most optimal ensemble result is achieved by
combining pairs of methods DB++[15] / SPIN [18] and SAST [17]/ ABINet[21].
This carefully curated selection of effective ensemble pairs sheds light on
the delicate interplay between methods and underscores the significance of
thoughtful model combination for achieving remarkable performance gains in
scene text spotting. Indeed, our ensemble framework delivers improved
performance in all metrics (See Fig. 3).

##  V CONCLUSION

In this paper, we have presented an ensemble learning framework for Vietnamese
scene text spotting in urban environments. The proposed technique demonstrates
the effectiveness of combining the strengths of different models. Through
meticulous experimentation, we identify models with promising performance,
strategically designing an ensemble framework that maximizes individual model
strengths. Nonetheless, our method does exhibit certain limitations, notably
an increase in computational complexity due to the integration of multiple
models and instances where certain model combinations yield suboptimal
results. In our future research, we aim to address these challenges by
focusing on enhancing spelling accuracy during word recognition and reducing
the computational complexity of the ensemble model.

## Acknowledgement

This research is supported by research funding from Faculty of Information
Technology, University of Science, Vietnam National University - Ho Chi Minh
City.

## References

  * [1] M. Huang, Y. Liu, Z. Peng, C. Liu, D. Lin, S. Zhu, N. Yuan, K. Ding, and L. Jin, “Swintextspotter: Scene text spotting via better synergy between text detection and text recognition,” arXiv:2203.10209, 2022.
  * [2] Y. Liu, J. Zhang, D. Peng, M. Huang, X. Wang, J. Tang, C. Huang, D. Lin, C. Shen, X. Bai, and L. Jin, “Spts v2: Single-point scene text spotting,” arXiv:2301.01635, 2023.
  * [3] J. Wang, C. Liu, L. Jin, G. Tang, J. Zhang, S. Zhang, Q. Wang, Y. Wu, and M. Cai, “Towards robust visual information extraction in real world: New dataset and novel solution,” arXiv:2102.06732, 2021.
  * [4] C. Zhang, Y. Tao, K. Du, W. Ding, B. Wang, J. Liu, and W. Wang, “Character-level street view text spotting based on deep multisegmentation network for smarter autonomous driving,” IEEE Transactions on Artificial Intelligence, vol. 3, no. 2, pp. 297–308, 2022.
  * [5] X. Rong, B. Li, J. P. Munoz, J. Xiao, A. Arditi, and Y. Tian, “Guided text spotting for assistive blind navigation in unfamiliar indoor environments,” in ISVC, pp. 11–22, 2016.
  * [6] H.-C. Wang, C. Finn, L. Paull, M. Kaess, R. Rosenholtz, S. Teller, and J. Leonard, “Bridging text spotting and slam with junction features,” in IROS, pp. 3701–3708, 2015.
  * [7] N. T. Pham, V. D. Pham, Q. Nguyen-Van, B. H. Nguyen, D. N. Minh Dang, and S. D. Nguyen, “Vietnamese scene text detection and recognition using deep learning: An empirical study,” in GTSD, pp. 213–218, 2022.
  * [8] M.-Q. Ha, V.-H. Phan, B. D. Q. Nguyen, H.-B. Nguyen, T.-H. Do, Q.-D. Pham, and N.-N. Dao, “Intelligent scene text recognition in streaming videos,” in Intelligence of Things: Technologies and Applications, pp. 356–365, 2022\.
  * [9] C. Tran, K. Nguyen-Trong, C. Pham, D. Tran-Anh, and T. Nguyen-Thi-Tan, “Improving text recognition by combining visual and linguistic features of text,” in SoICT, pp. 329–335, 2022.
  * [10] N. Nguyen, T. Nguyen, V. Tran, M.-T. Tran, T. D. Ngo, T. H. Nguyen, and M. Hoai, “Dictionary-guided scene text recognition,” in CVPR, pp. 7383–7392, 2021.
  * [11] A. D. Le, H. T. Nguyen, and M. Nakagawa, “End to end recognition system for recognizing offline unconstrained vietnamese handwriting,” arXiv:1905.05381, 2019.
  * [12] D. Yu, X. Li, C. Zhang, J. Han, J. Liu, and E. Ding, “Towards accurate scene text recognition with semantic reasoning networks,” arXiv:2003.12294, 2020\.
  * [13] S. Fang, H. Xie, Y. Wang, Z. Mao, and Y. Zhang, “Read like humans: Autonomous, bidirectional and iterative language modeling for scene text recognition,” arXiv:2103.06495, 2021.
  * [14] M. Liao, Z. Wan, C. Yao, K. Chen, and X. Bai, “Real-time scene text detection with differentiable binarization,” in AAAI, pp. 11474–11481, 2020.
  * [15] M. Liao, Z. Zou, Z. Wan, C. Yao, and X. Bai, “Real-time scene text detection with differentiable binarization and adaptive scale fusion,” IEEE T-PAMI, 2022.
  * [16] P. Wang, C. Zhang, F. Qi, Z. Huang, M. En, J. Han, J. Liu, E. Ding, and G. Shi, “A single-shot arbitrarily-shaped text detector based on context attended multi-task learning,” in ACM MM, pp. 1277–1285, 2019.
  * [17] X. Zhou, C. Yao, H. Wen, Y. Wang, S. Zhou, W. He, and J. Liang, “East: an efficient and accurate scene text detector,” in CVPR, pp. 5551–5560, 2017\.
  * [18] Zhang, Chengwei, Xu, Yunlu, Cheng, Zhanzhan, Pu, Shiliang, Niu, Yi, Wu, Fei, Zou, and Futai, “Spin: structure-preserving inner offset network for scene text recognition,” in AAAI, vol. 35, pp. 3305–3314, 2021.
  * [19] D. Yu, X. Li, C. Zhang, T. Liu, J. Han, J. Liu, and E. Ding, “Towards accurate scene text recognition with semantic reasoning networks,” in CVPR, pp. 12113–12122, 2020.
  * [20] Y. Du, Z. Chen, C. Jia, X. Yin, T. Zheng, C. Li, Y. Du, and Y.-G. Jiang, “Svtr: Scene text recognition with a single visual model,” in IJCAI, 2022\.
  * [21] S. Fang, H. Xie, Y. Wang, Z. Mao, and Y. Zhang, “Abinet: Read like humans: Autonomous, bidirectional and iterative language modeling for scene text recognition,” in CVPR, pp. 7098–7107, 2021.
  * [22] M. Liao, B. Shi, X. Bai, X. Wang, and W. Liu, “Textboxes: A fast text detector with a single deep neural network,” in AAAI, 2017.
  * [23] D. Peng, X. Wang, Y. Liu, J. Zhang, M. Huang, S. Lai, S. Zhu, J. Li, D. Lin, C. Shen, X. Bai, and L. Jin, “Spts: Single-point text spotting,” arXiv preprint arXiv:2112.07917, 2022.
  * [24] P. Wang, C. Zhang, F. Qi, S. Liu, X. Zhang, P. Lyu, J. Han, J. Liu, E. Ding, and G. Shi, “Pgnet: Real-time arbitrarily-shaped text spotting with point gathering network,” arXiv preprint arXiv:2104.05458, 2021.
  * [25] B. E. Rosen, “Ensemble learning using decorrelated neural networks,” No. 3-4 in 8, pp. 373–384, Taylor & Francis, 1996.
  * [26] L. Deng and J. Platt, “Ensemble deep learning for speech recognition,” in Interspeech, 2014.
  * [27] Á. Casado-García and J. Heras, “Ensemble methods for object detection,” in ECAI, pp. 2688–2695, IOS Press, 2020.
  * [28] D. Karatzas, L. Gomez-Bigorda, A. Nicolaou, S. Ghosh, A. Bagdanov, M. Iwamura, J. Matas, L. Neumann, V. R. Chandrasekhar, S. Lu, F. Shafait, S. Uchida, and E. Valveny, “Icdar 2015 competition on robust reading,” in ICDAR, pp. 1156–1160, 2015.
  * [29] M. Jaderberg, K. Simonyan, A. Vedaldi, and A. Zisserman, “Synthetic data and artificial neural networks for natural scene text recognition,” in Workshop on Deep Learning, NIPS, 2014.
  * [30] M. Jaderberg, K. Simonyan, A. Vedaldi, and A. Zisserman, “Reading text in the wild with convolutional neural networks,” No. 1 in 116, pp. 1–20, Springer, jan 2016.
  * [31] A. Gupta, A. Vedaldi, and A. Zisserman, “Synthetic data for text localisation in natural images,” in CVPR, 2016.
  * [32] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv:1412.6980, 2014.
  * [33] S. Wang, X. Xia, L. Ye, and B. Yang, “Automatic detection and classification of steel surface defect using deep convolutional neural networks,” No. 3 in 11, p. 388, MDPI, 2021.
  * [34] C. K. Ch’ng and C. S. Chan, “Total-text: A comprehensive dataset for scene text detection and recognition,” in ICDAR, vol. 1, pp. 935–942, 2017.
  * [35] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in CVPR, pp. 770–778, 2016.
  * [36] A. Howard, M. Sandler, G. Chu, L.-C. Chen, B. Chen, M. Tan, W. Wang, Y. Zhu, R. Pang, V. Vasudevan, et al., “Searching for mobilenetv3,” in ICCV, pp. 1314–1324, 2019.
  * [37] S. Wu, S. Zhong, and Y. Liu, “Deep residual learning for image steganalysis,” Multimedia tools and applications, vol. 77, pp. 10437–10453, 2018.
  * [38] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” arXiv:1512.03385, 2015.
  * [39] X. Yue, Z. Kuang, C. Lin, H. Sun, and W. Zhang, “Robustscanner: Dynamically enhancing positional clues for robust text recognition,” in ECCV, pp. 135–151, 2020.
  * [40] H. Li, P. Wang, C. Shen, and G. Zhang, “Show, attend and read: A simple and strong baseline for irregular text recognition,” in AAAI, no. 01 in 33, pp. 8610–8617, 2019.
  * [41] C. Li, W. Liu, R. Guo, X. Yin, K. Jiang, Y. Du, Y. Du, L. Zhu, B. Lai, X. Hu, et al., “Pp-ocrv3: More attempts for the improvement of ultra lightweight ocr system,” arXiv preprint arXiv:2206.03001, 2022.
  * [42] F. Sheng, Z. Chen, and B. Xu, “Nrtr: A no-recurrence sequence-to-sequence model for scene text recognition,” in ICDAR, pp. 781–786, 2019.
  * [43] H. Jiang, Y. Xu, Z. Cheng, S. Pu, Y. Niu, W. Ren, F. Wu, and W. Tan, “Reciprocal feature learning via explicit and implicit tasks in scene text recognition,” in ICDAR, 2021.