Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$

TL;DR

This paper classifies all Lie super-bialgebra structures on a specific generalized super $W$-algebra, showing they are all triangular coboundary by analyzing its cohomology.

Contribution

It proves the triviality of the first cohomology group for the algebra, leading to a complete classification of its Lie super-bialgebra structures as triangular coboundary.

Findings

All Lie super-bialgebra structures are triangular coboundary.

The first cohomology group with coefficients in the adjoint tensor module is trivial.

Provides a cohomological approach to classifying super-bialgebra structures.

Abstract

In this paper, Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$ are investigated. By proving the first cohomology group of $\mathfrak{L}$ with coefficients in its adjoint tensor module is trivial, namely, $H^1(\mathfrak{L},\mathfrak{L}\otimes {\mathfrak{L}})=0$, we obtain that all Lie super-bialgebra structures on $\mathfrak{L}$ are triangular coboundary.

Equations

$$\displaystyle\begin{array}[]{lllll}&[L_{p},L_{q}]=(p-q)L_{p+q},&[L_{p},I_{q}]=(p-q)I_{p+q},\vspace*{6pt}\\ &[L_{p},H_{r}]=(\frac{p}{2}-r)H_{p+r},&[G_{r},G_{t}]=I_{r+t},\vspace*{6pt}\\ &[L_{p},G_{r}]=(\frac{p}{2}-r)G_{p+r},&[I_{p},G_{r}]=(p-2r)H_{p+r}.\end{array}$$

$$\tau (x\otimes y)=(-1{)}^{[x][y]}y\otimes x,\forall x,y\in L.$$

$$\xi =(1\otimes \tau )\cdot (\tau \otimes 1):x_1\otimes x_2\otimes x_3\mapsto (-1{)}^{[x_1]([x_2]+[x_3])}{x}_2\otimes x_3\otimes x_1,$$

$$\phi (L_i,L_j)\subset L_{i+j},\mathrm{Ker}(1\otimes 1-\tau )\subset \mathrm{Ker}\phi ,\text{vspace}\ast 6pt$$

$$\phi \cdot (1\otimes \phi )\cdot (1\otimes 1\otimes 1+\xi +{\xi }^2)=0:L\otimes L\otimes L\to L.$$

$$\displaystyle\Delta(L_{i})\subset\mbox{$\sum\limits_{j+k=i}$}L_{j}\otimes L_{k},\ \ \ \ {\rm Im}\Delta\subset{\rm Im}(1\otimes 1-\tau),\vspace*{6pt}$$

$$(1\otimes 1\otimes 1+\xi +{\xi }^2)\cdot (1\otimes \Delta )\cdot \Delta =0:L⟶L\otimes L\otimes L.$$

$$x\circ(\mbox{$\sum\limits_{i}$}(a_{i}\otimes b_{i}))=\mbox{$\sum\limits_{i}$}([x,\,a_{i}]\otimes b_{i}+(-1)^{[x][a_{i}]}a_{i}\otimes[x,\,b_{i}]),\ \ \forall\,x,\,a_{i},\,b_{i}\in L.$$

$${\Delta }_r(x)=(-1{)}^{[r][x]}x\circ r,\forall x\in L.$$

$$c(r):=[r^{12},r^{13}]+[r^{12},r^{23}]+[r^{13},r^{23}]=0,$$

$$x\circ c(r)=0,\forall x\in L.$$

$$\displaystyle\begin{array}[]{lll}&&r^{12}=\mbox{$\sum\limits_{i}$}a_{i}\otimes b_{i}\otimes 1=r\otimes 1,\vspace*{6pt}\\ &&r^{13}=\mbox{$\sum\limits_{i}$}a_{i}\otimes 1\otimes b_{i}=(\tau\otimes 1)(1\otimes r),\vspace*{6pt}\\ &&r^{23}=\mbox{$\sum\limits_{i}$}1\otimes a_{i}\otimes b_{i}=1\otimes r\end{array}$$

$$\displaystyle\begin{array}[]{lll}&&[r^{12},r^{13}]=\mbox{$\sum\limits_{i,j}$}(-1)^{[a_{j}][b_{i}]}[a_{i},a_{j}]\otimes b_{i}\otimes b_{j},\vspace*{6pt}\\ &&[r^{12},r^{23}]=\mbox{$\sum\limits_{i,j}$}a_{i}\otimes[b_{i},a_{j}]\otimes b_{j},\vspace*{6pt}\\ &&[r^{13},r^{23}]=\mbox{$\sum\limits_{i,j}$}(-1)^{[a_{j}][b_{i}]}a_{i}\otimes a_{j}\otimes[b_{i},b_{j}]\end{array}$$

$$d([x,y])=(-1{)}^{[d][x]}x\circ d(y)-(-1{)}^{[y]([d]+[x])}y\circ d(x),\forall x,y\in L,$$

$$a_{\mathrm{Inn}}(x)=(-1{)}^{[a][x]}x\circ a,\forall x\in L,[a]=p.$$

$$H^1(L,V)\cong \mathrm{Der}(L,V)/\mathrm{Inn}(L,V).$$

$${\mathrm{Der}}_r(\mathfrak{L},\mathfrak{V})=\{d\in \mathrm{Der}(\mathfrak{L},\mathfrak{V})|d({\mathfrak{L}}_p)\subset {\mathfrak{V}}_{p+r},\forall p\in {\mathbb{Z}}_s\},$$

$$\mathrm{deg}(L_p)>\mathrm{deg}(I_q)>\mathrm{deg}(G_r)>\mathrm{deg}(H_t)$$

$$\mathrm{deg}(A_p\otimes B_q)=(\mathrm{deg}(A_p),\mathrm{deg}(B_q)),$$

$$\{L_p(Resp.I_p,G_p,H_p)|p\in {\mathbb{Z}}_s\}.$$

$$u=\mbox{$\sum\limits$}a_{p,q}^{A,B}A_{p}\otimes B_{q},$$

$$\displaystyle\begin{array}[]{lllll}&[L_{p},L_{q}]=(p-q)L_{p+q},&[L_{p},I_{q}]=(p-q)I_{p+q},\vspace*{6pt}\\ &[L_{p},H_{q}]=(\frac{p}{2}-q)H_{p+q},&[G_{p},G_{q}]=I_{p+q},\vspace*{6pt}\\ &[L_{p},G_{q}]=(\frac{p}{2}-q)G_{p+q},&[I_{p},G_{q}]=(p-2q)H_{p+q}.\end{array}$$

$${\mathfrak{L}}_p=\{x\in \mathfrak{L}|[L_0,x]=-px\}.$$

$$L_0\circ d_{t,\overline{0}}(L_p)-L_p\circ d_{t,\overline{0}}(L_0)=-p{d}_{t,\overline{0}}(L_p),$$

$$-(p+t)d_{t,\overline{0}}(L_p)-L_p\circ d_{t,\overline{0}}(L_0)=-p{d}_{t,\overline{0}}(L_p),d_{t,\overline{0}}(L_p)=L_p\circ (-\frac{d_{t,\overline{0}}(L_0)}{t}).$$

$$d_{t,\overline{0}}(I_p)=I_p\circ (-\frac{d_{t,\overline{0}}(L_0)}{t}),\text{vspace}\ast 6pt$$

$$d_{t,\overline{0}}(G_p)=G_p\circ (-\frac{d_{t,\overline{0}}(L_0)}{t}),\text{vspace}\ast 6pt$$

$$d_{t,\overline{0}}(H_p)=H_p\circ (-\frac{d_{t,\overline{0}}(L_0)}{t}).$$

$$L_0\circ d_0(L_p)-L_p\circ d_0(L_0)=-p{d}_0(L_p),$$

$$L_p\circ d_0(L_0)=0,\forall p\in \Gamma .$$

$$d_{0,\overline{0}}(L_{ϵ})$$

$$d_{0,\overline{1}}(L_{-ϵ})$$

$$d_{0}(L_{\epsilon})\cap(\textbf{L}\otimes\textbf{L})=\mbox{$\sum\limits_{{\beta}_{j}}$}\mbox{$\sum\limits_{i}$}a_{i}L_{{\beta}_{j}+i\epsilon}\otimes L_{-{\beta}_{j}-(i-1)\epsilon},$$

Full text

Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$

Hao Wang1), Huanxia Fa2), Junbo Li2)

*1)*Wu Wen-Tsun Key Laboratory of Mathematics and School of Mathematical Sciences,

University of Science and Technology of China, Hefei 230026, China

*2)*School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China

Abstract. In this paper, Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$ are investigated. By proving the first cohomology group of $\mathfrak{L}$ with coefficients in its adjoint tensor module is trivial, namely, $H^{1}(\mathfrak{L},\mathfrak{L}\otimes{\mathfrak{L}})=0$, we obtain that all Lie super-bialgebra structures on $\mathfrak{L}$ are triangular coboundary.

Key words: cohomology group, generalized super $W$-algebras, Lie super-bialgebra

*MR(2000) Subject Classification: * 17B10, 17B65, 17B68

1 Introduction

It is well known that the Virasoro algebra (named after the physicist Miguel Angel Virasoro) is a very important infinite dimensional Lie algebra and is widely used in conformal field theory and string theory. After that much attention has been paid to the Virasoro type Lie algebras or superalgebras (which contains the Virasoro algebra as its subalgebra), including their construction, structures and representations. The $W$-algebra $W(2,2)$ is certainly a Virasoro type Lie algebra, which plays important rolls in many areas of mathematics and physics (which was introduced in [24] during the study of vertex operator algebras). It possesses a basis $\{L_{m},\,I_{m}|m\in\mathbb{Z}\}$ as a vector space over the complex field $\mathbb{C}$, with the Lie brackets $[L_{m},L_{n}]=(m-n)L_{m+n}$, $[L_{m},I_{n}]=(m-n)I_{m+n}$, $[I_{m},I_{n}]=0$. Structures and representations of $W(2,2)$ are extensively investigated in many references, such as [1], [6], [8], [9], [10] and [25].

Some Lie superalgebras with $W$-algebra $W(2,2)$ as their even parts were constructed in [20] as an application of the classification of Balinsky-Novikov super-algebras with dimension $2|2$. The generalized super $W$-algebra $W(2,2)$ whose even part is the generalized $W$-algebra $W(2,2)$ is an infinite-dimensional Lie super algebra with the $\mathbb{C}$-basis $\{L_{p},\,I_{p},\,G_{r},\,H_{r}\,|\,p\in\Gamma,\,r\in s+\Gamma\}$, where $\Gamma$ is an nontrivial abelian subgroup of $\mathbb{R}$, $2s\in\Gamma$, admitting the following non-vanishing super-brackets:

[TABLE]

For convenience, we denote such an algebra by $\mathfrak{L}$.

In this paper, we investigated the Lie super-bialgebra structures of $\mathfrak{L}$, and proved that all Lie super bialgebra structures on $\mathfrak{L}$ are triangular coboundary (see Theorem 2.3). Our motivations mainly originate from the following.

• •

To construct Lie super bialgebras and their quantizations is an important approach to produce new quantum groups. Since the notion of Lie bialgebras was introduced by Drinfeld in 1983 (Refs.[2, 3]), there have appeared several papers on Lie coalgebras or Lie (super) bialgebras (e.g., Refs. [4, 5, 9, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23]).

• •

Though the result that all Lie super bialgebra structures on $\mathfrak{L}$ are triangular coboundary is not surprising, and coboundary triangular Lie super bialgebras have relatively simple structures, it seems to us that it is still worth paying more attention on them, as one can see from [18] that by considering dual structures of Lie bialgebras, one may expect to obtain some new Lie algebras, which is our next goal.

Finally we would like to make some remarks. We observe that many papers are forced on the Virasoro type Lie superalgebras which contain the super Virasoro algebra as their subalgebra (e.g., Refs. [4, 23]), especially the $N=2$ super Virasoro algebras. It is easy to find that the algebra $\mathfrak{L}$ doesn’t contain the super Virasoro Lie algebra as it’s subalgebra, so the methods developed there are not applicable to $\mathfrak{L}$. What’s more, the subscript set $\Gamma$ in our algebra is an arbitrary nontrivial abelian subgroup of $\mathbb{R}$, not necessarily discrete. In other words, it is possible that we can’t find a minimal positive element in $\Gamma$. All these make the study of $\mathfrak{L}$ more challengeable and attractive (this is also one of our motivations to present our results here), we need to find some new methods to handel these problems. For instance, one of our strategies used in the present paper is to introduce the length of a derivation so that the determination of derivations can be done by induction on the length. We would also like to mention that although central extension makes the representations of $\mathfrak{L}$ more interesting, it makes only little difference about the bialgebra structures. So we only consider the centerless case here. The investigation of central extension of $\mathfrak{L}$ and its representation theory is also our next goal.

2 The main results

We briefly recall some notations of Lie super-bialgebras. Let $L=L_{\overline{0}}\oplus L_{\overline{1}}$ be a vector space over $\mathbb{C}$, and all elements below are assumed to be $\mathbb{Z}_{2}$-homogeneous in this subsection, where $\mathbb{Z}_{2}=\{\overline{0},\overline{1}\}$. For any homogeneous element $x\in L$, we always denote by $[x]\in\mathbb{Z}_{2}$ the parity of $x$ , i.e., $x\in L_{[x]}$. Throughout what follows, if $[x]$ occurs in an expression, then it is assumed that $x$ is homogeneous and that the expression extends to the other elements by linearity. Denote by $\tau$ the super-twist map of $L\otimes L$:

[TABLE]

Denote by $\xi$ the super-cyclic map which cyclically permutes the coordinates in $L\otimes L\otimes L$:

[TABLE]

for all $x_{1},x_{2},x_{3}\in L$, where $1$ is the identity map of $L$. Then we can rewrite the definition of Lie super-algebra as follows: A Lie super-algebra is a pair $(L,\varphi)$ consisting of super-vector space $L$ and a bilinear map $\varphi:L\otimes L\rightarrow L$ (the super-bracket) satisfying the following conditions:

[TABLE]

Definition 2.1**.**

(1)  A Lie super-coalgebra is a pair $(L,\Delta)$ consisting of a super-vector space $L$ and a linear map $\Delta:L\rightarrow L\otimes L$ (the super-cobracket) satisfying

[TABLE]

(2)  A Lie super-bialgebra is a triple $(L,\,\varphi,\,\Delta)$ satisfying

(i)  $(L,\,\varphi)$ is a Lie super-algebra,

(ii)  $(L,\,\Delta)$ is a Lie super-coalgebra,

(iii)  $\Delta\varphi(x\otimes y)=x\circ\Delta(y)-(-1)^{[x][y]}y\circ\Delta(x)$ for all $x,\,y\in L$, where the symbol $\circ$ means the adjoint diagonal action:

[TABLE]

Definition 2.2**.**

(1)  A coboundary Lie super-bialgebra is a quadruple $(L,\,\varphi,\,\Delta,\,r)$ where $(L,\,\varphi,\,\Delta)$ is a Lie super-bialgebra and $r\in{\rm Im}(1\otimes 1-\tau)\subset L\otimes L$ such that $\Delta=\Delta_{r}$ is a coboundary of $r$, where in general $\Delta_{r}$ is defined by

[TABLE]

(2)  A coboundary Lie super-bialgebra $(L,\,\varphi,\,\Delta,\,r)$ is called triangular if it satisfies the classical Yang-Baxter equation (CYBE):

[TABLE]

where $r^{ij}$ are defined by (2.8).

An element $r$ in a Lie super-algebra $L$ is said to satisfy the modified Yang-Baxter equation (MYBE) if

[TABLE]

Denote by $\mathcal{U}(L)$ the universal enveloping algebra of $L$. If $r=\sum_{i}a_{i}\otimes b_{i}\in L\otimes L$, then (here we also use $1$ to denote the unit element in $\mathcal{U}(L)$):

[TABLE]

are elements of $\mathcal{U}(L)\otimes\mathcal{U}(L)\otimes\mathcal{U}(L)$. Obviously,

[TABLE]

are elements of $L\otimes L\otimes L$.

For a Lie super-algebra $L=L_{\overline{0}}\oplus L_{\overline{1}}$, let $V=V_{\overline{0}}\oplus V_{\overline{1}}$ be an $L$-module. An $\mathbb{Z}_{2}$-homogeneous linear map $d:L\longrightarrow V$ such that there exists $[d]\in\mathbb{Z}_{2}$ with $d(L_{i})\in V_{i+[d]}$ for $i\in\mathbb{Z}_{2}$ and

[TABLE]

is called a homogeneous derivation of parity $[d]\in\mathbb{Z}_{2}$. The derivation $d$ is called even if $[d]=\overline{0}$ and odd if $[d]=\overline{1}$. Denote by ${\rm Der}_{p}(L,V)$ the set of homogeneous derivations of parity $p$ ($p\in\{\overline{0},\overline{1}\}$) and ${\rm Der}(L,V)={\rm Der}_{\overline{0}}(L,V)\oplus{\rm Der}_{\overline{1}}(L,V)$ the set of derivations from $L$ to $V$. Denote by ${\rm Inn}(L,V)={\rm Inn}_{\overline{0}}(L,V)\oplus{\rm Inn}_{\overline{1}}(L,V)$ the set of all inner derivations from $L$ to $V$, where ${\rm Inn}_{p}(L,V)$ is the set of homogeneous inner derivations of parity $p$ consisting of $a_{{\rm Inn}}$ $(a\in V_{p})$ defined by

[TABLE]

Denote by $H^{1}(L,V)$ the first cohomology group of $L$ with coefficients in $V$, then it is known

[TABLE]

The main results of this article can be formulated as the following theorem.

Theorem 2.3**.**

*(1)  $H^{1}(\mathfrak{L},\mathfrak{V})=0$, where $\mathfrak{V}=\mathfrak{L}\otimes\mathfrak{L}$.

(2)  All Lie super-bialgebra structures on $\mathfrak{L}$ are triangular coboundary.*

3 The proof of Theorem 2.3

The proof of Theorem 2.3 mainly depends on the following proposition.

Proposition 3.1**.**

${\rm Der}(\mathfrak{L},\mathfrak{V})={\rm Inn}(\mathfrak{L},\mathfrak{V})$.

Denote ${\mathbb{Z}_{s}}=\Gamma\cup(s+\Gamma)$, which is an abelian subgroup of $\mathbb{R}$. A Lie superalgebra $L$ is called ${\mathbb{Z}_{s}}$-graded if $L=\oplus_{r\in{\mathbb{Z}_{s}}}L_{r}$ and $[L_{p},L_{q}]\subset L_{p+q}$. Then the algebra $\mathfrak{L}$ is ${\mathbb{Z}_{s}}$-graded with $\mathfrak{L}_{p}=\mathbb{C}L_{p}\oplus\mathbb{C}I_{p}\oplus\mathbb{C}G_{p}\oplus\mathbb{C}H_{p}$ if $s\in\Gamma$ and $\mathfrak{L}_{p}=\mathbb{C}L_{p}\oplus\mathbb{C}I_{p}$, $\mathfrak{L}_{p+s}=\mathbb{C}G_{p+s}\oplus\mathbb{C}H_{p+s}$ if $s\notin\Gamma$, $2s\in\Gamma$. Denote $\mathfrak{V}_{r}=\bigoplus_{p+q=r}\mathfrak{L}_{p}\otimes\mathfrak{L}_{q}$. Then $\mathfrak{V}$ is a ${\mathbb{Z}_{s}}$-graded vector space. For any $r\in{\mathbb{Z}_{s}}$, denote

[TABLE]

An element $d\in{\rm Der}_{r}(\mathfrak{L},\mathfrak{V})$ is called a homogeneous derivation of degree $r$, usually denoted by $d_{r}$. Similarly, we can define ${\rm Inn}_{r}(\mathfrak{L},\mathfrak{V})$, whose elements are called homogeneous inner derivations of degree $r$. Then ${\rm Der}(\mathfrak{L},\mathfrak{V})$=\prod\limits_{r\in{\mathbb{Z}_{s}}}$${\rm Der}_{r}(\mathfrak{L},\mathfrak{V}). For any $d=\sum_{r\in{\mathbb{Z}_{s}}}d_{r}\in{\rm Der}(\mathfrak{L},\mathfrak{V})$, the formal sum on the right hand side is not necessarily finite, while for any $x\in\mathfrak{L}$, $d(x)=\sum_{r\in{\mathbb{Z}_{s}}}d_{r}(x)$, in which there are finitely many nonzero summands.

A homogeneous element $L_{p}\in\mathfrak{L}$ $({\rm Resp.}\ \,I_{q},\,G_{r},\,H_{t})$ is called a homogeneous element of degree $p_{L}$ $({\rm Resp.}\ \,q_{I},\,r_{G},\,t_{H})$, and denoted by $\deg(L_{p})$ $({\rm Resp.}\ \,\deg(I_{q}),\,\deg(G_{r}),\,\deg(H_{t}))$. Define an order for homogeneous elements in $\mathfrak{L}$ as follows:

[TABLE]

and $\deg(A_{p})>\deg(A_{q})\Longleftrightarrow p>q$, $\deg(B_{r})>\deg(B_{t})\Longleftrightarrow r>t$, for $p,\,q\in\Gamma$, $r,\,t\in{s+\Gamma}$, $A\in\{L,\,I\}$ and $B\in\{G,\,H\}$. Then define degree of homogeneous elements $A_{p}\otimes B_{q}$ in $\mathfrak{V}$ by:

[TABLE]

where $A,\,B\in\{L,\,I,\,G,\,H\}$, $p,\,q\in\mathbb{Z}_{s}$.

Denote by L (Resp. $\textbf{I},\,\textbf{G},\,\textbf{H}$) the $\mathbb{C}$-vector space spanned by

[TABLE]

Any $u\in\mathfrak{V}$ can be written as the following formal sum of homogeneous summands:

[TABLE]

where $A_{p},\,B_{q}$ are homogeneous elements in $\{\textbf{L},\,\textbf{I},\,\textbf{G},\,\textbf{H}\}$ and $a_{p,q}^{A,B}\in\mathbb{C}$.

Definition 3.2**.**

For any nonzero element $u\!\in\!\mathfrak{V}$, with a formal sum of homogeneous summands given above, we define the degree of $u$ as follows: $\deg(u)={\rm max}\{\deg\,(X\otimes Y)\,|\,X\otimes Y$ is a homogeneous summand of $u$ given in (3.4) whose coefficient is nonzero}.

Fix a positive element $\epsilon\in\Gamma$, and denote by $\mathfrak{A}$ the subalgebra of $\mathfrak{L}$ spanned by $\{L_{\epsilon k}\,|\,k\in\mathbb{Z}\}$ as a vector space over $\mathbb{C}$, which is isomorphic to the centerless Virasoro algebra $Vir$: $\{L_{k}\,|\,[L_{m},L_{n}]=(m-n)L_{m+n}\}$ with the isomorphism $\sigma(L_{k})={\epsilon}^{-1}L_{\epsilon k}$.

Proposition 3.3**.**

If $x\in\mathfrak{L}$ satisfies $[L_{\epsilon m},x]=0$ for infinitely many $m>0$ or $m<0$, then $x=0$.

Proof.

We first consider the case that there are infinitely many $m>0$ satisfying $[L_{\epsilon m},x]=0$. If $x\neq 0$, we can write $x$ in the form of linear combinations of homogeneous elements in $\mathfrak{L}$. The highest degree summand must be a nonzero multiple of $A_{p}$ in which $A\in\{L,\,I\}$, $p\in\Gamma$ or $A\in\{G,\,H\}$, $p\in{s+\Gamma}$. From the definition of brackets given in (1.4), we can find a suitable $N>0$, for all $m>N$, $[L_{\epsilon m},A_{p}]\neq 0$. Then the highest degree summand of $[L_{\epsilon m},x]$ is a nonzero multiple of $A_{\epsilon m+p}$, Contradiction! As for the case there are infinitely many $m<0$ satisfying $[L_{\epsilon m},x]=0$, we can consider the lowest degree summands. We similarly get a contradiction. Thus $x=0$. ∎

Proposition 3.4**.**

If $u\in\mathfrak{V}$ satisfies $L_{\epsilon m}\circ u=0$ for infinitely many $m>0$ or $m<0$, then $u=0$.

Proof.

If $u\neq 0$, we can give a formal sum of $u$ as in (3.4). If $L_{\epsilon m}\circ u=0$ for infinitely many $m>0$, one can consider the highest degree summand of $u$, which must be a nonzero multiple of $X\otimes Y$, where $X,\,Y$ are homogeneous elements in $\{\textbf{L},\,\textbf{I},\,\textbf{G},\,\textbf{H}\}$. The highest degree summand of $L_{\epsilon m}\circ u$ must be a nonzero multiple of $[L_{\epsilon m},X]\otimes Y$, which implies $[L_{\epsilon m},X]=0$ for infinitely many $m>0$. According to Proposition 3.3, we get $X=0$. Contradiction! As for the case $L_{\epsilon m}\circ u=0$ for infinitely many $m<0$, we consider the lowest degree summand. We similarly get a contradiction. Thus $u=0$. ∎

The following proposition follows from Proposition 3.4.

Proposition 3.5**.**

An element $r\in{\rm Im}(1\otimes 1-\tau)\in\mathfrak{V}$ satisfies CYBE in (2.3) if and only if it satisfies MYBE in (2.4).

We first prove Theorem 2.3 for the case $s\in\Gamma$. Then $\mathfrak{L}={\rm Span}_{\mathbb{C}}\{L_{p},\,I_{p},\,G_{p},\,H_{p}\,|\,p\in\Gamma\}$ admits the following non-vanishing Lie brackets

[TABLE]

It is easy to see that $\mathfrak{h}={\rm Span}_{\mathbb{C}}\{L_{0}\}$ is the Cartan Subalgebra (CSA) of $\mathfrak{L}$. And

[TABLE]

Denote $\Gamma^{*}=\{t\in\Gamma\,|\,t\neq 0\}$.

Lemma 3.6**.**

${\rm Der}_{t}(\mathfrak{L},\mathfrak{V})={\rm Inn}_{t}(\mathfrak{L},\mathfrak{V})$, $\forall\,\,t\,\in\Gamma^{*}$.

Proof.

For any $d_{t}\in{\rm Der}_{t}(\mathfrak{L},\mathfrak{V})$ ($t\,\in\Gamma^{*}$), we can write $d_{t}=d_{t,\overline{0}}+d_{t,\overline{1}}$, where $d_{t,\overline{i}}\in{\rm Der}_{\overline{i}}(\mathfrak{L},\mathfrak{V})$ for $i=0,\,1$. Using $[L_{0},L_{p}]=-pL_{p}$, one can deduce

[TABLE]

which implies

[TABLE]

Using $[L_{0},I_{p}]=-pI_{p}$, $[L_{0},G_{p}]=-pG_{p}$ and $[L_{0},H_{p}]=-pH_{p}$, we obtain

[TABLE]

Hence $d_{t,\overline{0}}=(-\frac{d_{t,\overline{0}}(L_{0})}{t})_{{\rm Inn}}$. Similarly, $d_{t,\overline{1}}=(-\frac{d_{t,\overline{1}}(L_{0})}{t})_{{\rm Inn}}$. Thus $d_{t}=(-\frac{d_{t,\overline{0}}(L_{0})}{t}-\frac{d_{t,\overline{1}}(L_{0})}{t})_{{\rm Inn}}$, which implies ${\rm Der}_{t}(\mathfrak{L},\mathfrak{V})={\rm Inn}_{t}(\mathfrak{L},\mathfrak{V})$, $\forall\,\,t\,\in\Gamma^{*}$. ∎

Lemma 3.7**.**

$d_{0}(L_{0})=0$* for $d_{0}\in{\rm Der}_{0}(\mathfrak{L},\mathfrak{V})$.*

Proof.

Using $[L_{0},L_{p}]=-pL_{p}$, we obtain

[TABLE]

which implies

[TABLE]

From Proposition 3.4, we know $d_{0}(L_{0})=0$. ∎

Lemma 3.8**.**

Replace $d_{0}$ by $d_{0}-u_{{\rm Inn}}$, where $u\in\mathfrak{V}_{0}$, this replacement does not affect the results we already obtain in Lemma 3.7. With a suitable replacement, we can suppose $d_{0}(L_{\epsilon})=0$.

Proof.

To prove such a replacement does not affect the results we already obtain in Lemma 3.7, it is enough to prove $L_{0}\circ u=0$ for all $u\in\mathfrak{V}_{0}$, which is obvious from the diagonal action defined in (2.1) and the Lie brackets defined in (3.8).

Write $d_{0}=d_{0,\overline{0}}+d_{0,\overline{1}}$, where $d_{0,\overline{0}}\in{\rm Der}_{\overline{0}}(\mathfrak{L},\mathfrak{V})$, $d_{0,\overline{1}}\in{\rm Der}_{\overline{1}}(\mathfrak{L},\mathfrak{V})$. One can suppose

[TABLE]

where $a,\,b,\,c,\,d\in\Gamma$, $e_{a},\,e_{b},\,e_{c},\,e_{d},\,f_{a},\,f_{b},\,f_{c},\,f_{d}\in\mathbb{C}$. Since $\textbf{A}\otimes\textbf{B}$ is invariant under the diagonal action of elements in L, in which $\textbf{A},\,\textbf{B}\in\{\textbf{L},\,\textbf{I},\,\textbf{G},\,\textbf{H}\}$, it is equal to prove our lemma as follows: By replacing $d_{0}$ by $d_{0}-u_{{\rm Inn}}$, where $u\in\mathfrak{V}_{0}\cap(\textbf{A}\otimes\textbf{B})$ , we can suppose $d_{0}(L_{\epsilon})\cap(\textbf{A}\otimes\textbf{B})=0$, in which $\textbf{A},\,\textbf{B}\in\{\textbf{L},\,\textbf{I},\,\textbf{G},\,\textbf{H}\}$.

Here we only consider $d_{0}(L_{\epsilon})\cap(\textbf{L}\otimes\textbf{L})$, and all others are similar. Suppose

[TABLE]

[TABLE]

in which $i\in\mathbb{Z}$, ${\beta}_{i}\in\Gamma$, ${\beta}_{i}\not\equiv{\beta}_{j}\mod\mathbb{Z}\epsilon$ for $i\neq j$. So we can assume only one $\beta$ appeared in this formal sum, i.e.,

[TABLE]

[TABLE]

in which $m\in\mathbb{Z}_{+},\,a_{i},\,b_{i}\in\mathbb{C}$, $(a_{1},b_{1})\neq(0,0)$, $(a_{m},b_{m})\neq(0,0)$. Using $[L_{\epsilon},L_{-\epsilon}]=2\epsilon L_{0}$ and Lemma 3.7, we obtain

[TABLE]

which implies

[TABLE]

Comparing the same degree summands on both sides, we obtain

[TABLE]

Recall the definition length of $u=\sum_{i=1}^{m}e_{i}L_{\beta+i\epsilon}\otimes L_{-\beta-(i-1)\epsilon}$. We say $u$ have length $m$ if $e_{1}\,e_{m}\neq 0$. Replacing $d_{0}$ by $d_{0}-u_{{\rm Inn}}$, in which $u=\frac{a_{m}}{\beta+(m+1)\epsilon}L_{\beta+m\epsilon}\otimes L_{-(\beta+m\epsilon)}$, we successfully reduce the length of $d_{0}(L_{\epsilon})\cap(\textbf{L}\otimes\textbf{L})$ at least one. By induction on the length of $d_{0}(L_{\epsilon})\cap(\textbf{L}\otimes\textbf{L})$, we arrive at $d_{0}(L_{\epsilon})\cap(\textbf{L}\otimes\textbf{L})=0$. Thus, we can assume $d_{0}(L_{\epsilon})=0$. ∎

From the proof of Lemma 3.8, we immediately get the following proposition.

Proposition 3.9**.**

$d_{0}(L_{-\epsilon})=0$.

Lemma 3.10**.**

$d_{0}(L_{k\epsilon})=0$, $\forall\,\,k\in\mathbb{Z}$.

Proof.

Since $L_{\epsilon},\,L_{2\epsilon},\,L_{-\epsilon},\,L_{-2\epsilon}$ generate $\mathfrak{A}$, we only need to prove $d_{0}(L_{2\epsilon})=0=d_{0}(L_{-2\epsilon})$. Using $[L_{-\epsilon},L_{2\epsilon}]=-3\epsilon L_{\epsilon}$, Lemma 3.8 and Proposition 3.9, we obtain $L_{-\epsilon}\circ d_{0}(L_{2\epsilon})=0$. From the brackets defined in (1.4), it is easy to verify that the homogeneous element $u\in\mathfrak{V}$ satisfying $L_{-\epsilon}\circ u=0$ must be a linear combination of $A\otimes B$, $A,\,B\in\{L_{-\epsilon},\,I_{-\epsilon},\,G_{-\frac{\epsilon}{2}},\,H_{-\frac{\epsilon}{2}}\}$ (If $\frac{\epsilon}{2}\not\in\Gamma$, then we treat $G_{-\frac{\epsilon}{2}}=H_{-\frac{\epsilon}{2}}=0$), but none of them can be a summand of $d_{0}(L_{2\epsilon})$, which implies $d_{0}(L_{2\epsilon})=0$. Similarly, we get $d_{0}(L_{-2\epsilon})=0$. Then $d_{0}(L_{k\epsilon})=0,\,\forall\,k\in\mathbb{Z}$. ∎

Lemma 3.11**.**

$d_{0}(A_{p})=0$* where $A\in\{L,\,I,\,G,\,H\}$ and $p\in\mathbb{Z}$.*

Proof.

Using $[L_{m\epsilon},[L_{-m\epsilon},G_{p}]]=-(\frac{m\epsilon}{2}+p)(\frac{3m\epsilon}{2}-p)G_{n}$ and Lemma 3.10, we obtain

[TABLE]

For a formal sum of $d_{0}(G_{p})\neq 0$ as in (3.4), one can suppose its highest degree summand is a nonzero multiple of $A_{\beta+p}\otimes B_{-\beta}$ where $A,\,B\in\{L,\,I,\,G,\,H\}$. We can choose a suitable $m>0$ (such $m$ can always be found since it is easy to get a contradiction from Proposition 3.4) such that the highest degree summand of $L_{m\epsilon}\circ(L_{-m\epsilon}\circ(d_{0}(G_{p})))$ is an nonzero multiple of $A_{\beta+p+m\epsilon}\otimes B_{-(\beta+m)\epsilon}$. We immediately get a contradiction by comparing the highest degree summands on both sides of (3.9).

Similarly, $d_{0}(L_{p})=d_{0}(I_{p})=d_{0}(H_{p})=0$. Then this Lemma follows. ∎

From Lemma 3.11 and the fact $\mathfrak{L}={\rm Span}_{\mathbb{C}}\{L_{p},\,I_{p},\,G_{p},\,H_{p}|p\in\Gamma\}$ we have ${\rm Der}_{0}(\mathfrak{L},\mathfrak{V})={\rm Inn}_{0}(\mathfrak{L},\mathfrak{V})$. This, together with Lemma 3.6, gives the following proposition.

Proposition 3.12**.**

For $s\in\Gamma$, ${\rm Der}(\mathfrak{L},\mathfrak{V})={\rm Inn}(\mathfrak{L},\mathfrak{V})$.

In the case $s\notin\Gamma$ and $2s\in\Gamma$, $\mathfrak{L}={\rm Span}_{\mathbb{C}}\{L_{p},\,I_{p},\,G_{p+s},\,H_{p+s}\,|\,p\in\Gamma\}$ admits the following non-vanishing Lie brackets

[TABLE]

It is easy to see that $\mathfrak{h}:=Span_{\mathbb{C}}\{L_{0}\}$ is the Cartan Subalgebra (CSA) of $\mathfrak{L}$ and $\mathfrak{L}_{p}$ can be given as follows

[TABLE]

Using the similar arguments as those presented in the proof of Proposition 3.12 , we can deduce the following results.

Lemma 3.13**.**

${\rm Der}_{t}(\mathfrak{L},\mathfrak{V})={\rm Inn}_{t}(\mathfrak{L},\mathfrak{V})$, $\forall\,\,t\in{\mathbb{Z}^{*}_{s}}$.

Lemma 3.14**.**

For $d_{0}\in{\rm Der}_{0}(\mathfrak{L},\mathfrak{V})$, $d_{0}(L_{0})$ = 0.

Lemma 3.15**.**

Replace $d_{0}$ by $d_{0}-u_{{\rm Inn}}$, where $u\in\mathfrak{V}_{0}$, this replacement does not affect the results we already obtain in Lemma 3.14. With a suitable replacement, we can suppose $d_{0}(L_{\epsilon})=0$.

Lemma 3.16**.**

$d_{0}(L_{-\epsilon})=0$.

Lemma 3.17**.**

$d_{0}(L_{k\epsilon})=0$, $\forall\,\,k\in\mathbb{Z}$.

Lemma 3.18**.**

$d_{0}(A_{p})=0$* where $A\in\{L,\,I\}$, $p\in\Gamma$ or $A\in\{G,\,H\}$, $p\in{s+\Gamma}$.*

Proposition 3.19**.**

For $s\notin\Gamma$ and $2s\in\Gamma$, ${\rm Der}(\mathfrak{L},\mathfrak{V})={\rm Inn}(\mathfrak{L},\mathfrak{V})$.

Proof of Theorem 2.3   Propositions 3.12 and 3.19 imply Proposition 3.1, which is a restatement of Theorem 2.3 $(1)$. Theorem 2.3 $(2)$ follows from Theorem 2.3 $(1)$ and Proposition 3.5.

Acknowledgements  This work was supported by a NSF grant BK20160403 of Jiangsu Province and NSF grants 11431010, 11371278, 11671056 of China.