Biderivations of the twisted Heisenberg-Virasoro algebra and their applications

TL;DR

This paper investigates the structure of biderivations in the twisted Heisenberg-Virasoro algebra, revealing non-inner forms and applications to linear maps and algebra structures, advancing understanding of its algebraic properties.

Contribution

It characterizes non-skew-symmetric biderivations and explores their applications to linear commuting maps and post-Lie algebra structures.

Findings

Identified non-inner, non-skew-symmetric biderivations.

Characterized linear commuting maps on the algebra.

Proved all biderivations of the graded algebra are trivial.

Abstract

In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg-Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg-Virasoro algebra are given. It also is proved that every biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra is trivial.

Equations

$$\varphi (x\circ y)=\varphi (x)\circ y+x\circ \varphi (y)$$

$$f(x\circ y,z)=x\circ f(y,z)+f(x,z)\circ y,$$

$$f(x,y\circ z)=f(x,y)\circ z+y\circ f(x,z)$$

$$\{L_n,I_n,C_1,C_2,C_3|n\in \mathbb{Z}\}$$

$$[L_n,L_m]=(n-m)L_{m+n}+\frac{n^3-n}{12}{\delta }_{n,-m}{C}_1,$$

$$[L_n,I_m]=-m{I}_{m+n}-(n^2+n){\delta }_{n,-m}{C}_2,$$

$$[I_n,I_m]=n{\delta }_{n,-m}{C}_3,[\mathcal{H},C_1]=[\mathcal{H},C_2]=[\mathcal{H},C_3]=0.$$

$$(i-m)k_i^{(n)}=(2n-m-i)h_{m-n+i}^{(m)}$$

$$k_{2n-m}^{(n)}=h_{2m-n}^{(m)}=0,k_n^{(n)}=h_m^{(m)},\forall m,n\in \mathbb{Z}\mathrm{with}m\ne n.$$

$$i{t}_i^{(n)}=-(m-n+i)g_{m-n+i}^{(m)},\text{for}i\ne 0,n-m$$

$$(m-1){\rho }_2(L_n)+m{\rho }_3(L_n)=-(m-n)g_{m-n}^{(m)},$$

$$(n-1){\theta }_2(L_m)+n{\theta }_3(L_m)=(n-m)t_{n-m}^{(n)}$$

$$t_{n+k}^{(n)}=-g_{n+k}^{(n)}=\frac{{\mu }_k}{n+k},\forall k\in \mathbb{Z}\backslash \{-n\},$$

$${\rho }_2(L_n)={\theta }_2(L_n)=-{\mu }_{-n},$$

$${\rho }_3(L_n)={\theta }_3(L_n)={\mu }_{-n}.$$

$$(n+k)t_{n+k}^{(n)}=-(m+k)g_{m+k}^{(m)}.$$

$$(m-1)({\rho }_2(L_n)+{\rho }_3(L_n))+{\rho }_3(L_n)={\mu }_{-n},\forall n\in \mathbb{Z}.$$

$$r_{\Omega }(L_m,L_n)=\sum _{k\in \mathbb{Z}}{\mu }_k{I}_{m+n+k}$$

$$f(x,y)=\lambda [x,y]+r_{\Omega }(x,y)$$

$$[L_m,L_n]=(m-n)L_{m+n},[L_m,I_n]=-n{I}_{m+n},[I_m,I_n]=0.$$

$$f(x,y)$$

$$f(L_n,L_m)$$

$$f(L_n,I_m)$$

$$f(I_n,L_m)$$

$$f(I_n,I_m)$$

$$\varphi (L_n)=\lambda L_n+\sum _{k\in \mathbb{Z}\setminus \{-n\}}\frac{{\mu }_k}{n+k}{I}_{n+k}+t_0^{(n)}{I}_0,$$

$$\psi (L_n)=\lambda L_n+\sum _{k\in \mathbb{Z}\setminus \{-n\}}\frac{-{\mu }_k}{n+k}{I}_{n+k}+g_0^{(n)}{I}_0,$$

$$f(L_n,L_m)=\lambda [L_n,L_m]+r_{\Omega }(L_n,L_m)$$

$$\varphi (L_n)=\sum _{i\in \mathbb{Z}}{k}_i^{(n)}{L}_i+\sum _{i\in \mathbb{Z}}{t}_i^{(n)}{I}_i,$$

$$\psi (L_n)=\sum _{j\in \mathbb{Z}}{h}_j^{(n)}{L}_j+\sum _{j\in \mathbb{Z}}{g}_j^{(n)}{I}_j$$

$$[\varphi (L_n),L_m]=\sum _{i\in \mathbb{Z}}(i-m)k_i^{(n)}{L}_{i+m}+\sum _{i\in \mathbb{Z}}i{t}_i^{(n)}{I}_{i+m},$$

$$[L_n,\psi (L_m)]$$

$$f(L_n,L_m)$$

Full text

Keywords: biderivation, twisted Heisenberg-Virasoro algebra, linear commuting maps, post-Lie algebra, left-symmetric algebra

AMS subject classifications: 17B05, 17B40, 17B68.

1. Introduction

Derivations and generalized derivations are very important subjects in algebra. Now let us recall the definitions of the derivation and biderivation of an algebra as follows. Suppose that $L$ be a vector space equipped with a binary operation $(x,y)\mapsto x\circ y$ for all $x,y\in L$. As usually, $(L,\circ)$ is said to be an algebra (not necessarily is an associative algebra, for example, as the Lie algebra, left-symmetric algebra, etc.).

• •

A linear map $\phi:L\rightarrow L$ is called a derivation if it satisfies

[TABLE]

for all $x,y\in L$. If $L$ is a Lie algebra, then we denote by $x\circ y=[x,y]$ for all $x,y\in L$. In this case, for $x\in L$, it is easy to see that $\phi_{x}:L\rightarrow L,y\mapsto{\rm ad}x(y)=[x,y],$ for all $y\in L$ is a derivation of $L$, which is called an inner derivation. Denote by ${\rm{Der}}(L)$ and by ${\rm{Inn}}(L)$ the space of derivations and the space of inner derivations of $L$ respectively.

• •

A bilinear map $f:L\times L\rightarrow L$ is called a biderivation of $L$ if it is a derivation with respect to both components. Namely, for each $x\in L$, both linear maps $\phi_{x}$ and $\psi_{x}$ form $L$ into itself given by $\phi_{x}=f(x,\cdot)$ and $\psi_{x}=f(\cdot,x)$ are derivations of $L$, i.e.,

[TABLE]

for all $x,y,z\in L$. Denote by $B(L)$ the set of all biderivations of $L$. For a Lie algebra $L$ and $\lambda\in\mathbb{C}$, it is easy to verify that the bilinear map $f:L\times L\rightarrow L$ given by $f(x,y)=\lambda[x,y]$ for all $y\in L$ is a biderivation of $L$. Such biderivation is said to be inner. Recall that $f$ is skew-symmetric if $f(x,y)=-f(y,x)$ for all $x,y\in L$.

In recent years, many authors put so much effort into the problems of biderivations [4, 10, 13, 15, 18, 19, 24, 29, 30, 31]. In [4], Brešar et al. showed that all biderivations on commutative prime rings are inner biderivations, and determine the biderivations of semiprime rings. The notion of biderivations of Lie algebras was introduced in [30]. In recent years, many authors put so much effort into this problem [4, 10, 13, 15, 18, 19, 24, 27, 29, 30]. In this paper, we shall study the biderivation of the twisted Heisenberg-Virasoro algebra. The twisted Heisenberg-Virasoro algebra is one of the most important Lie algebras both in mathematics and in mathematical physics. The structure and representation theories of the twisted Heisenberg-Virasoro algebra and its various extended Lie algebras have been extensively investigated (see, e.g.,[2, 11, 12, 23, 28]). Now we give the definition of the twisted Heisenberg-Virasoro algebra and an example of its biderivation as follows.

Definition 1.1**.**

The twisted Heisenberg-Virasoro algebra is the Lie algebra $(\mathcal{H},[,])$ which has $\mathbb{C}$-basis

[TABLE]

and the following Lie brackets, for all $m,n\in\mathbb{Z}$,

[TABLE]

Example 1.2**.**

Let $f:\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{H}$ be a bilinear map determined by $f(L_{m},L_{n})=I_{m+n}$, $f(L_{m},I_{n})=f(I_{n},L_{m})=f(I_{n},I_{m})=0$ and $f(x,C_{i})=f(C_{i},y)=0,i=1,2,3$ for all $m,n\in\mathbb{Z}$ and $x,y\in\mathcal{H}$. Then it is easy to verify that $f$ is a biderivation of $\mathcal{H}$. Note that it is non-inner and non-skew-symmetric.

Throughout the paper, the symbols $\mathbb{C}$ and $\mathbb{Z}$ represent for the sets of complex numbers and integers, respectively. Recall that a Lie algebra $L$ is said to be perfect if $L=[L,L]$. Note that the twisted Heisenberg-Virasoro algebra is perfect. Denote by $Z(L)$ the center of a Lie algebra $L$. Obviously, $Z(\mathcal{H})={\rm Span}\{I_{0},C_{1},C_{2},C_{3}\}$. We denote a subset of $Z(\mathcal{H})$ by $Z^{\prime}(\mathcal{H})={\rm Span}\{C_{1},C_{2},C_{3}\}$.

Our results can be briefly summarized as follows: In Section 2, we give some lemmas which will be used to our proof. In Section 3, we characterize the biderivations without the skew-symmetric condition of the twisted Heisenberg-Virasoro algebra. In Section 4, we give three applications of biderivation of the twisted Heisenberg-Virasoro algebra, i.e., the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg-Virasoro algebra are given. As a simple corollary, we prove that every biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra is trivial.

2. Preliminary results

Lemma 2.1**.**

Suppose that $k_{i}^{(n)},h_{i}^{(m)}\in\mathbb{C},i,m,n\in\mathbb{Z}$ such that

[TABLE]

for all $m,n,i\in\mathbb{Z}$. Then there is $\lambda\in\mathbb{C}$ such that $k_{i}^{(m)}=h_{i}^{(m)}=\delta_{m,i}\lambda$ for all $m,i\in\mathbb{Z}$.

Proof.

For any $m,n$ with $m\neq n$, by taking $i=2n-m$,$m$ and $i=n$ in (3) respectively, we have

[TABLE]

Let $m,n$ run all integers with $m\neq n$, then by (4) we see that $k_{i}^{(n)}=0,h_{j}^{(m)}=0$ for all $i\neq n,\ j\neq m$ and $k_{n}^{(n)}=h_{n}^{(n)}=h_{0}^{(0)}$ for all $n\in\mathbb{Z}$. By letting $h_{0}^{(0)}=\lambda$, the proof is completed. ∎

Lemma 2.2**.**

Suppose that $t_{i}^{(n)},g_{i}^{(m)}\in\mathbb{C},\ i,m,n\in\mathbb{Z}$ and $\rho_{2},\rho_{3},\theta_{2},\theta_{3}$ are linear complex valued functions on $\widetilde{\mathcal{H}}$ satisfy

[TABLE]

for all $i,m,n\in\mathbb{Z}$. Then there is a set of complex numbers $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ such that

[TABLE]

[TABLE]

[TABLE]

Proof.

By taking $i=n+k$ in (5) with $k\neq-m$, $k\neq-n$, we have

[TABLE]

Let $m=n=1-k$ in (11), one has $t_{1}^{(1-k)}=-g_{1}^{(1-k)}$. Denote $\mu_{k}=t_{1}^{(1-k)}$. By using (11) with $m=1-k$, we deduce that $t_{n+k}^{(n)}=\frac{\mu_{k}}{n+k}$ for all $k\in\mathbb{Z}\backslash\{-n\}$. Similarly, by using (11) with $n=1-k$ we see that $g_{m+k}^{(m)}=-\frac{\mu_{k}}{m+k}$ for all $k\in\mathbb{Z}\backslash\{-m\}$. This proves that (8) holds. Notice that (8) implies $(m-n)g_{m-n}^{(m)}=-\mu_{-n}.$ This, together with (6), yields that

[TABLE]

This indicates that $\rho_{3}(L_{n})=-\rho_{2}(L_{n})=\mu_{-n}$. Similarly, notice that (8) implies that $(n-m)t_{n-m}^{n}=\mu_{-m}$ and then by (7) we deduce that $-\theta_{2}(L_{m})=\theta_{3}(L_{m})=\mu_{-m}$. The proof is completed. ∎

Lemma 2.3**.**

Suppose that $s_{i}^{(n)},l_{i}^{(n)}\in\mathbb{C}$ for all $i,n\in\mathbb{Z}$.

(i) If $(m-i)s_{i}^{(n)}=0$ for all $i,m,n\in\mathbb{Z}$, then $s_{i}^{(n)}=0$ for all $i,n\in\mathbb{Z}.$

(ii) If $jl_{j}^{(m)}=0$ for all $j\neq m,m-n$, then $l_{j}^{(m)}=0$ for all $j\neq 0$.

Proof.

The proof is similar to Lemma 2.1. ∎

Lemma 2.4**.**

Let $L$ be a perfect Lie algebra and $f$ be a biderivation of $L$. If $\alpha\in Z(L)$, then $f(x,\alpha)=f(\alpha,x)=0$ for all $x\in L$.

Proof.

For each $x\in L$, since $\phi_{x}=f(x,\cdot)$ is a derivation of $L$, so $0=\phi_{x}([\alpha,y])=[\phi_{x}(\alpha),y]+[\alpha,\phi_{x}(y)]=[\phi_{x}(\alpha),y]$ for all $y\in L$. In the other words, $\phi_{x}(\alpha)=f(x,\alpha)\in Z(L)$. Similarly, we have $f(\alpha,x)\in Z(L)$. Now from (1) and (2) one has $f([y,z],\alpha)=[y,f(z,\alpha)]+[f(y,\alpha),z]=0$ and $f(\alpha,[y,z])=[f(\alpha,y),z]+[y,f(\alpha,z)]=0$ for all $y,z\in L$. Thanks to $L=[L,L]={\rm Span}\{[y,z]|y,z\in L\}$, forcing $f(x,\alpha)=f(\alpha,x)=0$ for all $x\in L$. ∎

Lemma 2.5**.**

Let $L$ be a perfect Lie algebra and $Z^{\prime}(L)\subset Z(L)$. Then the linear map $\pi:B(L)\rightarrow B(L/Z^{\prime}(L))$ defined by $\pi(f)(x+Z^{\prime}(L),y+Z^{\prime}(L))=f(x,y)+Z^{\prime}(L)$ is injective. Furthermore, $\pi$ is bijective if and only if any biderivation of $L/Z^{\prime}(L))$ can be extended to a biderivation of $L$.

Proof.

From Lemma 2.4, $\pi$ is well-defined. Now for any $f\in\ker\pi$, i.e., $f(x,y)\in Z^{\prime}(L),\forall x,y\in L$. Then from $f([x,y],z)=[f(x,z),y]+[x,f(y,z)]=0$ and $L$ is perfect, we have $f=0$. The second part of the lemma is trivial. ∎

For a Lie algebra $\widetilde{L}$, let $L=\widetilde{L}\oplus Z^{\prime}(L)$ be a central extension of $\widetilde{L}$. Then it is well known that $\widetilde{L}\cong L/Z^{\prime}(L)$. Furthermore, Lemmas 2.4 and 2.5 tell us that if $L$ is perfect then we only need to characterize the biderivation of $\widetilde{L}$.

3. Biderivation of the twisted Heisenberg-Virasoro algebra

We fist give a class of biderivations of the twisted Heisenberg-Virasoro algebra $\mathcal{H}$ as follows.

Definition 3.1**.**

Let $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ be a set satisfying $|\Omega|<+\infty$, $i.e.,\Omega$ only contains finitely many nonzero numbers. For such $\Omega$, we define a bilinear map $r_{\Omega}:\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{H}$ given by

[TABLE]

for all $m,n\in\mathbb{Z}$ and $r_{\Omega}(x,y)=0$ if either of $x,y$ is contained in $\{C_{1},C_{2},C_{3},I_{n}|n\in\mathbb{Z}\}$.

It is easy to verify that the above $r_{\Omega}$ is a biderivation of twisted Heisenberg-Virasoro algebra. Note that $r_{\Omega}$ is symmetric and non-inner. The case in which $\Omega=\{\mu_{0}=1,\mu_{i}=0|i\in\mathbb{Z}\setminus\{0\}\}$ gives just Example 1.2. Our main result is following.

Theorem 3.2**.**

$f$* is a biderivation of $\mathcal{H}$ if and only if there are $\lambda\in\mathbb{C}$ and a finite set $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ such that*

[TABLE]

for all $x,y\in\mathcal{H}$, where $r_{\Omega}$ is given by Definition 3.1.

The proof of Theorem 3.2 will be completed by several lemmas as follows. Let $\widetilde{\mathcal{H}}=\mathcal{H}/Z^{\prime}(\mathcal{H})$ where $Z^{\prime}(\mathcal{H})={\rm Span}\{C_{1},C_{2},C_{3}\}$. Then $\widetilde{\mathcal{H}}$ just is the Lie algebra $W(0,0)$ [17], which has the basis $\{L_{n},I_{n}\lvert n\in\mathbb{Z}\}$ and the Lie bracket

[TABLE]

Lemma 3.3**.**

[17]** ${\rm{Der}}\widetilde{\mathcal{H}}={\rm{I{nn}}}\widetilde{\mathcal{H}}\oplus\mathbb{C}D_{1}\oplus\mathbb{C}D_{2}^{0,0}\oplus\mathbb{C}D_{3}$, where $D_{1},D_{2}^{0,0},D_{3}$ are outer derivations defined by $D_{1}(L_{m})=0$, $D_{1}(I_{m})=I_{m}$, $D_{2}^{0,0}(L_{m})=(m-1)I_{m},$ $D_{2}^{0,0}(I_{m})=0,$ $D_{3}(L_{m})=mI_{m},$ $D_{3}(I_{m})=0$ for all $m\in\mathbb{Z}$.

Lemma 3.4**.**

Suppose that $f$ is a biderivation of $\widetilde{\mathcal{H}},$ Then there are linear maps $\phi$ and $\psi$ from $\widetilde{\mathcal{H}}$ into itself such that

[TABLE]

for all $x,y\in\widetilde{\mathcal{H}}$, where $\rho_{1},\rho_{2},\rho_{3}$ and $\theta_{1},\theta_{2},\theta_{3}$ are linear complex value function on $\widetilde{\mathcal{H}}$ and $D_{1},D_{2}^{0,0},D_{3}$ is given by Lemma 3.3

Proof.

For the biderivation $f$ on $\widetilde{\mathcal{H}}$ and a fixed element $x\in\widetilde{\mathcal{H}}$, we define a map $\phi_{x}:\widetilde{\mathcal{H}}\rightarrow\widetilde{\mathcal{H}}$ is given by $\phi_{x}(y)=f(x,y)$. Then we know by (2) that $\phi_{x}$ is a derivation of $\widetilde{\mathcal{H}}$. By Lemma 3.3, there are three complex value function $\rho_{1},\rho_{2},\rho_{3}$ on $\widetilde{\mathcal{H}}$ and a linear map $\phi$ from $\widetilde{\mathcal{H}}$ into itself such that $\phi_{x}=\rho_{1}(x)D_{1}+\rho_{2}(x)D_{2}^{0,0}+\rho_{3}(x)D_{3}+{\rm ad}\phi(x)$. Namely, $f(x,y)=\rho_{1}(x)D_{1}(y)+\rho_{2}(x)D_{2}^{0,0}(y)+\rho_{3}(x)D_{3}(y)+[\phi(x),y]$. Due to $f$ is bilinear, $\rho_{1},\rho_{2},\rho_{3}$ are linear. Similarly, if we define a map $\psi_{z}$ from $\widetilde{\mathcal{H}}$ into itself is given by $\widetilde{\mathcal{H}}$ for all $y\in\widetilde{\mathcal{H}}$, then it can obtained three linear complex valued functions $\theta_{1},\theta_{2},\theta_{3}$ on $\widetilde{\mathcal{H}}$ and a map $\psi$ from $\widetilde{\mathcal{H}}$ into itself such that

[TABLE]

The proof is completed. ∎

Lemma 3.5**.**

Let $f$ be a biderivation of $\widetilde{\mathcal{H}}$ and $\phi,\psi,\rho_{i},\theta_{i},i=1,2,3$ be given by Lemma 3.4. Then the following equations hold.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

It will follow by Lemmas 3.3 and 3.4. ∎

Lemma 3.6**.**

Let $f$ be a biderivation of $\widetilde{\mathcal{H}}$ and $\phi,\psi,\rho_{i},\theta_{i},i=1,2,3$ be given by Lemma 3.4. Then there are $\lambda\in\mathbb{C}$ and a finite set $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ such that (9), (10) and

[TABLE]

for all $n\in\mathbb{Z}$, where $t_{0}^{(n)}$ and $g_{0}^{(n)}$ are complex numbers related to $n$. And then, we have

[TABLE]

where $r_{\Omega}(L_{n},L_{m})=\sum_{k\in\mathbb{Z}}\mu_{k}I_{m+n+k}$ is given by Definition 3.1.

Proof.

For any fixed $n\in\mathbb{Z}$, suppose that

[TABLE]

where $k_{i}^{(n)},t_{i}^{(n)},h_{i}^{(n)},g_{i}^{(n)}\in\mathbb{C}$ for every $i\in\mathbb{Z}$. With (24) and (25), direct computations show that

[TABLE]

[TABLE]

Note that (17) together with (26), (27) tells us that all (3), (5), (6) and (7) hold. Therefore, by Lemmas 2.1 and 2.2, we know that there are $\lambda\in\mathbb{C}$ and a finite set $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ such that (9), (10), (21) and (22) are established. And then, by (17), (9), (10) and (21) one can obtain that

[TABLE]

which gives (23). The proof is completed. ∎

Lemma 3.7**.**

Let $f$ be a biderivation of $\widetilde{\mathcal{H}}$, $\lambda$ be given by Lemma 3.6 and $\phi,\psi,\rho_{i},\theta_{i},i=1,2,3$ be given by Lemma 3.4. Then

[TABLE]

and $\rho_{1}(L_{n})=\theta_{1}(L_{n})=\rho_{i}(I_{n})=\theta_{i}(I_{n})=0,i=1,2,3$, for all $n\in\mathbb{Z}$ where $t_{0}^{(n)}$ and $q_{0}^{(n)}$ are complex numbers related to $n$.

Proof.

For any fixed $n\in\mathbb{Z}$, we suppose that

[TABLE]

where $p_{i}^{(n)},q_{i}^{(n)},s_{j}^{(n)},l_{j}^{(n)}\in\mathbb{C}$ for every $i\in\mathbb{Z}$. By using (28), (29) and Lemma 3.6 we have

[TABLE]

This, together with (18), yields that $l_{m}^{(m)}=\lambda$ and

[TABLE]

From this, it follows by Lemma 2.3 that $s_{i}^{(m)}=l_{j}^{(m)}=\theta_{2}(I_{m})=\theta_{3}(I_{m})=\rho_{1}(L_{m})=0$ for all $i,j,m\in\mathbb{Z}$ with $j\neq 0,m$. Similarly, by (19) we deduce that $q_{m}^{(m)}=\lambda$ and $p_{i}^{(m)}=q_{j}^{(m)}=\rho_{2}(I_{m})=\rho_{3}(I_{m})=\theta_{1}(L_{m})=0$ for all $i,j,m\in\mathbb{Z}$ with $j\neq 0,m$. Finally, by (20) we have that

[TABLE]

for all $m,n\in\mathbb{Z}$, which yields that $\rho_{1}(I_{m})=\theta_{1}(I_{m})=0$ for all $m\in\mathbb{Z}$. The proof is completed. ∎

Lemma 3.8**.**

Let $f$ be a biderivation of $\widetilde{\mathcal{H}}$. Then there are $\lambda\in\mathbb{C}$ and a finite set $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ such that $f(x,y)=\lambda[x,y]+r_{\Omega}(x,y)$ for all $x,y\in\widetilde{\mathcal{H}}$, where $r_{\Omega}$ is given by Definition 3.1.

Proof.

Notice Lemmas 3.6, 3.7 and (18), (19), (20), by a simple computation we have

[TABLE]

here we use the fact $r_{\Omega}(L_{n},I_{m})=r_{\Omega}(I_{n},L_{m})=r_{\Omega}(I_{n},I_{m})=0$ by Definition 3.1. This, together with (23) and the bilinearity of $f$, completes the proof. ∎

The proof of Theorem 3.2: The “if” part is easy to verify, we now prove the “only if” part.

Now we assume that $f$ is a biderivation of $\mathcal{H}$. Define a linear map $\pi:B(\mathcal{H})\rightarrow B(\mathcal{H}/Z^{\prime}(\mathcal{H}))$ defined by $\pi(f)(x+Z^{\prime}(\mathcal{H}),y+Z^{\prime}(\mathcal{H}))=f(x,y)+Z^{\prime}(\mathcal{H})$ where $Z^{\prime}(\mathcal{H})={\rm Span}\{C_{1},C_{2},C_{3}\}$. Note that $\mathcal{H}/Z^{\prime}(\mathcal{H})\cong\widetilde{\mathcal{H}}$. By Lemmas 2.5 and 3.8, we can assume that

[TABLE]

where $\lambda\in\mathbb{C}$ and $\Omega=\{\mu_{k}\in\mathbb{C}|k\in\mathbb{Z}\}$ is a finite set. For any $x,y\in{\rm Span}\{L_{i},I_{i}|i\in\mathbb{Z}\}$ and $\alpha,\beta\in{\rm Span}\{C_{1},C_{2},C_{3}\}$, we have by Lemma 2.4 that

[TABLE]

The proof is completed.

4. Applications

In this section, we give some applications of biderivations of the twisted Heisenberg-Virasoro algebra.

4.1. Linear commuting maps on Lie algebras

Recall that a linear commuting map $\phi$ on a Lie algebra $L$ subject to $[\phi(x),x]=0$ for any $x\in L$. The first important result on linear (or additive) commuting maps is Posner’s theorem [22] from 1957. Then many scholars study commuting maps on all kinds of algebra structures. Brešar [6] briefly discuss various extensions of the notion of a commuting map. About the recent articles on commuting maps we can reference [3, 5, 6, 10, 14, 19, 29, 31].

Obviously, if $\phi$ on $L$ is such a map, then $[\phi(x),y]=[x,\phi(y)]$ for any $x,y\in L$. Define by $f(x,y)=[\phi(x),y]=[x,\phi(y)]$, then it is easy to check that $f$ is a biderivation of $L$.

Theorem 4.1**.**

Any linear map $\phi$ on $\mathcal{H}$ is commuting if and only if there are $\lambda\in\mathbb{C}$ and a linear map $\tau:\mathcal{H}\rightarrow Z(\mathcal{H})$ such that $\phi(x)=\lambda x+\tau(x)$ for all $x\in\mathcal{H}$.

Proof.

Notice that the “if” part is easy to verify. we now prove the “only if” part.

By the above discussion, we see that if we let $f(x,y)=[\phi(x),y],\ x,y\in\mathcal{H}$, then $f$ is a biderivation of $L$. From Theorem 3.2, we have

[TABLE]

since $f(x,x)=0$, so $f(x,y)=-f(y,x)$. This implies that $\lambda[x,y]+r_{\Omega}(x,y)=-\lambda[y,x]-r_{\Omega}(y,x)$. Therefore, we see that $r_{\Omega}(x,y)=r_{\Omega}(y,x)=-r_{\Omega}(y,x)$ for all $x,y\in\mathcal{H}$. Thus, $r_{\Omega}(x,y)=0$. From this with (30), we deduce that $[\phi(x)-\lambda(x),y]=0$ for all $x,y\in\mathcal{H}$. In other words, $\phi(x)-\lambda(x)\in Z(\mathcal{H})$. Hence we can find a linear map $\tau:\mathcal{H}\rightarrow Z(\mathcal{H})$ such that $\phi(x)=\lambda x+\tau(x)$ for all $x\in\mathcal{H}$. The proof is completed. ∎

4.2. Post-Lie algebra

Post-Lie algebras have been introduced by Valette in connection with the homology of partition posets and the study of Koszul operads [27]. As [8] pointed out, post-Lie algebras are natural common generalization of pre-Lie algebras and LR-algebras in the geometric context of nil-affine actions of Lie groups. Recently, many authors study some post-Lie algebras and post-Lie algebra structures [7, 8, 20, 21, 26]. In particular, the authors [8] study the commutative post-Lie algebra structure on Lie algebra. By using our results, we can characterize the commutative post-Lie algebra structure on $\mathcal{H}$. Let us recall the following definition of a commutative post-Lie algebra.

Definition 4.2**.**

[25]** Let $(L,[,])$ be a complex Lie algebra. A commutative post-Lie algebra structure on $L$ is a $\mathbb{C}$-bilinear product $x\cdot y$ on $L$ satisfying the following identities:

[TABLE]

for all $x,y,z\in L$. We also say $(L,[,],\cdot)$ a commutative post-Lie algebra.

A post-Lie algebra $(L,[,],\cdot)$ is said to be trivial if $x\cdot y=0$ for all $x,y\in L$. The following lemma shows the connection between commutative post-Lie algebra and biderivation of a Lie algebra.

Lemma 4.3**.**

[25]** Suppose that $(L,[,],\cdot)$ is a commutative post-Lie algebra. If we define a bilinear map $f:L\times L\rightarrow L$ by $f(x,y)=x\cdot y$ for all $x,y\in L$, then $f$ is a biderivation of $L$.

Theorem 4.4**.**

Any commutative post-Lie algebra structure on the twisted Heisenberg-Virasoro algebra $\mathcal{H}$ is trivial.

Proof.

Suppose that $(\mathcal{H},[,],\cdot)$ is a commutative post-Lie algebra. By Lemma 4.3 and Theorem 3.2, we know that there are $\lambda\in\mathbb{C}$ and a finite set $\Omega=\{\mu_{i}\in\mathbb{C}|i\in\mathbb{Z}\}$ such that $x\cdot y=\lambda[x,y]+r_{\Omega}(x,y)$ for all $x,y\in\mathcal{H}$, where $r_{\Omega}$ is given by Definition 3.1. Because the product $\cdot$ is commutative, we have $\lambda[L_{1},L_{2}]+\chi_{\Omega}(L_{1},L_{2})=\lambda[L_{2},L_{1}]+\chi_{\Omega}(L_{2},L_{1})$, which implies $\lambda=0$. By (31), we see that

[TABLE]

If there is $\mu_{k}\in\Omega$ such that $\mu_{k}\neq 0$, then it is easy to see that the left-hand side of the above equation contains an item $\mu_{k}I_{6+k}\neq 0$, whereas the right-hand side is equal to zero, which is a contradiction. Thus, we have $\Omega=\{0\}$, i.e., $\mu_{i}=0$ for any $i\in\mathbb{Z}$. In other words, $r_{\Omega}=0$. That is, $x\cdot y=0$ for all $x,y\in\mathcal{H}$. ∎

4.3. Biderivation of left-symmetric algebras

Recall that a left-symmetric algebra [1, 9] is an algebra $(A,\circ)$ such that

[TABLE]

for all $x,y,z\in A$. Note that an associative algebra is a left-symmetric algebra. A good ting is that a Left-symmetric algebra is Lie admissible, i.e., the commutator $[x,y]=x\circ y-y\circ x$ defines a Lie algebra $\mathcal{G}(A)$, which is called the sub-adjacent Lie algebra of $A$, and conversely $A$ is called a compatible left-symmetric algebra structure on $\mathcal{G}(A)$. The authors [11] gave a class of left-symmetric algebra structures $A$ on the twisted Heisenberg-Virasoro algebra $\mathcal{H}=\mathcal{G}(A)$ which here is called a graded twisted Heisenberg-Virasoro left-symmetric algebra as follows.

Definition 4.5**.**

A graded twisted Heisenberg-Virasoro left-symmetric algebra is an algebra $(\mathcal{A},\circ)$ with $\mathbb{C}$-basis $\{L_{n},I_{n},C_{1},C_{2},C_{3}|n\in\mathbb{Z}\}$ such that

[TABLE]

[TABLE]

for all $m,n,i,j\in\mathbb{Z}$, where $\alpha,\beta,\epsilon\in\mathbb{C}$ such that ${\rm{Re}}\epsilon>0,\epsilon^{-1}\notin\mathbb{Z}$ or ${\rm{Re}}\epsilon=0,{\rm Im}\epsilon>0.$

The following lemma is easy to verify by a direct computation.

Lemma 4.6**.**

Let $(A,\circ)$ be a left-symmetric algebra and $(L,[,])$ be the sub-adjacent Lie algebra of $A$. Then every derivation (resp. biderivation) of $A$ is also a derivation (resp. biderivation) of $L$.

Theorem 4.7**.**

Any biderivation $f$ of the graded twisted Heisenberg-Virasoro left-symmetric algebra $\mathcal{A}$ is trivial, i.e., $f=0$.

Proof.

Note that the twisted Heisenberg-Virasoro algebra $\mathcal{H}$ is the sub-adjacent Lie algebra of the left-symmetric algebra $\mathcal{A}$ given by Definition 4.5. Let $f$ be any biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra $\mathcal{A}$. Then by Lemma 4.6 we see that $f$ is also a biderivation of $\mathcal{H}$. From Theorem 3.2, there are $\lambda\in\mathbb{C}$ and a finite set $\Omega=\{\mu_{i}\in\mathbb{C}|i\in\mathbb{Z}\}$ such that $f(x,y)=\lambda[x,y]+r_{\Omega}(x,y)$ for all $x,y\in\mathcal{A}=\mathcal{H}$. It is not difficult to verify by Definition 4.5 that $\lambda=0$ and $\Omega=0$. Therefore, $f(x,y)=0$ for all $x,y\in\mathcal{A}$. ∎

Remark 4.8**.**

As far as we know, there is no any result on biderivation of left-symmetric algebras unless associative algebras. Theorem 4.7 tells us that any biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra is trivial. Is there a non-graded twisted Heisenberg-Virasoro left-symmetric algebra which has a non-trivial biderivation? This problem will be interesting.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China [grant number 11171294], the Natural Science Foundation of Heilongjiang Province of China [grant number A2015007], the fund of the Heilongjiang Education Committee [grant number 12531483], and the special fund of Heilongjiang University of the Fundamental Research Funds for Universities in Heilongjiang province (grant number HDJCCX-2016211).