Lie Bialgebra Structures and Quantization of Generalized Loop Planar Galilean Conformal Algebra

Abstract

In this paper, we analyze the Lie bialgebra (LB) and quantize the generalized loop planar-Galilean conformal algebra (GLPGCA) $W(\Gamma )$. Additionally, we prove that all LB structures on $W(\Gamma )$ possess a triangular coboundary. We also quantize $W(\Gamma )$ using the Drinfeld-twist quantization technique and identify a group of noncommutative algebras and noncocommutative Hopf algebras.

1. Introduction

Lie bialgebra (LB) structure is closely linked with the Yang–Baxter equation [1]. Many researchers have studied the structural theory of Lie super-bialgebras (LSBs) [2,3,4,5,6,7,8]. For instance, the LB structures of a generalized loop Virasoro algebra were investigated in [7]. Also, in [5], the researcher identified the LB structures on the Schrödinger–Virasoro Lie algebra (LA).

Drinfeld described the “quantum group” as an LA’s universal enveloping algebraic deformation in the Hopf algebra category. Moreover, in quantum group applications, quantizations of LSB structures were comprehensively studied. Drinfeld raised the question of whether a general method exists for LSB quantization. Etingof and Kazhdan provided a clear answer [9]. Additionally, some researchers have recently quantized several algebras, including [10,11,12,13,14,15,16,17].

In [18], the researchers initiated the study of a different nonrelativistic limit, proposing that Galilean conformal algebras have a distinct nonrelativistic limit on the ADS/CFT conjecture. The finite-dimensional Galilean conformal algebra is linked with some non-semisimple LA—considered a nonrelativistic analog of conformal algebras. Moreover, a finite Galilean conformal algebra can undergo infinite-dimensional lift in all space-time dimensions (see [18,19,20]).

In [18], the researchers comprehensively examined nonrelativistic versions of the ADS/CFT conjecture. Such studies have mainly been performed in the Schrödinger symmetry group context. The researchers started a study based on a different nonrelativistic conformal symmetry, with one being identified by the relativistic conformal group’s parametric contraction. They obtained that Galilean conformal symmetry has as many generators as the relativistic symmetry group, making it distinct from the Schrödinger group. Interestingly, a Galilean conformal algebra permits extending an infinite-dimensional symmetry algebra (possibly through dynamic means). The latter features a Virasoro–Kac–Moody subalgebra. The researchers commented on the realizations of this extended symmetry in their boundary field theory. They established rather unique geometric structure for bulk gravity, second to any realization of this symmetry.

In [18], Bagchi and Gopakumar first established the planar-Galilean conformal algebra, and others have since obtained meaningful results in this area [21,22,23,24,25,26].

Loop algebras are closely linked with theoretical and mathematical physics [27]. These algebras were also found to be more instrumental to Kac–Moody affine LAs in [28]. Affine LAs assist in both string theory and two-dimensional (2D) conformal field theory [29]. Recently, in [27,30,31], the researchers examined some properties of a loop algebra. Specifically, a generalized loop planar-Galilean conformal algebra (GLPGCA) $W(\Gamma )$ is the tensor product $W\otimes \mathbb{F}\left[t^{\pm 1}\right]$ of a generalized planar-Galilean conformal algebra (GPGCA) W with a Laurent polynomial algebra $\mathbb{F}\left[t^{\pm 1}\right]$. The structural theory of $W(\Gamma )$ has been assessed in [32].

Suppose $\mathbb{F}$ is an algebraic closed field of the variable 0, and $\Gamma$ is an appropriate additive subgroup of $\mathbb{F}$. A generalized planar-Galilean conformal algebra (GPGCA) W is an infinite-dimensional LA with $\mathbb{F}$-basis $\{L_m,H_m,I_m,J_m|m\in \Gamma \}$ subject to the following equations:

$$\begin{array}{cc}\hfill & [L_m,L_n]=(m-n)L_{m+n},[L_m,H_n]=-n{H}_{m+n},\hfill \\ \hfill & [L_m,I_n]=(m-n)I_{m+n},[L_m,J_n]=(m-n)J_{m+n},\hfill \\ \hfill & [H_m,I_n]=J_{m+n},[H_m,J_n]=-I_{m+n},\hfill \\ \hfill & [H_m,H_n]=[I_m,I_n]=[I_m,J_n]=[J_m,J_n]=0,\hfill \end{array}$$

(1)

for any $m,n\in \Gamma$.

GLPGCA $W(\Gamma )$ is generated by $\{L_{m,i},H_{m,i},I_{m,i},J_{m,i}|m\in \Gamma ,i\in \mathbb{Z}\}$ over $\mathbb{F}$, subject to the following equations:

$$\begin{array}{c}[L_{m,i},L_{n,j}]=(m-n)L_{m+n,i+j},[L_{m,i},H_{n,j}]=-n{H}_{m+n,i+j},\hfill \\ [L_{m,i},I_{n,j}]=(m-n)I_{m+n,i+j},[L_{m,i},J_{n,j}]=(m-n)J_{m+n,i+j},\hfill \\ [H_{m,i},I_{n,j}]=J_{m+n,i+j},[H_{m,i},J_{n,j}]=-I_{m+n,i+j},\hfill \\ [H_{m,i},H_{n,j}]=[I_{m,i},I_{n,j}]=[I_{m,i},J_{n,j}]=[J_{m,i},J_{n,j}]=0,\hfill \end{array}$$

where $X_{m;i}=X_m\otimes x^i$ for any $X\in \{L,H,I,J\}$, $m,n\in \Gamma$ and $i,j\in \mathbb{Z}$. The centers of W and $W(\Gamma )$ are always $\left\{0\right\}$.

In this paper, we analyze LB structures and $W(\Gamma )$ quantization. In Section 2, we study the LB structure of $W(\Gamma )$. Evidently, all the LB structures on $W(\Gamma )$ possess a triangular coboundary. Section 3 employs the Drinfeld-twist general quantization method to specifically quantize LB structures on $W(\Gamma )$. Moreover, we identify a group of noncommutative algebras and noncocommutative Hopf algebras. The major results are captured in Theorems 2 and 3.

2. LB Structures of $\mathbf{W}\mathbf{(}\mathbf{\Gamma }\mathbf{)}$

Definition 1 

([2]). Let L be a vector space over $\mathbb{F}$ and τ be the twist map of $L\otimes L$. That is,

$$\tau (x\otimes y)=y\otimes x,\forall x,y\in L.$$

where ε is a map cyclically permuting $L\otimes L\otimes L$’s coordinates, meaning that

$$\epsilon (x_1\otimes x_2\otimes x_3)=x_2\otimes x_3\otimes x_1,\forall x_1,x_2,x_3\in L.$$

Definition 2 

([2]). Let L be a vector space over $\mathbb{F}$ and $\varphi :L\otimes L\to L$ be a linear map. The pair ($L,\varphi$) is termed an LA if the following criteria apply:

$$\begin{array}{c}\hfill \mathrm{Ker}(1\otimes 1-\tau )\subset \mathrm{Ker}\varphi ,\varphi (1\otimes \varphi )(1\otimes 1\otimes 1+\epsilon +{\epsilon }^2)=0,\end{array}$$

where 1 represents the identity map on L.

Definition 3 

([2]). Let L be a vector space over $\mathbb{F}$ and $\Delta :L\to L\otimes L$ be a linear map. The pair ($L,\Delta$) is called a Lie co-algebra if the following criteria apply:

$$\begin{array}{c}\hfill \mathrm{Im}\Delta \subset \mathrm{Im}(1\otimes 1-\tau ),(1\otimes 1\otimes 1+\epsilon +{\epsilon }^2)(1\otimes \Delta )\Delta =0.\end{array}$$

The map Δ is considered the cobracket of L.

Definition 4 

([2]). An LB is a triple ($L,\varphi ,\Delta$) such that ($L,\varphi$) is an LA, and ($L,\Delta$) is a Lie co-algebra, and

$$\Delta \varphi (x\otimes y)=x·\Delta y-y·\Delta x,\forall x,y\in L,$$

where the symbol ‘·’ represents the adjoint diagonal action:

$$x·(\sum _i{a}_i\otimes b_i)=\sum _i([x,a_i]\otimes b_i+a_i\otimes [x,b_i]),\forall x,a_i,b_i\in L.$$

Definition 5 

([2]). A coboundary LB is a quadruple ($L,\varphi ,\Delta ,r$), where ($L,\varphi ,\Delta$) is an LB and $r\in \mathrm{Im}(1\otimes 1-\tau )$ such that $\Delta ={\Delta }_r$ is a coboundary of r, for arbitrary $x\in L$, and ${\Delta }_r$ is defined as follows:

$${\Delta }_r\left(x\right)=x·r$$

where $U\left(L\right)$ is the universal enveloping algebra of L. For

$$r=\sum _i{a}_i\otimes b_i\in L\otimes L,$$

$r^{ij}$ and $c\left(r\right)$ represent the variables of $U\left(L\right)\otimes U\left(L\right)\otimes U\left(L\right)$ by the following:

$$r^{12}=\sum _i{a}_i\otimes b_i\otimes 1,r^{13}=\sum _i{a}_i\otimes 1\otimes b_i,r^{23}=\sum _{i}1\otimes a_i\otimes b_i,$$

and

$$c\left(r\right)=[r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13},r^{23}].$$

Definition 6 

([2]). A coboundary LB ($L,\varphi ,\Delta ,r$) is considered triangular if r aligns with the following classical Yang–Baxter equation (CYBE):

$$c\left(r\right)=0.$$

Theorem 1 

([33]). Let ($L,\varphi$) be an LA. Then, for certain $r\in Im(1\otimes 1-\tau )$, $\Delta ={\Delta }_r$ grants ($L,\varphi ,\Delta$) an LB structure if and only if r satisfies the modified Yang–Baxter equation (MYBE) given as follows:

$$x·c\left(r\right)=0,\forall x\in L.$$

Let $W(\Gamma )={\oplus }_{m\in \Gamma }W{(\Gamma )}_m$, where

$$W{(\Gamma )}_m=spa{n}_{\mathbb{F}}\{L_{m,i},H_{m,i},I_{m,i},J_{m,i}|i\in \mathbb{Z}\},m\in \Gamma .$$

Then, $W(\Gamma )$ is a $\Gamma$-graded LA.

We use V to convey $W(\Gamma )\otimes W(\Gamma )$, and $W(\Gamma )$ is a $\Gamma$-graded LA. Thus, V permits a natural $\Gamma$-graded $W(\Gamma )$-module structure under the adjoint diagonal action of $W(\Gamma )$. Specifically,

$$V={\Sigma }_{\alpha \in \Gamma }{V}_{\alpha },$$

where

$$V_{\alpha }={\Sigma }_{u+v=\alpha }{W}_u\otimes W_v,u,v\in \Gamma ,$$

which is $\Gamma$-graded.

We employ $Der\left(W\right(\Gamma ),V)$ and $Inn\left(W\right(\Gamma ),V)$ to express all derivations’ vector spaces and inner derivations from $W(\Gamma )$ to V. The first cohomology group of $W(\Gamma )$ with coefficients in module V is $Der\left(W\right(\Gamma ),V)/Inn\left(W\right(\Gamma ),V)$. A derivation $D\in Der\left(W\right(\Gamma ),V)$ is considered a degree $\alpha$ if $D\left(W{(\Gamma )}_m\right)\subset V_{\alpha +m}$ for all $\alpha ,m\in \Gamma$. $Der{(W(\Gamma ),V)}_{\alpha }$ represents the space of derivations of degree $\alpha$. For any derivation $D\in Der\left(W\right(\Gamma ),V)$, we obtain the following:

$$D=\sum _{\alpha \in \Gamma }{D}_{\alpha },$$

which is true if for every $x\in W(\Gamma )$, many $D_{\alpha }\left(x\right)\ne 0$, and $D\left(x\right)={\sum }_{\alpha \in \Gamma }{D}_{\alpha }\left(x\right)$ exist finitely.

Lemma 1. 

If $\alpha \ne 0$, every $D\in Der{(W(\Gamma ),V)}_{\alpha }$ is an inner derivation.

Proof. 

For any $x_m\in W{(\Gamma )}_m$, applying $D_{\alpha }$ to $[L_{0,0},x_m]=-m{x}_m$, and $D_{\alpha }\left(x_m\right)\in V_{\alpha +m}$, we obtain $D_{\alpha }\left(x_m\right)=-x_m·\left({\alpha }^{-1}{D}_{\alpha }\left(L_{0,0}\right)\right)$. Thus, $D_{\alpha }$ is deemed an inner derivation. □

Lemma 2. 

Let $D\in Der\left(W\right(\Gamma ),V)$ be a derivation. Then, if $D\left(L_{0,0}\right)=0$, $D\in Der{(W(\Gamma ),V)}_0$.

Proof. 

We apply D to the equation $[L_{0,0},L_{\alpha ,i}]=-\alpha L_{\alpha ,i}$. Afterward, we realize $[L_{0,0},D\left(L_{\alpha ,i}\right)]=-\alpha D\left(L_{\alpha ,i}\right)$. So $D\left(L_{\alpha ,i}\right)\in V_{\alpha }$. □

Definition 7. 

The GPGCA W is an infinite-dimensional LA with $\mathbb{F}$-basis $\{L_m,H_m,I_m,J_m|m\in \Gamma \}$, as described in (1). As a vector space,

$$V\cong (W\otimes W)\otimes \mathbb{F}[x,x^{-1},y,y^{-1}]\cong (W\otimes W)[x,x^{-1},y,y^{-1}],$$

which is the space of formal Laurant polynomials in two variables with coefficients in $W\otimes W$. We then fix $i,j\in \mathbb{Z}$, and we let

$$V_{i,j}=spa{n}_{\mathbb{F}}\{X_{m,i}\otimes Y_{n,j}|m,n\in \Gamma \}=spa{n}_{\mathbb{F}}\{X_m\otimes Y_n{x}^i{y}^j|m,n\in \Gamma \}\cong W\otimes W,$$

where $X\in \{L,H,I,J\}$, $Y\in \{L,H,I,J\}$. We find that $V_{i,j}$ is a W-module.

$$V=\underset{i,j\in \mathbb{Z}}{⨁}{V}_{i,j},$$

which is considered a ${\mathbb{Z}}^2$-graded space. The $W(\Gamma )$-module on V can be expressed:

$$X_{m,i}·(P_{\beta }\otimes Q_{\gamma }{x}^j{y}^k)=[X_m,P_{\beta }]\otimes Q_{\gamma }{x}^{i+j}{y}^k+P_{\beta }\otimes [X_m,Q_{\gamma }]x^j{y}^{i+k},$$

where $X,P,Q\in \{L,H,I,J\}$.

Lemma 3. 

For any $D\in Der\left(W\right(\Gamma ),V)$, the vector $v\in V$ exists such that $D_W=D_v$, where $D_v$ represents an inner derivation of $W(\Gamma )$ defined by $D_v\left(X_{r,i}\right)=X_{r,i}·v$ for any $r\in \Gamma$, $i\in \mathbb{Z}$, $X\in \{L,H,I,J\}$.

Proof. 

We examine the restriction of D to W and represent $D_W$ by D for ease. As $V={⨁}_{i,j\in \mathbb{Z}}{V}_{i,j}$ as a W-module, where $V_{i,j}\cong W\otimes W$. We let $D_{i,j}\in Der(W,V_{i,j})$, and we obtain $D={\sum }_{i,j\in \mathbb{Z}}{D}_{i,j}$, which is true in that only finitely many terms $D_{i,j}\left(x\right)\ne 0$ are available when we apply $D={\sum }_{i,j\in \mathbb{Z}}{D}_{i,j}$ to any $x\in W(\Gamma )$. By [34] (Section 3.2), we obtain $H^1(W,V_{i,j})=H^1(W,W\otimes W)=0$, meaning that $D_{i,j}$ is an inner derivation for any $i,j\in \mathbb{Z}$. Thus, there is some $v_{i,j}\in V_{i,j}$ such that $D_{i,j}=D_{v_{i,j}}$, where $D_{v_{i,j}}$ represents an inner derivation in relation to $v_{i,j}$. In the equation, $D={\sum }_{i,j\in \mathbb{Z}}{D}_{i,j}$ is a finite sum and $D=D_v$ in the rest. Afterward, we deduce that D is an inner derivation.

We then examine finite set $X=\left\{(i,j)\right|(L_{0,0}·v_{i,j},L_{1,0}·v_{i,j},L_{2,0}·v_{i,j})\ne (0,0,0)\}$. The complement of $X\in \mathbb{Z}\times \mathbb{Z}$ is $\overline{X}=\left\{(i,j)\right|L_{0,0}·v_{i,j}=L_{1,0}·v_{i,j}=L_{2,0}·v_{i,j}=0\}$. For given pair $(i,j)$, we assume the following:

$$\begin{array}{cc}\hfill v_{i,j}=& \sum _{p,q\in \Gamma }(a_1(p,q)L_p\otimes L_q{x}^i{y}^j+a_2(p,q)L_p\otimes H_q{x}^i{y}^j+a_3(p,q)L_p\otimes I_q{x}^i{y}^j+a_4(p,q)L_p\otimes J_q{x}^i{y}^j\hfill \\ & +a_5(p,q)H_p\otimes L_q{x}^i{y}^j+a_6(p,q)H_p\otimes H_q{x}^i{y}^j+a_7(p,q)H_p\otimes I_q{x}^i{y}^j+a_8(p,q)H_p\otimes J_q{x}^i{y}^j\hfill \\ & +a_9(p,q)I_p\otimes L_q{x}^i{y}^j+a_{10}(p,q)I_p\otimes H_q{x}^i{y}^j+a_{11}(p,q)I_p\otimes I_q{x}^i{y}^j+a_{12}(p,q)I_p\otimes J_q{x}^i{y}^j\hfill \\ & +a_{13}(p,q)J_p\otimes L_q{x}^i{y}^j+a_{14}(p,q)J_p\otimes H_q{x}^i{y}^j+a_{15}(p,q)J_p\otimes I_q{x}^i{y}^j+a_{16}(p,q)J_p\otimes J_q{x}^i{y}^j),\hfill \end{array}$$

where $a_1(p,q),\cdots ,a_{16}(p,q)\in \mathbb{F}$.

By $L_{0,0}{v}_{i,j}=0$, we can obtain

$$\begin{array}{c}\hfill (p+q)a_1(p,q)=(p+q)a_2(p,q)\cdots =(p+q)a_{16}(p,q)=0.\end{array}$$

Thus, we surmise the following:

$$\begin{array}{cc}\hfill v_{i,j}=& \sum _{k\in \Gamma }(a_1\left(k\right)L_k\otimes L_{-k}{x}^i{y}^j+a_2\left(k\right)L_k\otimes H_{-k}{x}^i{y}^j+a_3\left(k\right)L_k\otimes I_{-k}{x}^i{y}^j+a_4\left(k\right)L_k\otimes J_{-k}{x}^i{y}^j\hfill \\ & +a_5\left(k\right)H_k\otimes L_{-k}{x}^i{y}^j+a_6\left(k\right)H_k\otimes H_{-k}{x}^i{y}^j+a_7\left(k\right)H_k\otimes I_{-k}{x}^i{y}^j+a_8\left(k\right)H_k\otimes J_{-k}{x}^i{y}^j\hfill \\ & +a_9\left(k\right)I_k\otimes L_{-k}{x}^i{y}^j+a_{10}\left(k\right)I_k\otimes H_{-k}{x}^i{y}^j+a_{11}\left(k\right)I_k\otimes I_{-k}{x}^i{y}^j+a_{12}\left(k\right)I_k\otimes J_{-k}{x}^i{y}^j\hfill \\ & +a_{13}\left(k\right)J_k\otimes L_{-k}{x}^i{y}^j+a_{14}\left(k\right)J_k\otimes H_{-k}{x}^i{y}^j+a_{15}\left(k\right)J_k\otimes I_{-k}{x}^i{y}^j+a_{16}\left(k\right)J_k\otimes J_{-k}{x}^i{y}^j),\hfill \end{array}$$

where $a_1\left(k\right),\cdots ,a_{16}\left(k\right)\in \mathbb{F}$.

By $L_{1,0}{v}_{i,j}=0$, we can attain the following:

$$\begin{array}{c}\hfill (k-2)a_A(k-1)=(1+k)a_A\left(k\right),\end{array}$$

(2)

$$\begin{array}{c}\hfill (k-2)a_B(k-1)=k{a}_B\left(k\right),\end{array}$$

(3)

$$\begin{array}{c}\hfill (k-1)a_C(k-1)=(1+k)a_C\left(k\right),\end{array}$$

(4)

$$\begin{array}{c}\hfill (k-1)a_D(k-1)=k{a}_D\left(k\right),\end{array}$$

(5)

where $A\in \{1,3,4,9,11,12,13,15,16\}$, $B\in \{2,10,14\}$, $C\in \{5,7,8\}$, $D\in \left\{6\right\}$.

Substituting k with $k-1$ in the above equations, we can obtain the following:

$$\begin{array}{c}\hfill (k-3)a_A(k-2)=k{a}_A(k-1),\end{array}$$

(6)

$$\begin{array}{c}\hfill (k-3)a_B(k-2)=(k-1)a_B(k-1),\end{array}$$

(7)

$$\begin{array}{c}\hfill (k-2)a_C(k-2)=k{a}_C(k-1),\end{array}$$

(8)

$$\begin{array}{c}\hfill (k-2)a_D(k-2)=(k-1)a_D(k-1).\end{array}$$

(9)

Similarly, by $L_{2,0}{v}_{i,j}=0$, we can calculate the following:

$$\begin{array}{c}\hfill (k-4)a_A(k-2)=(2+k)a_A\left(k\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill (k-4)a_B(k-2)=k{a}_B\left(k\right),\end{array}$$

(11)

$$\begin{array}{c}\hfill (k-2)a_C(k-2)=(2+k)a_C\left(k\right),\end{array}$$

(12)

$$\begin{array}{c}\hfill (k-2)a_D(k-2)=k{a}_D\left(k\right).\end{array}$$

(13)

By (2), (5), (10), we have $a_A\left(k\right)=0$. By (3), (6), (11), we obtain $a_B\left(k\right)=0$. By (4), (7), (12), we have $a_C\left(k\right)=0$. By (5), (9), (13), we obtain $a_D\left(k\right)=0$.

Because $v_{i,j}$ is a finite sum, we calculate that $a_1\left(k\right)=\cdots =a_{16}\left(k\right)=0$ for $k\in \Gamma$. Then, we obtain $v_{i,j}=0$ for all $(i,j)\in \overline{X}$ and $\left\{(i,j)\right|v_{i,j}\ne 0\}\subset X$. In sum, $D={\sum }_{i,j\in \mathbb{Z}}{D}_{i,j}$ is a finite sum, and $D=D_v$ for $v={\sum }_{i,j\in \mathbb{Z}}{v}_{i,j}$. Thus,

$Der(W,V)={⨁}_{i,j\in \mathbb{Z}}Der(W,V_{i,j})=Inn(W,V).$ □

Lemma 4. 

Every derivation from $W(\Gamma )$ to V is an inner derivation, meaning that $H^1(W(\Gamma ),V)=0$.

Proof. 

We apply D from $W(\Gamma )$ to V. Through Lemma 5, substituting D with $D-D_v$ for some $v\in V$, we suppose that $D\left(W\right)=0$ by Lemma 4. By Lemma 3, we have $D\in Der{(W(\Gamma ),V)}_0$. By Lemma 2, we only need to examine degree 0’s derivation. If we can prove that

$$D\left(L_{m,i}\right)=D\left(H_{m,i}\right)=D\left(I_{m,i}\right)=D\left(J_{m,i}\right)=0,$$

and we can deduce that $H^1(W(\Gamma ),V)=0$.

First, we assume the following:

$$\begin{array}{cc}\hfill D\left(L_{m,i}\right)=& \sum _{r\in \Gamma }(a_{m,i,r}^1{L}_r\otimes L_{m-r}+a_{m,i,r}^2{L}_r\otimes H_{m-r}+a_{m,i,r}^3{L}_r\otimes I_{m-r}+a_{m,i,r}^4{L}_r\otimes J_{m-r}\hfill \\ & +a_{m,i,r}^5{H}_r\otimes L_{m-r}+a_{m,i,r}^6{H}_r\otimes H_{m-r}+a_{m,i,r}^7{H}_r\otimes I_{m-r}+a_{m,i,r}^8{H}_r\otimes J_{m-r}\hfill \\ & +a_{m,i,r}^9{I}_r\otimes L_{m-r}+a_{m,i,r}^{10}{I}_r\otimes H_{m-r}+a_{m,i,r}^{11}{I}_r\otimes I_{m-r}+a_{m,i,r}^{12}{I}_r\otimes J_{m-r}\hfill \\ & +a_{m,i,r}^{13}{J}_r\otimes L_{m-r}+a_{m,i,r}^{14}{J}_r\otimes H_{m-r}+a_{m,i,r}^{15}{J}_r\otimes I_{m-r}+a_{m,i,r}^{16}{J}_r\otimes J_{m-r}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill D\left(H_{m,i}\right)=& \sum _{r\in \Gamma }(b_{m,i,r}^1{L}_r\otimes L_{m-r}+b_{m,i,r}^2{L}_r\otimes H_{m-r}+b_{m,i,r}^3{L}_r\otimes I_{m-r}+b_{m,i,r}^4{L}_r\otimes J_{m-r}\hfill \\ & +b_{m,i,r}^5{H}_r\otimes L_{m-r}+b_{m,i,r}^6{H}_r\otimes H_{m-r}+b_{m,i,r}^7{H}_r\otimes I_{m-r}+b_{m,i,r}^8{H}_r\otimes J_{m-r}\hfill \\ & +b_{m,i,r}^9{I}_r\otimes L_{m-r}+b_{m,i,r}^{10}{I}_r\otimes H_{m-r}+b_{m,i,r}^{11}{I}_r\otimes I_{m-r}+b_{m,i,r}^{12}{I}_r\otimes J_{m-r}\hfill \\ & +b_{m,i,r}^{13}{J}_r\otimes L_{m-r}+b_{m,i,r}^{14}{J}_r\otimes H_{m-r}+b_{m,i,r}^{15}{J}_r\otimes I_{m-r}+b_{m,i,r}^{16}{J}_r\otimes J_{m-r}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill D\left(I_{m,i}\right)=& \sum _{r\in \Gamma }(c_{m,i,r}^1{L}_r\otimes L_{m-r}+c_{m,i,r}^2{L}_r\otimes H_{m-r}+c_{m,i,r}^3{L}_r\otimes I_{m-r}+c_{m,i,r}^4{L}_r\otimes J_{m-r}\hfill \\ & +c_{m,i,r}^5{H}_r\otimes L_{m-r}+c_{m,i,r}^6{H}_r\otimes H_{m-r}+c_{m,i,r}^7{H}_r\otimes I_{m-r}+c_{m,i,r}^8{H}_r\otimes J_{m-r}\hfill \\ & +c_{m,i,r}^9{I}_r\otimes L_{m-r}+c_{m,i,r}^{10}{I}_r\otimes H_{m-r}+c_{m,i,r}^{11}{I}_r\otimes I_{m-r}+c_{m,i,r}^{12}{I}_r\otimes J_{m-r}\hfill \\ & +c_{m,i,r}^{13}{J}_r\otimes L_{m-r}+c_{m,i,r}^{14}{J}_r\otimes H_{m-r}+c_{m,i,r}^{15}{J}_r\otimes I_{m-r}+c_{m,i,r}^{16}{J}_r\otimes J_{m-r}),\hfill \end{array}$$

$$\begin{array}{cc}\hfill D\left(J_{m,i}\right)=& \sum _{r\in \Gamma }(d_{m,i,r}^1{L}_r\otimes L_{m-r}+d_{m,i,r}^2{L}_r\otimes H_{m-r}+d_{m,i,r}^3{L}_r\otimes I_{m-r}+d_{m,i,r}^4{L}_r\otimes J_{m-r}\hfill \\ & +d_{m,i,r}^5{H}_r\otimes L_{m-r}+d_{m,i,r}^6{H}_r\otimes H_{m-r}+d_{m,i,r}^7{H}_r\otimes I_{m-r}+d_{m,i,r}^8{H}_r\otimes J_{m-r}\hfill \\ & +d_{m,i,r}^9{I}_r\otimes L_{m-r}+d_{m,i,r}^{10}{I}_r\otimes H_{m-r}+d_{m,i,r}^{11}{I}_r\otimes I_{m-r}+d_{m,i,r}^{12}{I}_r\otimes J_{m-r}\hfill \\ & +d_{m,i,r}^{13}{J}_r\otimes L_{m-r}+d_{m,i,r}^{14}{J}_r\otimes H_{m-r}+d_{m,i,r}^{15}{J}_r\otimes I_{m-r}+d_{m,i,r}^{16}{J}_r\otimes J_{m-r}),\hfill \end{array}$$

where $X_{m,i,r}^1\cdots X_{m,i,r}^{16}\in \mathbb{F}[x,x^{-1},y,y^{-1}]$, $X_{m,0,r}^1=\cdots =X_{m,0,r}^{16}=0$, $X\in \{a,b,c,d\}$.

Applying D to the $[L_{m,i},L_{n,j}]=(m-n)L_{m+n,i+j}$, by comparing the coefficients on either side of the equation, we can obtain the following four equation types:

$$(m-n)a_{m+n,i+j,r}^A=a_{m,i,r-n}^A(r-2n)x^j+a_{m,i,r}^A(m-n-r)y^j+a_{n,j,r-m}^A(2m-r)x^i+a_{n,j,r}^A(m-n+r)y^i,$$

(14)

$$(m-n)a_{m+n,i+j,r}^B=a_{m,i,r-n}^B(r-2n)x^j+a_{m,i,r}^B(m-r)y^j+a_{n,j,r-m}^B(2m-r)x^i+a_{n,j,r}^B(r-n)y^i,$$

(15)

$$(m-n)a_{m+n,i+j,r}^C=a_{m,i,r-n}^C(r-n)x^j+a_{m,i,r}^C(m-n-r)y^j+a_{n,j,r-m}^C(m-r)x^i+a_{n,j,r}^C(m-n+r)y^i,$$

(16)

$$(m-n)a_{m+n,i+j,r}^D=a_{m,i,r-n}^D(r-n)x^j+a_{m,i,r}^D(m-r)y^j+a_{n,j,r-m}^D(m-r)x^i+a_{n,j,r}^D(r-n)y^i,$$

(17)

where $A\in \{1,3,4,9,11,12,13,15,16\}$, $B\in \{2,10,14\}$, $C\in \{5,7,8\}$, $D\in \left\{6\right\}$.

We then demonstrate that $a_{m,i,r}^A=a_{m,i,r}^B=a_{m,i,r}^C=a_{m,i,r}^D=0$.

Observing (14) and setting $m=j=0$, we obtain

$$n{a}_{n,i,r}^A=a_{0,i,r-n}^A(2n-r)+a_{0,i,r}^A(n+r).$$

(18)

Setting $m=-n$ and $i=0$ in (14), we obtain

$$2n{a}_{0,j,r}^A=(2n+r)a_{n,j,r+n}^A+(r-2n)a_{n,j,r}^A.$$

(19)

Substituting Formula (18) into (19) results in

$$2n(n+r)a_{0,i,r}^A={(2n+r)}^2{a}_{0,i,n+r}^A-{(2n-r)}^2{a}_{0,i,r-n}^A.$$

(20)

Setting $m=n=0$ and $j=1$ in (14), we find that

$$(x-y)a_{0,i,r}^A=(x^i-y^i)a_{0,1,r}^A.$$

(21)

Now, we can multiply $x-y$ on both sides of (20), and we obtain

$$2n(n+r)a_{0,1,r}^A={(2n+r)}^2{a}_{0,1,n+r}^A-{(2n-r)}^2{a}_{0,1,r-n}^A.$$

(22)

By definition, in a derivation, limited nonzero terms exist in $D\left(L_{m,i}\right)$. We decide on some $n\ne 0$ such that $a_{0,1,r+n}^A=a_{0,1,r-n}^A=0$. Then, we deduce $a_{0,1,r}^A=0$. Therefore, through (18) and (21), we obtain $a_{m,i,r}^A=0$.

In a similar vein, through (15)–(17), we can calculate $a_{m,i,r}^B=a_{m,i,r}^C=a_{m,i,r}^D=0$. Therefore, we figure that $D\left(L_{m,i}\right)=0$.

Subsequently, we attempt to prove that $D\left(H_{m,i}\right)=D\left(I_{m,i}\right)=D\left(J_{m,i}\right)=0$. We only prove $D\left(I_{m,i}\right)=0$, but the proof of $D\left(H_{m,i}\right)=D\left(J_{m,i}\right)=0$ is same.

We apply D to the $[L_{m,i},I_{n,j}]=(m-n)I_{m+n,i+j}$, and then we obtain following equations:

$$(m-n)c_{m+n,i+j,r}^A=c_{n,j,r-m}^A(2m-r)x^i+c_{n,j,r}^A(m-n+r)y^i,$$

(23)

$$(m-n)c_{m+n,i+j,r}^B=c_{n,j,r-m}^B(2m-r)x^i+c_{n,j,r}^B(r-n)y^i,$$

(24)

$$(m-n)c_{m+n,i+j,r}^C=c_{n,j,r-m}^C(m-r)x^i+c_{n,j,r}^C(m-n+r)y^i,$$

(25)

$$(m-n)c_{m+n,i+j,r}^D=c_{n,j,r-m}^D(m-r)x^i+c_{n,j,r}^D(r-n)y^i,$$

(26)

where $A\in \{1,3,4,9,11,12,13,15,16\}$, $B\in \{2,10,14\}$, $C\in \{5,7,8\}$, $D\in \left\{6\right\}$.

Based on Definition 4, we realize that $x\ne y$, $x\ne 0$, $y\ne 0$. Through (23) with $m=n=0$, we calculate the following:

$$c_{0,j,r}^A=0(r\ne 0),$$

(27)

for any $j\in \mathbb{Z}$.

Through (23) with $n=i=0$, $r\ne 0$ and $r\ne m$, we obtain $m{c}_{m,j,r}^A=0$, meaning that

$$c_{m,j,r}^A=0(m\ne 0,r\ne 0,r\ne m),$$

(28)

for any $j\in \mathbb{Z}$.

Through (27) and (28), we obtain

$$c_{m,j,r}^A=0(r\ne 0,r\ne m).$$

(29)

Through (23) with $r=m=-n\ne 0$, according to (29), we obtain $m{x}^i{c}_{-m,j,0}^A=0$, meaning that

$$c_{m,j,0}^A=0(m\ne 0).$$

(30)

Through (29) and (30), we obtain the following:

$$c_{m,j,r}^A=0(r\ne m),$$

(31)

for any $j\in \mathbb{Z}$.

Through (23) with $r=n$ and $m\ne 0$, based on (31), we deduce that $m{y}^i{c}_{n,j,n}^A=0$, yielding

$$c_{n,j,n}^A=0,$$

(32)

for any $j\in \mathbb{Z}$, $n\in \Gamma$.

Through (27)–(32), we obtain $c_{n,j,r}^A=0$ for any $j\in \mathbb{Z}$ and $n,r\in \Gamma$.

Next, we suppose that

$$\begin{array}{cc}\hfill D\left(I_{m,i}\right)=& \sum _{r\in \Gamma }(c_{m,i,r}^2{L}_r\otimes H_{m-r}+c_{m,i,r}^5{H}_r\otimes L_{m-r}+c_{m,i,r}^6{H}_r\otimes H_{m-r}+c_{m,i,r}^7{H}_r\otimes I_{m-r}\hfill \\ & +c_{m,i,r}^8{H}_r\otimes J_{m-r}+c_{m,i,r}^{10}{I}_r\otimes H_{m-r}+c_{m,i,r}^{14}{J}_r\otimes H_{m-r}),\hfill \end{array}$$

Next, applying D to $[I_{m,i},I_{n,j}]=0$, we obtain

$$c_{m,i,r}^B{y}^j=c_{n,j,r}^B{y}^i$$

(33)

$$c_{m,i,r}^C{x}^j=c_{n,i,r+n-m}^C{x}^i$$

(34)

$$c_{m,i,r}^D{y}^j=c_{n,j,r}^D{y}^i,$$

(35)

$$c_{m,i,r}^D{x}^j=c_{n,j,r+n-m}^D{x}^i$$

(36)

Through (24), with $m=n=0$, we find

$$c_{0,j,r}^B=0(r\ne 0),$$

(37)

for any $j\in \mathbb{Z}$.

Through (24) with $n=i=0$, $r\ne 0$ and $r\ne m$, we obtain $m{c}_{m,j,r}^B=0$, meaning that

$$c_{m,j,r}^B=0(m\ne 0,r\ne 0,r\ne m),$$

(38)

for any $j\in \mathbb{Z}$.

Through (37) and (38), we obtain

$$c_{m,j,r}^B=0(r\ne 0,r\ne m).$$

(39)

Through (24) with $r=m=-n\ne 0$, based on (39), we figure $m{x}^i{c}_{-m,j,0}^B=0$, meaning that

$$c_{m,j,0}^B=0(m\ne 0).$$

(40)

Through (39) and (40), we obtain

$$c_{m,j,r}^B=0(r\ne m),$$

(41)

for any $j\in \mathbb{Z}$.

Through (33) with $r=m\ne n$, we find

$$c_{m,j,m}^B=0,$$

(42)

for any $j\in \mathbb{Z}$, $m\in \Gamma$.

Through (41) and (42), we find $c_{m,j,r}^B=0$ for any $j\in \mathbb{Z}$, $m,r\in \Gamma$.

By the same token, through (25), (26), (34)–(36), we obtain $c_{n,j,r}^C=c_{n,j,r}^D=0$ for any $j\in \mathbb{Z}$, $n,r\in \Gamma$, we surmise that $D\left(I_{m,i}\right)=0$. □

Lemma 5. 

We assume that $\gamma \in V$ satisfies $u·\gamma \in \mathrm{Im}(1\otimes 1-\tau )$ for all $u\in W(\Gamma )$. Then, we obtain $\gamma \in \mathrm{Im}(1\otimes 1-\tau )$.

Proof. 

We let $\gamma ={\sum }_{\alpha \in \Pi }{\gamma }_{\alpha }$, where ${\gamma }_{\alpha }\in V_{\alpha }$, and $\Pi$ is a finite subset. As $1\otimes 1-\tau$ is homogeneous, $u·\gamma \in \mathrm{Im}(1\otimes 1-\tau )\iff u·{\gamma }_{\alpha }\in \mathrm{Im}(1\otimes 1-\tau )$. Since $L_{0,0}·{\gamma }_{\alpha }=\alpha {\gamma }_{\alpha }\in \mathrm{Im}(1\otimes 1-\tau )$, we have ${\gamma }_{\alpha }\in \mathrm{Im}(1\otimes 1-\tau )$ for any $\alpha \ne 0$. If $\alpha \ne 0$, we suppose that $\gamma ={\gamma }_0={\sum }_{\alpha \in \Gamma }{M}_{\alpha }\otimes N_{-\alpha }{f}_{\alpha }^{M,N}(x,y)$, where $M,N\in \{L,H,I,J\}$, $f_{\alpha }^{M,N}(x,y)\in \mathbb{F}[x^{\pm 1},y^{\pm 1}]$. Consider, also, that $L_{0,i}·{\gamma }_0=\sum \alpha M_{\alpha }\otimes N_{-\alpha }(y^i-x^i)\in \mathrm{Im}(1\otimes 1-\tau )$ and $\tau (M_{\alpha }\otimes N_{-\alpha }{f}_{\alpha }^{M,N}(x,y))=N_{-\alpha }\otimes M_{\alpha }{f}_{\alpha }^{M,N}(y,x)$. Then, we surmise that if $\alpha \ne 0$, we obtain $f_{\alpha }^{M,N}(x,y)+f_{-\alpha }^{M,N}(y,x)=0$. Hence ${\gamma }_0^{\prime }={\sum }_{\alpha \ne 0}{M}_{\alpha }\otimes N_{-\alpha }{f}_{\alpha }^{M,N}(x,y)\in \mathrm{Im}(1\otimes 1-\tau )$.

Let ${\gamma }_0^{″}={\gamma }_0-{\gamma }_0^{\prime }={\Sigma }_{M,N}{M}_0\otimes N_0{f}_0^{M,N}(x,y)$. Clearly, $L_{\alpha ,0}·{\gamma }_0^{″}\in \mathrm{Im}(1\otimes 1-\tau )$. Thus, logically, $f_0^{M,N}(x,y)+f_0^{M,N}(y,x)=0$. So ${\gamma }_0^{″}\in \mathrm{Im}(1\otimes 1-\tau )$. As such, ${\gamma }_0={\gamma }_0^{\prime }+{\gamma }_0^{″}\in \mathrm{Im}(1\otimes 1-\tau )$. □

Lemma 6 

([33]). Let $W(\Gamma )$ be an LA and $r\in \mathrm{Im}(1\otimes 1-\tau )$. As such, $(1\otimes 1\otimes 1+\epsilon +{\epsilon }^2)(1\otimes {\Delta }_r){\Delta }_r\left(x\right)=x·c\left(r\right),\forall x\in W(\Gamma )$. Specifically, a triple $(W(\Gamma ),[·,·],{\Delta }_r)$ is an LB if and only if r meets the criterion $x·c\left(r\right)=0$ for all $x\in W(\Gamma )$.

Lemma 7. 

If a variable $c\in W(\Gamma )\otimes W(\Gamma )\otimes W(\Gamma )$ meets the criterion $x·c=0$ for all $x\in W(\Gamma )$, then we obtain $c=0$.

Proof. 

We let $c={\sum }_{\alpha ,\beta ,\gamma }{M}_{\alpha }\otimes N_{\beta }\otimes P_{\gamma }{f}_{\alpha ,\beta ,\gamma }^{M,N,P}(x,y,z)$, where $f_{\alpha ,\beta ,\gamma }^{M,N,P}(x,y,z)$$\in \mathbb{F}[x^{\pm 1},y^{\pm 1},z^{\pm 1}]$, $M,N,P\in \{L,H,I,J\}$. We choose a total order on $\Gamma$, which aligns with the group structure of $\Gamma$. Then, we can deduce the lexicographic order on $\Gamma \times \Gamma \times \Gamma$. If we suppose that $c\ne 0$, the nonzero homogenerous term $M_{\alpha 0}\otimes N_{\beta 0}\otimes P_{\gamma 0}{f}_{\alpha 0,\beta 0,\gamma 0}^{M,N,P}(x,y,z)$ exists. Moreover, this term is assumed to be maximal. After, we choose a variable $\delta \in \Gamma$ such that $[L_{\delta },M_{\alpha 0}]=h(\delta ,\alpha 0)M_{\alpha 0+\delta }\ne 0$, where $h(\delta ,\alpha 0)\in \mathbb{F}$. Therefore, $h(\delta ,\alpha 0)M_{\alpha 0+\delta }\otimes N_{\beta 0}\otimes P_{\gamma 0}{f}_{\alpha 0,\beta 0,\gamma 0}^{M,N,P}(x,y,z)$ is a maximal term of $L_{\delta ,0}·c$, which contradicts $L_{\delta ,0}·c=0$. □

From the above discussion, we obtain the following:

Theorem 2. 

$Der\left(W\right(\Gamma ),V)=Inn\left(W\right(\Gamma ),V)$. Thus, every LB structure on $W(\Gamma )$ possesses a triangular coboundary.

Proof. 

We assume ($W(\Gamma ),\varphi ,\Delta$) is an LB structure on $W(\Gamma )$. By Definition 4, we have $\Delta \in Der\left(W\right(\Gamma ),V)$. By Lemma 4, we obtain $Der\left(W\right(\Gamma ),V)=Inn\left(W\right(\Gamma ),V)$. Thus, there exist $D\in Inn\left(W\right(\Gamma ),V)$ such that $\Delta =D$, $\Delta \left(x\right)=x·r$ for any $x\in W(\Gamma )$. By Definition 5, we have $\Delta ={\Delta }_r$ and $\Delta \left(x\right)={\Delta }_r\left(x\right)=x·r\in \mathrm{Im}\Delta$ for some $r\in W(\Gamma )\otimes W(\Gamma )$. Because $\mathrm{Im}\Delta \subset \mathrm{Im}(1\otimes 1-\tau )$, we obtain $x·r\in \mathrm{Im}(1\otimes 1-\tau )$ for all $x\in W(\Gamma )$. From Lemma 5, we can obtain $r\in \mathrm{Im}(1\otimes 1-\tau )$. Based on Lemma 6, r meets the criterion of MYBE $x·c\left(r\right)=0$. Through Lemma 7, we obtain $c\left(r\right)=0$. Thus, ($L,\varphi ,\Delta$) is a coboundary triangular LB. □

3. Quantization of $\mathbf{W}\mathbf{(}\mathbf{\Gamma }\mathbf{)}$

Definition 8 

([13,14]). For any variable x of a unital R-algebra (R represents ring), $a\in R$, $r,k\in {\mathbb{N}}^{+}$, we set the following:

$$x_a^{\left(r\right)}=(x+a)(x+a+1)\cdots (x+a+r-1),$$

$$x_a^{\left[r\right]}=(x+a)(x+a-1)\cdots (x+a-r+1),$$

$$\left({\displaystyle \genfrac{}{}{0pt}{}{a}{r}}\right)={\displaystyle \frac{a(a-1)\cdots (a-r+1)}{r!}},$$

$${\left({\displaystyle \genfrac{}{}{0pt}{}{a}{r}}\right)}_k={\displaystyle \frac{a(a-k)\cdots (a-(r-1\left)k\right)}{r!}}.$$

Specifically, we set $x_0^{\left(r\right)}=x^{\left(r\right)}$, $x_0^{\left[r\right]}=x^{\left[r\right]}$, $x_a^{\left(0\right)}=1$, $x_a^{\left[0\right]}=1$.

Lemma 8 

([13,14]). For any variable x of a unital $\mathbb{F}$-algebra, $a\in \mathbb{F}$, $r,s,t\in {\mathbb{N}}^{+}$, we know the following:

$$x_a^{(s+t)}=x_a^{\left(s\right)}{x}_{a+s}^{\left(t\right)},x_a^{[s+t]}=x_a^{\left[s\right]}{x}_{a+s}^{\left[t\right]},x_a^{\left[s\right]}=x_{a-s+1}^{\left(s\right)},$$

$$\sum _{s+t=r}{\displaystyle \frac{{(-1)}^t}{s!t!}}{x}_a^{\left[s\right]}{x}_b^{\left(t\right)}=\left({\displaystyle \genfrac{}{}{0pt}{}{a-b}{r}}\right)={\displaystyle \frac{(a-b)\cdots (a-b-r+1)}{r!}},$$

$$\sum _{s+t=r}{\displaystyle \frac{{(-1)}^t}{s!t!}}{x}_a^{\left[s\right]}{x}_{b-s}^{\left[t\right]}=\left({\displaystyle \genfrac{}{}{0pt}{}{a-b+r-1}{r}}\right)={\displaystyle \frac{(a-b)\cdots (a-b+r-1)}{r!}}.$$

Definition 9 

([33]). Let ($H,\mu ,\iota ,{\Delta }_0,S_0,ϵ$) represent a Hopf algebra. Moreover, the Drinfeld twist ϝ on H represents an invertible variable in $H\otimes H$ such that

$$(\mathsf{ϝ}\otimes 1)({\Delta }_0\otimes Id)\left(\mathsf{ϝ}\right)=(1\otimes \mathsf{ϝ})(Id\otimes {\Delta }_0)\left(\mathsf{ϝ}\right),(ϵ\otimes Id)\left(\mathsf{ϝ}\right)=1\otimes 1=(Id\otimes ϵ)\left(\mathsf{ϝ}\right).$$

Lemma 9 

([35]). Let ($H,\mu ,\iota ,{\Delta }_0,S_0,ϵ$) be a Hopf algebra, ϝ be a Drinfeld twist on H. It follows that $w=\mu (Id\otimes S_0)\mathsf{ϝ}$ is invertible in H with $w^{-1}=\mu (Id\otimes S_0){\mathsf{ϝ}}^{-1}$. Additionally, we establish $\Delta :H\to H\otimes H$ and $S:H\to H$ through

$$\Delta \left(x\right)=\mathsf{ϝ}{\Delta }_0\left(x\right){\mathsf{ϝ}}^{-1},S\left(x\right)=w{S}_0\left(x\right)w^{-1},\forall x\in H.$$

Afterward, ($H,\mu ,\iota ,\Delta ,S,ϵ$) becomes a new Hopf algebra, which we label as a twisting of H via the Drinfeld twist ϝ.

Lemma 10 

([13,14]). For any variables $x,y$ in an associated algebra, $p\in \mathbb{N}$, we know the following:

$$x{y}^p=\sum _{k=0}^p{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{p}{k}}\right)y^{p-k}{\left(ady\right)}^k\left(x\right).$$

Definition 10. 

Let $U\left(W\right(\Gamma \left)\right)$ be the universal enveloping algebra of $W(\Gamma )$ and ($U\left(W(\Gamma )\right),\mu ,\iota ,{\Delta }_0,S_0,ϵ$) be the standard Hopf algebra structure on $U\left(W\right(\Gamma \left)\right)$. As such, the coproduct ${\Delta }_0$, the antipode $S_0$, and the counit ϵ are defined

$${\Delta }_0\left(X\right)=X\otimes 1+1\otimes X,S_0\left(X\right)=-X,ϵ\left(X\right)=0,\forall X\in W(\Gamma ).$$

Specifically, ${\Delta }_0\left(1\right)=1\otimes 1$, $S_0\left(1\right)=ϵ\left(1\right)=1$.

Lemma 11. 

Let $X=-m^{-1}{L}_{0,0}$($m\in \Gamma \setminus \left\{0\right\}$) and $Y=I_{m,i}$. Then, we obtain $[X,Y]=Y$, and then X and Y can be used to generate a 2D nonabelian LA of $W(\Gamma )$.

Proof. 

For any $m\in \Gamma \setminus \left\{0\right\}\subseteq {\mathbb{F}}^{*}$, by $[L_{0,0},I_{m,i}]=-m{I}_{m,i}$, we can obtain $[X,Y]=Y$. □

Lemma 12. 

For any $a\in \mathbb{F}$, $n\in \Gamma$, $m\in \Gamma \setminus \left\{0\right\}$, $j\in \mathbb{Z}$ and $r\in {\mathbb{N}}^{+}$, we obtain

$$L_{n,j}{X}_a^{\left(r\right)}=X_{a-m^{-1}n}^{\left(r\right)}{L}_{n,j},L_{n,j}{X}_a^{\left[r\right]}=X_{a-m^{-1}n}^{\left[r\right]}{L}_{n,j},$$

(43)

$$H_{n,j}{X}_a^{\left(r\right)}=X_{a-m^{-1}n}^{\left(r\right)}{H}_{n,j},H_{n,j}{X}_a^{\left[r\right]}=X_{a-m^{-1}n}^{\left[r\right]}{H}_{n,j},$$

(44)

$$I_{n,j}{X}_a^{\left(r\right)}=X_{a-m^{-1}n}^{\left(r\right)}{I}_{n,j},I_{n,j}{X}_a^{\left[r\right]}=X_{a-m^{-1}n}^{\left[r\right]}{I}_{n,j},$$

(45)

$$J_{n,j}{X}_a^{\left(r\right)}=X_{a-m^{-1}n}^{\left(r\right)}{J}_{n,j},J_{n,j}{X}_a^{\left[r\right]}=X_{a-m^{-1}n}^{\left[r\right]}{J}_{n,j}.$$

(46)

Proof. 

Compared with the proof (43), the proof of (44)–(46) is similar. We prove (43) via induction on r. This applies in the case of $r=1$. Assuming that the case of r is also true, we consider the case of $r+1$, and we obtain

$$L_{n,j}{X}_a^{(r+1)}=L_{n,j}{X}_a^{\left(r\right)}(X+a+r)=X_{a-m^{-1}n}^{\left(r\right)}{L}_{n,j}(X+a+r),$$

(47)

$$[X,L_{n,j}]=-m^{-1}[L_{0,0},L_{n,j}].$$

(48)

Through (47) and (48), we obtain

$$\begin{array}{c}{X}_{a-m^{-1}n}^{\left(r\right)}{L}_{n,j}(X+a+r)=X_{a-m^{-1}n}^{\left(r\right)}(X{L}_{n,j}-m^{-1}n{L}_{n,j}+(a+r)L_{n,j})\hfill \\ =X_{a-m^{-1}n}^{\left(r\right)}(X+a-m^{-1}n+r)L_{n,j}=X_{a-m^{-1}n}^{(r+1)}{L}_{n,j}.\hfill \end{array}$$

Therefore, we deduce that $L_{n,j}{X}_a^{(r+1)}=X_{a-m^{-1}n}^{(r+1)}{L}_{n,j}$, meaning that $L_{n,j}{X}_a^{\left(r\right)}=X_{a-m^{-1}n}^{\left(r\right)}{L}_{n,j}$ is true. The proof of $L_{n,j}{X}_a^{\left[r\right]}=X_{a-m^{-1}n}^{\left[r\right]}{L}_{n,j}$ is similar. Then, we can prove (44)–(46). □

Lemma 13. 

For any $a\in \mathbb{F}$, $s,r\in {\mathbb{N}}^{+}$, we obtain

$$Y^s{X}_a^{\left(r\right)}=X_{a-s}^{\left(r\right)}{Y}^s,Y^s{X}_a^{\left[r\right]}=X_{a-s}^{\left[r\right]}{Y}^s.$$

(49)

Proof. 

The example of $s=r=1$ is clear. If $s=1$, we prove (49) by induction on r. Then, we obtain the following:

$$\begin{array}{c}Y{X}_a^{(r+1)}=Y{X}_a^{\left(r\right)}(X+a+r)=X_{a-1}^{\left(r\right)}Y(X+a+r)=X_{a-1}^{\left(r\right)}(XY-Y+(a+r)Y)\hfill \\ =X_{a-1}^{\left(r\right)}(X+a+r-1)Y=X_{a-1}^{(r+1)}Y,\hfill \end{array}$$

and this means that

$$Y{X}_a^{\left(r\right)}=X_{a-1}^{\left(r\right)}Y.$$

(50)

Suppose that $Y^s{X}_a^{\left(r\right)}=X_{a-s}^{\left(r\right)}{Y}^s$. Through (50), we calculate

$$Y^{s+1}{X}_a^{\left(r\right)}=Y{X}_{a-s}^{\left(r\right)}{Y}^s=X_{a-(s+1)}^{r}Y{Y}^s=X_{a-(s+1)}^r{Y}^{s+1}.$$

□

Lemma 14. 

For any $m\in \Gamma \setminus \left\{0\right\}$, $n\in \Gamma$, $j\in \mathbb{Z}$, $r\in {\mathbb{N}}^{+}$, we obtain

$$L_{n,j}{Y}^r=Y^r{L}_{n,j}+r{Y}^{r-1}(n-m)I_{m+n,i+j},$$

(51)

$$H_{n,j}{Y}^r=Y^r{H}_{n,j}+r{Y}^{r-1}{J}_{m+n,i+j},$$

(52)

$$I_{n,j}{Y}^r=Y^r{I}_{n,j},$$

(53)

$$J_{n,j}{Y}^r=Y^r{I}_{n,j}.$$

(54)

Proof. 

We only prove (51) and (53).

$$L_{n,j}{Y}^r=\sum _{k=0}^r{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{r}{k}}\right)I_{m,i}^{r-k}{\left(ad{I}_{m,i}\right)}^k{L}_{n,j}=Y^r{L}_{n,j}+r{Y}^{r-1}(n-m)I_{m+n,i+j},$$

and

$$I_{n,j}{Y}^r=\sum _{k=0}^r{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{r}{k}}\right)I_{m,i}^{r-k}{\left(ad{I}_{m,i}\right)}^k{I}_{n,j}=Y^r{I}_{n,j}.$$

□

Definition 11 

([13,14]). For $a\in \mathbb{F}$, set

$${\mathsf{ϝ}}_a=\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_a^{\left[r\right]}\otimes Y^r{t}^r,F_a=\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{\left(r\right)}\otimes Y^r{t}^r.$$

(55)

$$U_a=\mu (S_0\otimes I_d)F_a,V_a=\mu (I_d\otimes S_0){\mathsf{ϝ}}_a.$$

(56)

In particular, we set $\mathsf{ϝ}={\mathsf{ϝ}}_0$, $F=F_0$, $u=u_0$, $v=v_0$. Since $S_0\left(X_a^{\left(r\right)}\right)={(-1)}^r{X}_{-a}^{\left[r\right]}$ and $S_0\left(Y^r\right)={(-1)}^r{Y}^r$, we obtain $U_a={\sum }_{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a}^{\left[r\right]}{Y}^r{t}^r$ and $V_a={\sum }_{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{\left[r\right]}{Y}^r{t}^r$.

Lemma 15 

([13,14]). For $a,b\in \mathbb{F}$, we have ${\mathsf{ϝ}}_a{F}_b=1\otimes {(1-Yt)}^{a-b}$ and $V_a{U}_b={(1-Yt)}^{-a-b}$, ${\mathsf{ϝ}}_a$, $F_a$, $U_a$, $V_a$ are invertible variables with ${\mathsf{ϝ}}_a=F_a^{-1}$ and $U_a=V_a^{-1}$.

Proof. 

$$\begin{array}{c}{\mathsf{ϝ}}_a{F}_b=\sum _{r,s=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!s!}}{X}_a^{\left[r\right]}{X}_b^{\left(s\right)}\otimes Y^r{Y}^s{t}^r{t}^s=\sum _{k=0}^{\infty }{(-1)}^k\sum _{r+s=k}{\displaystyle \frac{{(-1)}^s}{r!s!}}{X}_a^{\left[r\right]}{X}_b^{\left(s\right)}\otimes Y^k{t}^k\hfill \\ =\sum _{k=0}^{\infty }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{a-b}{k}}\right)\otimes Y^k{t}^k=1\otimes {(1-Yt)}^{a-b}\hfill \end{array}$$

From (27), (44) and Lemma 8, we obtain

$$\begin{array}{c}{V}_a{U}_b=\sum _{r,s=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!s!}}{X}_a^{\left[r\right]}{Y}^r{X}_{-b}^{\left[s\right]}{Y}^s{t}^{r+s}\otimes =\sum _{k=0}^{\infty }{(-1)}^k\sum _{r+s=k}{\displaystyle \frac{{(-1)}^s}{r!s!}}{X}_a^{\left[r\right]}{X}_{-b-r}^{\left[s\right]}\otimes Y^k{t}^k\hfill \\ =\sum _{k=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{a+b+k-1}{k}}\right)Y^k{t}^k=1\otimes {(1-Yt)}^{-a-b}\hfill \end{array}$$

Then, we can deduce that ${\mathsf{ϝ}}_a=F_a^{-1}$, $U_a=V_a^{-1}$, $\mathsf{ϝ}=F^{-1}$ and $U=V^{-1}$. □

Lemma 16. 

For any $m\in \Gamma \setminus \left\{0\right\}$, $n\in \Gamma$, $i,j\in \mathbb{Z}$, $a\in \mathbb{F}$, we obtain

$$(L_{n,j}\otimes 1)F_a=F_{a-m^{-1}n}(L_{n,j}\otimes 1),$$

(57)

$$(H_{n,j}\otimes 1)F_a=F_{a-m^{-1}n}(H_{n,j}\otimes 1),$$

(58)

$$(I_{n,j}\otimes 1)F_a=F_{a-m^{-1}n}(I_{n,j}\otimes 1),$$

(59)

$$(J_{n,j}\otimes 1)F_a=F_{a-m^{-1}n}(J_{n,j}\otimes 1),$$

(60)

$$(1\otimes L_{n,j})F_a=F_a(1\otimes L_{n,j})+(n-m)F_{a+1}(X_a^{\left(1\right)}\otimes I_{m+n,i+j}t),$$

(61)

$$(1\otimes H_{n,j})F_a=F_a(1\otimes H_{n,j})+F_{a+1}(X_a^{\left(1\right)}\otimes J_{m+n,i+j}t),$$

(62)

$$(1\otimes I_{n,j})F_a=F_a(1\otimes I_{n,j}),$$

(63)

$$(1\otimes J_{n,j})F_a=F_a(1\otimes J_{n,j}).$$

(64)

Proof. 

We only prove (57), (61), (63); the proofs of other equations are similar.

Through (43) and (55), we obtain

$$(L_{n,j}\otimes 1)F_a=\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{L}_{n,j}{X}_a^{\left(r\right)}\otimes Y^r{t}^r=\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_{a-m^{-1}n}^{\left(r\right)}{L}_{n,j}\otimes Y^r{t}^r=F_{a-m^{-1}n}(L_{n,j}\otimes 1)$$

The proof of (58)–(60) is similar to that of (57).

Through (51) and (55), we obtain

$$\begin{array}{cc}\hfill (1\otimes L_{n,j})F_a& =\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{\left(r\right)}\otimes (Y^r{L}_{n,j}+r{Y}^{r-1}(n-m)I_{m+n,i+j})t^r\hfill \\ & =\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{\left(r\right)}\otimes Y^r{L}_{n,j}{t}^r+(n-m)\sum _{r=1}^{\infty }{\displaystyle \frac{1}{(r-1)!}}{X}_a^{\left(r\right)}\otimes Y^{r-1}{I}_{m+n,i+j}{t}^r\hfill \\ & =(\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{\left(r\right)}\otimes Y^r{t}^r)(1\otimes L_{n,j})+(n-m)\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{(r+1)}\otimes Y^r{I}_{m+n,i+j}{t}^{r+1}\hfill \\ & =F_a(1\otimes L_{n,j})+(n-m)\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_a^{\left(1\right)}{X}_{a+1}^{\left(r\right)}\otimes Y^r{I}_{m+n,i+j}{t}^r·t\hfill \\ & =F_a(1\otimes L_{n,j})+(n-m)(\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_{a+1}^{\left(r\right)}\otimes Y^r{t}^r)(X_a^{\left(1\right)}\otimes I_{m+n,i+j}t)\hfill \\ & =F_a(1\otimes L_{n,j})+(n-m)F_{a+1}(X_a^{\left(1\right)}\otimes I_{m+n,i+j}t).\hfill \end{array}$$

Through (53) and (55), we obtain

$$\begin{array}{cc}\hfill (1\otimes I_{n,j})F_a& =(1\otimes I_{n,j})(\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_{a+1}^{\left(r\right)}\otimes Y^r{t}^r)=\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_{a+1}^{\left(r\right)}\otimes I_{n,j}{Y}^r{t}^r=\sum _{r=0}^{\infty }{\displaystyle \frac{1}{r!}}{X}_{a+1}^{\left(r\right)}\otimes Y^r{I}_{n,j}{t}^r\hfill \\ & =F_a(1\otimes I_{n,j}).\hfill \end{array}$$

□

Lemma 17. 

For any $m\in \Gamma \setminus \left\{0\right\}$, $n\in \Gamma$, $j\in \mathbb{Z}$, $a\in \mathbb{F}$, we obtain

$$L_{n,j}{U}_a=U_{a+m^{-1}n}{L}_{n,j}+(m-n)U_{a+m^{-1}n}{X}_{-a-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}t,$$

(65)

$$H_{n,j}{U}_a=U_{a+m^{-1}n}{H}_{n,j}-U_{a+m^{-1}n}{X}_{-a-m^{-1}n}^{\left[1\right]}{J}_{m+n,i+j}t,$$

(66)

$$I_{n,j}{U}_a=U_{a+m^{-1}n}{I}_{n,j},$$

(67)

$$J_{n,j}{U}_a=U_{a+m^{-1}n}{J}_{n,j}.$$

(68)

Proof. 

We only prove (65) and (67). Through (43), (49), (51) and (56), we obtain

$$\begin{array}{cc}\hfill L_{n,j}{U}_a& =L_{n,j}\left(\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a}^{\left[r\right]}{Y}^r{t}^r\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{L}_{n,j}{X}_{-a}^{\left[r\right]}{Y}^r{t}^r\hfill \\ & =\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{L}_{n,j}{Y}^r{t}^r=\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}(Y^r{L}_{n,j}+r{Y}^{r-1}(n-m)I_{m+n,i+j})t^r\hfill \\ & =\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{Y}^r{L}_{n,j}{t}^r+\sum _{r=1}^{\infty }{\displaystyle \frac{{(-1)}^r}{(r-1)!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{Y}^{r-1}(n-m)I_{m+n,i+j}{t}^r\hfill \\ & =U_{a+m^{-1}n}{L}_{n,j}+\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^{r+1}}{r!}}{X}_{-a-m^{-1}n}^{[r+1]}{Y}^r(n-m)I_{m+n,i+j}{t}^{r+1}\hfill \\ & =U_{a+m^{-1}n}{L}_{n,j}+(m-n)\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{X}_{-a-m^{-1}n-r}^{\left[1\right]}{Y}^r{I}_{m+n,i+j}{t}^r·t\hfill \\ & =U_{a+m^{-1}n}{L}_{n,j}+(m-n)\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{Y}^r{X}_{-a-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}{t}^r·t\hfill \\ & =U_{a+m^{-1}n}{L}_{n,j}+(m-n)U_{a+m^{-1}n}{X}_{-a-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}t.\hfill \end{array}$$

$$\begin{array}{cc}\hfill I_{n,j}{U}_a& =I_{n,j}\left(\sum _{r=0}^{\infty }{\displaystyle \frac{(-1^r)}{r!}}{X}_{-a}^{\left[r\right]}{Y}^r{t}^r\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{(-1^r)}{r!}}{I}_{n,j}{X}_{-a}^{\left[r\right]}{Y}^r{t}^r=\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{I}_{n,j}{Y}^r{t}^r\hfill \\ & =\sum _{r=0}^{\infty }{\displaystyle \frac{{(-1)}^r}{r!}}{X}_{-a-m^{-1}n}^{\left[r\right]}{Y}^r{I}_{n,j}{t}^r=U_{a+m^{-1}n}{I}_{n,j}.\hfill \end{array}$$

□

Theorem 3. 

Among the two distinct variables $X=-m^{-1}{L}_{0,0}$ ($m\in \Gamma \setminus \left\{0\right\}$) and $Y=I_{m,i}$ such that $[X,Y]=Y$ in $W(\Gamma )$, noncommutative algebra and noncocommutative Hopf algebra structures exist ($U\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right],\mu ,\iota ,\Delta ,S,ϵ$) on $U\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right]$, meaning that $U\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right]/tU\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right]\cong U\left(W\right(\Gamma \left)\right)$, which retains the product and counit of $U\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right]$. The coproduct and antipode are defined as follows:

$$\Delta \left(L_{n,j}\right)=L_{n,j}\otimes {(1-Yt)}^{m^{-1}n}+1\otimes L_{n,j}+(n-m)(X^{\left(1\right)}\otimes {(1-Yt)}^{-1}{I}_{m+n,i+j}t),$$

(69)

$$\Delta \left(H_{n,j}\right)=H_{n,j}\otimes {(1-Yt)}^{m^{-1}n}+1\otimes H_{n,j}+X^{\left(1\right)}\otimes {(1-Yt)}^{-1}{J}_{m+n,i+j}t,$$

(70)

$$\Delta \left(I_{n,j}\right)=I_{n,j}\otimes {(1-Yt)}^{m^{-1}n}+1\otimes I_{n,j},$$

(71)

$$\Delta \left(J_{n,j}\right)=J_{n,j}\otimes {(1-Yt)}^{m^{-1}n}+1\otimes J_{n,j},$$

(72)

$$S\left(L_{n,j}\right)=-{(1-Yt)}^{-m^{-1}n}(L_{n,j}+(m-n)X_{-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}t),$$

(73)

$$S\left(H_{n,j}\right)=-{(1-Yt)}^{-m^{-1}n}(H_{n,j}-X_{-m^{-1}n}^{\left[1\right]}{J}_{m+n,i+j}t),$$

(74)

$$S\left(I_{n,j}\right)=-{(1-Yt)}^{-m^{-1}n}{I}_{n,j},$$

(75)

$$S\left(J_{n,j}\right)=-{(1-Yt)}^{-m^{-1}n}{J}_{n,j},$$

(76)

where $m\in \Gamma \setminus \left\{0\right\}$, $n\in \Gamma$, $i,j\in \mathbb{Z}$.

Proof. 

We only prove (69), (71), (73), (75), but the cases of (70), (72), (74), and (76) are similar. Through Definition 11 and Lemmas 15–17, we obtain

$$\begin{array}{cc}\hfill \Delta \left(L_{n,j}\right)& =\mathsf{ϝ}{\Delta }_0\left(L_{n,j}\right){\mathsf{ϝ}}^{-1}=\mathsf{ϝ}(L_{n,j}\otimes 1+1\otimes L_{n,j}){\mathsf{ϝ}}^{-1}=\mathsf{ϝ}(L_{n,j}\otimes 1)F+\mathsf{ϝ}(1\otimes L_{n,j})F\hfill \\ & =\mathsf{ϝ}{F}_{-m^{-1}n}(L_{n,j}\otimes 1)+\mathsf{ϝ}(F(1\otimes L_{n,j})+(n-m)F_1(X^{\left(1\right)}\otimes I_{m+n,i+j}t))\hfill \\ & =(1\otimes {(1-Yt)}^{m^{-1}n})(L_{n,j}\otimes 1)+1\otimes L_{n,j}+(n-m)(X^{\left(1\right)}\otimes {(1-Yt)}^{-1}{I}_{m+n,i+j}t)\hfill \\ & =L_{n,j}\otimes {(1-Yt)}^{m^{-1}n}+1\otimes L_{n,j}+(n-m)(X^{\left(1\right)}\otimes {(1-Yt)}^{-1}{I}_{m+n,i+j}t).\hfill \end{array}$$

$$\begin{array}{cc}\hfill \Delta \left(I_{n,j}\right)& =\mathsf{ϝ}{\Delta }_0\left(I_{n,j}\right){\mathsf{ϝ}}^{-1}=\mathsf{ϝ}(I_{n,j}\otimes 1+1\otimes I_{n,j}){\mathsf{ϝ}}^{-1}=\mathsf{ϝ}(I_{n,j}\otimes 1)F+\mathsf{ϝ}(1\otimes I_{n,j})F\hfill \\ & =\mathsf{ϝ}{F}_{-m^{-1}n}(I_{n,j}\otimes 1)+\mathsf{ϝ}F(1\otimes I_{n,j})=I_{n,j}\otimes {(1-Yt)}^{m^{-1}n}+1\otimes I_{n,j}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill S\left(L_{n,j}\right)& =-V{L}_{n,j}U=-V(U_{m^{-1}n}{L}_{n,j}+(m-n)U_{m^{-1}n}{X}_{-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}t)\hfill \\ & =-V{U}_{m^{-1}n}(L_{n,j}+(m-n)X_{-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}t)\hfill \\ & =-{(1-Yt)}^{-m^{-1}n}(L_{n,j}+(m-n)X_{-m^{-1}n}^{\left[1\right]}{I}_{m+n,i+j}t).\hfill \end{array}$$

$$\begin{array}{cc}\hfill S\left(I_{n,j}\right)& =-V{I}_{n,j}U=-VU{m}^{-1}n{I}_{n,j}=-{(1-Yt)}^{-m^{-1}n}{I}_{n,j}.\hfill \end{array}$$

□

Conclusion 1. 

We find that all LB structures on $W(\Gamma )$ possess a triangular coboundary.

Conclusion 2. 

Noncommutative algebra and noncocommutative Hopf algebra structures ($U\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right],\mu ,\iota ,\Delta ,S,ϵ$) exist on $U\left(W\right(\Gamma \left)\right)\left[\right[t\left]\right]$.

In future research, we can examine the LB structures of other algebras. Moreover, if their LB structures possess a triangular coboundary, we can also use the Drinfeld-twist quantization technique to quantize them.