---
abstract: 'As a natural generalization of derivations, 2-local derivations of a Lie algebra play an important role to the study of local properties of the structure of the Lie algebra. This paper initiates the study of 2-local derivations of Lie algebras over fields of prime characteristic. Let $\ggg$ be a Jacobson-Witt algebra over an infinite field of characteristic $p>2$. In this paper, we study properties of 2-local derivations on $\ggg$, and show that every 2-local derivation on $\ggg$ is a derivation.'
address:
- 'Department of Mathematics, Shanghai Maritime University, Shanghai, 201306, China.'
- 'Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5, and School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050016, Hebei, China.'
author:
- Yufeng Yao and Kaiming Zhao
title: '2-local derivations on the Jacobson-Witt algebras in prime characteristic'
---

[^1]

Introduction
============

As is known to all, the derivation algebra of an associative algebra $A$ plays an important role to the study of the structure of $A$. In the theory of Lie algebras, a well-known result due to H. Zassenhaus states that all derivations on a finite dimensional Lie algebra with nondegenerate Killing form are inner (cf. [@Jac]). In particular, finite dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero admit only inner derivations. Hence, they are isomorphic to their derivation algebras.

As a generalization of derivation, $\check{\text{S}}$emrl introduced the notion of 2-local derivations on algebras in [@Se]. The concept of 2-local derivation is actually an important and interesting property for an algebra. The main problem in this subject is to determine all 2-local derivations, and to see whether they are automatically (global) derivations. All 2-local derivations on several important classes of Lie algebras have been determined. In [@AKR], it was shown that each 2-local derivation on a finite dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation and each finite dimensional nilpotent Lie algebra with dimension larger than two admits a 2-local derivation which is not a derivation. Furthermore, the authors in [@WCN] proved that all 2-local derivations on finite dimensional basic classical Lie superalgebras except $A(n,n)$ over an algebraically closed field of characteristic zero are derivations. Similar results on 2-local derivations on simple Leibniz algebras were obtained in [@AKO]. All 2-local derivation on Witt algebras and some of their subalgebras were shown to be derivations in [@ZCZ; @AKY]. Similar result was obtained quite recently for the $W$-algebra $W(2,2)$ in [@Ta]. In the present paper, we initiate the study of 2-local derivations on finite dimensional Lie algebras over an infinite field of positive characteristic. The algebras we concern are the so-called Jacobson-Witt algebras, which are the modular version of some generalized Witt algebras. Let us briefly introduce them below.

Different from the situation of characteristic zero, besides classical simple Lie algebras, there is another variety of simple Lie algebras, the so-called simple Lie algebras of Cartan type, in the classification of finite dimensional simple Lie algebras over an algebraically closed field $\mathbb{F}$ of prime characteristic $p>5$ (cf. [@PS]). The Lie algebras of Cartan type consist of four families $W, S, H, K$ (cf. [@SF; @St]). The algebras we focus on in the present paper are the first series. The Jacobson-Witt algebra $W_n$ is the derivation algebra of the divided power algebra $\mathfrak{A}_n=\bbf[x_1,\cdots, x_n]/(x_1^p,\cdots, x_n^p)$, where $(x_1^p,\cdots, x_n^p)$ is the ideal of $\bbf[x_1,\cdots, x_n]$ generated by $x_i^p$, $1\leq i\leq n$. Over the past decades, the representation theory of the Jacobson-Witt algebras was extensively studied (see [@Sh; @HZ; @SY]). The derivation algebra of $W_n$ was completely determined (see [@SF; @St]). This paper is devoted to studying 2-local derivations on $W_n$. We determine all 2-local derivations on the Jacobson-Witt algebras, and show that each 2-local derivation is a (global) derivation.

This paper is organized as follows. In section 2, we recall the basic notations, definitions, structure and some important properties of the Jacobson-Witt algebras. Section 3 is devoted to studying 2-local derivations on the Jacobson-Witt algebras. We present some properties of 2-local derivations, and show that every 2-local derivation on any Jacobson-Witt algebra is a derivation.

Similar to the study on structure of simple Lie algebras of positive characteristic, the study on 2-local derivations of Lie algebras of positive characteristic is very different and more difficult than the case of characteristic $0$. We have to establish new and different methods to achieve our goal.

Notations and preliminaries
===========================

In this paper, we always assume that $\bbf$ is an infinite field of characteristic $p>2$, and let $\bbf_p$ denote the prime subfield of $\bbf$. Throughout this paper, all algebras and vector spaces are over $\bbf$ and finite dimensional. We denote by $\mathbb{Z}, \mathbb{N}, \mathbb{Z}_+$ the set of all integers, nonnegative integers and positive integers respectively. For a set $S$, we use $|S|$ or $\#S$ to denote the cardinality of $S$.

Derivations and 2-local derivations on a Lie algebra
----------------------------------------------------

A [**derivation**]{} on a Lie algebra $\ggg$ is a linear transformation $D:\ggg\longrightarrow\ggg$ such that the following Leibniz law holds: $$D([x, y])=[D(x), y]+[x, D(y)],\,\,\forall\,x,y\in\ggg.$$ The set of all derivations of $\ggg$ is denoted by $\Der(\ggg)$, which is a Lie algebra under the usual commutant operation. For each $x\in\ggg$, let $$\ad x:\ggg\longrightarrow\ggg,\,\,\,\ad x (y)=[x, y], \,\,\forall\,y\in\ggg.$$ Then $\ad x$ is a derivation on $\ggg$ for any $x\in\ggg$, which is called an inner derivation. The set of all inner derivations of $\ggg$ is denoted by $\Inn(\ggg)$, which is an ideal of $\Der(\ggg)$.

A map $\Delta: \ggg\longrightarrow\ggg$ (not necessarily linear) is called a [**2-local derivation**]{} if for any $x,y\in\ggg$, there exists a derivation $D_{xy}\in\Der(\ggg)$ (depending on $x, y$) such that $\Delta(x)=D_{xy}(x)$ and $\Delta(y)=D_{xy}(y)$. In particular, for any $x\in\ggg$ and $k\in\bbf$, there exists $D_{xx}\in\Der(\ggg)$ such that $$\Delta(kx)=D_{xx}(kx)=kD_{xx}(x)=k\Delta(x).$$ In particular, $$\label{a property}
\Delta(0)=0.$$ Hence, a 2-local derivation $\Delta$ on $\ggg$ is a derivation if and only if $\Delta$ is additive and satisfies the Leibniz law, i.e., $$\Delta(x+y)=\Delta(x)+\Delta(y), \,\Delta([x,y])=[\Delta(x), y]+[x, \Delta(y)],\,\,\forall\,x,y\in\ggg.$$

The Jacobson-Witt algebras
--------------------------

In this subsection, we recall the basic definitions and properties of the Jacobson-Witt algebras which we concern in this paper. We use the terminology and notations in [@SF; @St]. For $n\in\mathbb{Z}_+$, set $$A_n=\{\alpha=(\alpha_1,\cdots, \alpha_n)\in\mathbb{N}^n :  0\leq \alpha_i\leq p-1, 1\leq i\leq n\},$$ $$\tau=(p-1, \cdots, p-1), \,\,\varepsilon_i=(\delta_{i1},\cdots, \delta_{in})\,\text{ for}\,\, 1\leq i\leq n,$$ where $$\begin{aligned}
\delta_{ij}=\begin{cases} 1, &\mbox{ if } i=j;\cr
0, &\mbox{ otherwise.}
\end{cases}\end{aligned}$$ Let ${\mathfrak{A}}_n=\bbf[x_1,\cdots, x_n]/(x_1^p,\cdots, x_n^p)$ be the divided power algebra of $n$ variables $x_1,\cdots, x_n$, where $(x_1^p,\cdots, x_n^p)$ denotes the ideal of $\bbf[x_1,\cdots, x_n]$ generated by $x_i^p, 1\leq i\leq n$. For $\alpha=(\alpha_1,\cdots,\alpha_n)\in A_n$, set $|\alpha|=\sum_{i=1}^n\alpha_i, $ and use $x^{\alpha}:=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ to denote its canonical image in $\mathfrak{A}_n$ for brevity. Then $\mathfrak{A}_n$ has a basis $\{x^{\alpha} :  \alpha\in A_n\}$ with the multiplication subject to $x^{\alpha}x^{\beta}=x^{\alpha+\beta}$ with the convention that $x^{\alpha+\beta}=0$ if $\alpha+\beta\notin A_n$. Moreover, $\mathfrak{A}_n$ has a natural $\mathbb{Z}$-grading $$\mathfrak{A}_n=\bigoplus\limits_{i=0}^{n(p-1)}(\mathfrak{A}_n)_{[i]},$$ where $(\mathfrak{A}_n)_{[i]}=\text{span}_{\mathbb{F}}\{x^{\alpha} :  |\alpha|=i\}$. For $1\leq i\leq n$, let $D_i$ be the linear transformation on $\mathfrak{A}_n$ with $D_i(x^{\alpha})=\alpha_i x^{\alpha-\varepsilon_i}$ for any $\alpha\in A_n$. Then it is easy to see that $D_i\in\Der(\mathfrak{A}_n)$ for $1\leq i\leq n$. The [**Jacobson-Witt algebra**]{} $W_n$ is defined as the derivation algebra of $\mathfrak{A}_n$, i.e., $W_n=\Der(\mathfrak{A}_n)$. Then by [@SF §4.2], $W_n$ is a free $\mathfrak{A}_n$-module of rank $n$ with a basis $\{D_1,\cdots, D_n\}$. The Lie bracket in $W_n$ is given by $$[fD_i, gD_j]=f(D_i(g))D_j-g(D_j(f))D_i,\,\,f,g\in\mathfrak{A}_n, 1\leq i,j\leq n.$$ Moreover, $W_n$ is a simple Lie algebra unless $n=1$ and $p=2$. The natural $\mathbb{Z}$-grading on $\mathfrak{A}_n$ induces the corresponding $\mathbb{Z}$-grading structure on $W_n$, $$W_n=\bigoplus\limits_{i=-1}^{n(p-1)-1}(W_n)_{[i]},$$ where $$(W_n)_{[i]}=\text{span}_{\mathbb{F}}\{x^{\alpha}D_j :  |\alpha|=i+1, 1\leq j\leq n\}.$$ Furthermore, $W_n$ has a canonical torus $T=\sum_{i=1}^n\bbf x_iD_i\in (W_n)_{[0]}$, and it has the following root space decomposition with respect to the torus $T$: $$\label{decomp for W}
W_n=T\oplus\big(\bigoplus\limits_{\alpha\in\Lambda}(W_n)_{\alpha}\big),$$ where $$(W_n)_{\alpha}=\text{span}_{\bbf}\{x^{\alpha+\varepsilon_j}D_j :  1\leq j\leq n\},$$ and $\Lambda=\{a_1\varepsilon_1+\cdots+a_n\varepsilon_{n}-\varepsilon_i :  0\leq a_1,\cdots, a_n\leq p-1, 1\leq i\leq n\}\setminus\{0\}$ is the set of all roots.

We need the following known result on the derivation algebras of the Jacobson-Witt algebras for later use.

\[derivation lem\] We have $\Der(W_n)=\Inn(W_n)$ for any $n\in\mathbb{Z}_+$.

The assertion follows from [@SF Theorems 8.5, Chapter 4] (also see [@St Theorem 7.1.2]).

By Lemma \[derivation lem\], we can reformulate the definition of 2-local derivation on the Jacobson-Witt algebra as follows. Let $\ggg=W_n$ be the Jacobson-Witt algebra. A map $\Delta$ on $\ggg$ is a 2-local derivation if for any two elments $x, y\in\ggg$, there exists an element $a_{x,y}\in\ggg$ such that $\Delta(x)=[a_{xy}, x]$ and $\Delta(y)=[a_{xy}, y]$.

2-local derivations on the Jacobson-Witt algebras
=================================================

In this section, we shall determine all 2-local derivations on the Jacobson-Witt algebras. In general, for an element $x$ in a Lie algebra $\ggg$, the centralizer of $x$ in $\ggg$ is defined as $\mathfrak{z}_{\ggg}(x)=\{y\in\ggg :  [x,y]=0\}.$ Then $\mathfrak{z}_{\ggg}(x)$ is a subalgebra of $\ggg$ containing $x$ itself.

From now on to the end of this section, we always assume that $\ggg=W_n$ is the Jacobson-Witt algebra. For $\lambda=(\lambda_1, \cdots, \lambda_n),
\mu=(\mu_1,\cdots, \mu_n)\in\bbf^n$, let $(\lambda, \mu)=\sum_{i=1}^n\lambda_i\mu_i$.

A vector $\lambda=(\lambda_1, \cdots, \lambda_n)\in\bbf^n$ is called [**regular**]{} if $\lambda_1,\cdots, \lambda_n$ are $\bbf_p$ linearly independent, that is, for $\mu\in\bbf_p^n$, $(\lambda,\mu)=0$ if and only if $\mu=(0,\cdots, 0)$.

For $\lambda=(\lambda_1,\cdots, \lambda_n)\in\bbf^n$ and $0\le k\le p-1$, let $$\frakd^{(k)}_{\lambda}=\sum_{i=1}^n\lambda_ix_i^kD_i,\quad$$ T\_[k-1]{}=\_{x\_i\^kD\_i:1in}.$$ For $1\leq i\leq n$, let $$\label{Di'}
\mathscr{D}_i=D_i+\sum\limits_{j=i}^{n-1}(\prod\limits_{k=i}^jx_k^{p-1})D_{j+1},$$ Then it follows from [@Pr Lemma 3(i)] that $\mathscr{D}_i=(-1)^{i-1}\mathscr{D}_1^{p^{i-1}}$ for $1\leq i\leq n$.

The following result on the structure of centralizers of some special elements in $\ggg$ is crucial to determine 2-local derivations on $\ggg$.

\[lem on centralizer\] Let $\lambda=(\lambda_1,\cdots, \lambda_n)\in\bbf^n$ be regular. Then the following statements hold.

-   $\mathfrak{z}_{\ggg}(\frakd^{(1)}_{\lambda})=T$.

-   $\mathfrak{z}_{\ggg}(\sum_{i=1}^n x_i^2D_i)\cap T=0$.

-   $\mathfrak{z}_{\ggg}(\mathscr{D}_1)=\sum\limits_{i=1}^{n}\bbf \mathscr{D}_i$.

\(1) Take any $D\in \mathfrak{z}_{\ggg}(\frakd^{(1)}_{\lambda})$. Thanks to (\[decomp for W\]), we can write $D=D_0+\sum\limits_{\alpha\in\Lambda}D_{\alpha}$ with $D_0\in T$ and $D_{\alpha}\in\ggg_{\alpha}$ for any $\alpha\in\Lambda$. Then $$0=[\frakd^{(1)}_{\lambda}, D]=\Big[\frakd^{(1)}_{\lambda}, D_0+\sum\limits_{\alpha\in\Lambda}D_{\alpha}\Big]=\sum\limits_{\alpha\in\Lambda}(\lambda,\alpha) D_{\alpha}.$$ Since $\lambda$ is regular, $(\lambda,\alpha)\neq 0$ for any $\alpha\in\Lambda$. It follows that $D_{\alpha}=0$ for any $\alpha\in\Lambda$. This implies that $\mathfrak{z}_{\ggg}(\frakd^{(1)}_{\lambda})\subseteq T$. On the other hand, it is obvious that $T\subseteq \mathfrak{z}_{\ggg}(\frakd^{(1)}_{\lambda})$. Hence, $\mathfrak{z}_{\ggg}(\frakd^{(1)}_{\lambda})=T$.

\(2) Take any $E=\sum\limits_{j=1}^nk_jx_jD_j\in\mathfrak{z}_{\ggg}(\sum_{i=1}^n x_i^2D_i)\cap T$ with $k_j\in\bbf$ for $1\leq j\leq n$, then $$0=\Big[\sum\limits_{j=1}^nk_jx_jD_j,\sum_{i=1}^n x_i^2D_i\Big]=\sum\limits_{i=1}^n k_i x_i^2D_i.$$ It follows that $k_i=0$ for any $1\leq i\leq n$. Consequently, $\mathfrak{z}_{\ggg}(\sum_{i=1}^n x_i^2D_i)\cap T=0$.

\(3) follows from [@Pr Lemma 7(ii)].

\[delta lem\] Let $\lambda=(\lambda_1,\cdots,\lambda_n)\in\bbf^n$ be regular, and $\Delta$ be a 2-local derivation on $\ggg$ such that $\Delta(\frakd^{(1)}_{\lambda})=0$. Then for any nonzero element $X= \sum_{\alpha\in S} x^{\alpha} d_\alpha\in W_n$, where $S\subseteq A_n$ with $d_\alpha\in \sum_{i=1}^n\bbf D_i$, we have $\Delta(X) \in \sum_{\alpha\in S} \bbf x^{\alpha}  d_\alpha$.

For $\frakd^{(1)}_{\lambda}$ and $X$, there exists an element $a\in\ggg$ such that $\Delta(\frakd^{(1)}_{\lambda})=[a, \frakd^{(1)}_{\lambda}]$ and $\Delta(X)=[a, X]$. Since $\Delta(\frakd^{(1)}_{\lambda})=0$, it follows that $a\in T$ by Lemma \[lem on centralizer\]. Thus $$\Delta(X)=\bigg[a, \sum_{\alpha\in S} x^{\alpha} d_\alpha\bigg]\in\sum_{\alpha\in S} \bbf x^{\alpha}  d_\alpha.$$

\[Try1\] Let $\nu=(1,1,\cdots, 1)\in\bbf^n$, and $X=\sum_{i\ge -1}X_{i}\in\ggg$, where $X_{i}\in\ggg_{[i]}$ for $i\geq -1$. Then the following statements hold.

-   If $X\in \mathfrak{z}_{\ggg}(\sum\limits_{i=1}^nx_i^2D_i)$, then $X_{-1}=X_{0}=0$.

-   If $X\in \mathfrak{z}_{\ggg}(\frakd_\nu^{(\frac{p+1}{2})})$, then $X_{k}=0$ for all $k< \frac{p-1}2$.

\(1) Since $\sum\limits_{i=1}^nx_i^2D_i\in\ggg_{[1]}$ and $$\label{eq1}
0=\bigg[X, \sum\limits_{i=1}^nx_i^2D_i\bigg]=\sum\limits_{k=-1}^{n(p-1)-1}\bigg[X_{k}, \sum\limits_{i=1}^nx_i^2D_i\bigg],$$ it follows that $$\label{eq2}
\bigg[X_{k}, \sum\limits_{i=1}^nx_i^2D_i\bigg]=0, \,\,-1\leq k\leq n(p-1)-1.$$

Write $X_{-1}=\sum\limits_{s=1}^n a_sD_s$ for $a_s\in\bbf, 1\leq s\leq n$. Then $$0=\bigg[X_{-1}, \sum\limits_{i=1}^nx_i^2D_i\bigg]=\sum\limits_{i=1}^n 2a_ix_iD_i,$$ which implies that $a_s=0$, $-1\leq s\leq n$. Hence, $X_{-1}=0$.

Write $$X_{0}=\sum\limits_{1\leq i,j\leq n}a_{ij}x_iD_j, a_{ij}\in\bbf, 1\leq i, j\leq n.$$ If there exists some $s\neq t$ such that $a_{st}\neq 0$, then $a_{st}x_sx_tD_t$ appears as a summand in $[X_{0}, \sum_{i=1}^nx_i^2D_i]$, and can not be cancelled by other summands. This contradicts with (\[eq2\]) in the case $k=0$. Hence, $X_{0}\in T$. Then it follows from (\[eq2\]) for the case $k=0$ and Lemma \[lem on centralizer\](2) that $X_{0}=0$.

\(2) Since $\frakd_\nu^{(\frac{p+1}{2})}\in\ggg_{[\frac{p-1}{2}]}$ and $$0=\Big[X, \frakd_\nu^{(\frac{p+1}{2})}\Big]=\sum\limits_{k=-1}^{n(p-1)-1}\bigg[X_{k}, \frakd_\nu^{(\frac{p+1}{2})}\bigg],$$ it follows that $$\Big[X_{k}, \frakd_\nu^{(\frac{p+1}{2})}\Big]=0, \,\,-1\leq k\leq n(p-1)-1.$$

Assume $X_{k}=\sum\limits_{i=1}^n f_i^{(k)}D_i$, where $f_i^{(k)}\in (\mathfrak{A}_n)_{[k+1]}, 1\leq i\leq n$. Then $$0=\Big[X_{k}, \frakd_\nu^{(\frac{p+1}{2})}\Big]=\sum\limits_{i=1}^n\Big({(\frac{p+1}{2})}x_i^{\frac{p-1}{2}}f_i^{(k)}-
\big(\sum\limits_{j=1}^nx_j^{\frac{p+1}{2}}D_j(f_i^{(k)})\big)\Big)D_i.$$ Hence, $${(\frac{p+1}{2})}x_i^{\frac{p-1}{2}}f_i^{(k)}-\big(\sum\limits_{j=1}^nx_j^{\frac{p+1}{2}}D_j(f_i^{(k)})\big)=0,\,\forall\,1\leq i\leq n.$$ It follows that $$f_i^{(k)}=0\,\text{ for\, any}\,-1\leq k<\frac{p-1}{2}, 1\leq i\leq n.$$ That is $$X_{k}=0\,\text{ for}\,-1\leq k<\frac{p-1}{2}.$$

As a direct consequence of Proposition \[Try1\], we have

\[Try2\] Suppose $\Delta$ is a 2-local derivation on $\ggg$ such that $\Delta(\frakd^{(1)}_{\lambda})=\Delta(\sum_{i=1}^nx_i^2D_i)=0$ for some regular vector $\lambda\in\bbf^n$. Then

-   $\Delta(\ggg_{[k]})=0$ for any $k$.

-   $\Delta(\sum_{k\ge r}\ggg_{[k]})\subseteq \sum_{k\ge r+\frac{p-1}2}\ggg_{[k]}$ for any $r$.

-   $\Delta(\mathscr{D}_i)=0$ for $1\leq i\leq n$.

\(1) Let $X\in\ggg_{[k]}$. For $\sum_{i=1}^nx_i^2D_i$ and $X$, there exists $Y=\sum_{i\geq -1} Y_{i}\in\ggg$ with $Y_{i}\in\ggg_{[i]}$ for $i\geq -1$ such that $0=\Delta(\sum_{i=1}^nx_i^2D_i)=[Y, \sum_{i=1}^nx_i^2D_i]$ and $\Delta(X)=[Y, X]$. It follows from Proposition \[Try1\] that $Y_{0}=0$. Since $X, \Delta(X)\in\ggg_{[k]}$, we further obtain that $$\Delta(X)=[Y, X]=[Y_{0}, X]=0.$$

\(2) Since $\frakd_\nu^{(\frac{p+1}{2})}\in\ggg_{[\frac{p-1}{2}]}$ for $\nu=(1,1,\cdots, 1)$, it follows from the statement (1) that $\Delta\big(\frakd_\nu^{(\frac{p+1}{2})}\big)=0$. Then for any $X\in\sum_{k\ge r}\ggg_{[k]}$, there exists $Y=\sum_{i\geq -1} Y_{i}\in\ggg$ with $Y_{i}\in\ggg_{[i]}$ for $i\geq -1$ such that $$0=\Delta(\frakd_\nu^{(\frac{p+1}{2})})=[Y, \frakd_\nu^{(\frac{p+1}{2})}],$$ and $\Delta(X)=[Y, X]$. Note that $Y_{k}=0$ for $k<\frac{p-1}{2}$ by Proposition \[Try1\], we further have $$\Delta(X)=[Y, X]=\sum_{i\geq\frac{p-1}{2}}[Y_i, X]\in\sum_{k\ge r+\frac{p-1}{2}}\ggg_{[k]}.$$

\(3) By Lemma \[delta lem\] and the statement (2), we see that $$\label{Eq1 for D}
\Delta(\mathscr{D}_i)=\sum\limits_{j=i}^{n-1}l_i^{(j+1)}(\prod\limits_{k=i}^jx_k^{p-1})D_{j+1},$$ where $l_i^{(j)}\in\bbf$ for $i< j\leq n$.

On the other hand, for $\sum_{i=1}^nx_i^2D_i$ and $\mathscr{D}_i$, there exists $b=\sum_{i\geq -1}b_{i}\in\ggg$ with $b_{i}\in\ggg_{[i]}$ for $i\geq -1$ such that $0=\Delta\big(\sum_{i=1}^nx_i^2D_i\big)=\big[b, \sum_{i=1}^nx_i^2D_i\big]$ and $\Delta(\mathscr{D}_i)=[b, \mathscr{D}_i]$. Then $b_{-1}=b_{0}=0$ by Proposition \[Try1\]. Hence $$\label{Eq2 for D}
\Delta(\mathscr{D}_i)=[b, \mathscr{D}_i]=\sum_{k\geq 1}[b_{k}, D_i+\sum\limits_{j=i}^{n-1}(\prod\limits_{k=i}^jx_k^{p-1})D_{j+1}].$$ By comparing the right hand sides of (\[Eq1 for D\]) and (\[Eq2 for D\]), we have $$l_i^{(j+1)}(\prod\limits_{k=i}^jx_k^{p-1})D_{j+1}=[b_{(p-1)(j-i+1)}, D_i]+
\sum\limits_{s=i}^{j}[b_{(p-1)(j-s)},(\prod\limits_{k=i}^sx_k^{p-1})D_{s+1}], \, i< j\leq n-1.$$ The right-hand-side of this equation does not produce any term of the form $(\prod_{k=i}^jx_k^{p-1})D_{j+1}$ since $b_0=0$. This implies that $l_i^{(j)}=0$ for $i< j\leq n$, i.e., $\Delta(\mathscr{D}_i)=0$, as desired.

We define the support of $X=\sum_{\alpha\in A_n}\sum_{1\le i\le n}a_{\alpha, i}x^\alpha D_i\in\ggg$, where $a_{\alpha, i}\in F$, as $${\rm supp}(X):=\{(\alpha,i):a_{\alpha, i}\ne0\}.$$ In this section, from now on, we take a regular vector $\lambda=(\lambda_1,\cdots, \lambda_n)\in\bbf^n$, and let $\Delta$ be a 2-local derivation on $\ggg$ such that $\Delta(\frakd^{(1)}_{\lambda})=\Delta(\sum\limits_{i=1}^nx_i^2D_i)=0$.

We want to show that $\Delta=0$. To the contrary, assume that there is $X=\sum_{i= r}^{r+s}X_{i}\in\ggg$ with $X_i\in\ggg_{[i]}$ for $r\leq i\leq r+s$, $ X_{r}\ne0$, $ X_{r+s}\ne0$ such that $\Delta(X)\ne0$. We need to deduce a contradiction. Thanks to Lemma \[delta lem\], we can write $$X':=\Delta(X)=\sum_{i= r}^{r+s}X'_{i},$$ where $X'_i\in\ggg_{[i]}$ for $r\leq i\leq r+s$. We may choose such an $X$ so that $r+s$ is maximal and $s$ is minimal. We may further asume that ${\rm supp}(X_{r+s})$ and ${\rm supp}(X_{r+s-1})$ are minimal in the sense that, for any $Y=\sum_{i=r}^{r+s}Y_{i}\in\ggg$, where $Y_i\in\ggg_{[i]}$ for any $i\geq -1$, if ${\rm supp}(X_{i})={\rm supp}(Y_{i})$ for $i\in\{r+s-1, r+s\}\setminus\{k\}$ and ${\rm supp}(X_{k})={\rm supp}(Y_{k})$, then $\Delta(Y)=0$.

The following observation is elementary.

\[X’\] We have $X'_i=0$ if $i< r+s-1$, that is, $X'= X'_{r+s}+X'_{r+s-1}.$

From the minimal conditions on $X$, we know that $\Delta(X-X_{r+s})=0$. There is an element $a\in\ggg$ such that $$0=\Delta(X-X_{r+s})=[a,  X-X_{r+s}],\,\,\,
X'=[a, X]=[a, X_{r+s}].$$ The statement follows from $a=a_{-1}+a_0+\cdots+a_m$ with $a_i\in\ggg_{[i]}$.

\[X\_[r+s]{}\] If $(\alpha, i)\in {\rm supp}(X_{r+s})$, then $$X'_{r+s}\in{\rm span}_{\bbf} \{x^\alpha D_i, x^{\alpha+\epsilon_k-\epsilon_j} D_i,
x^\alpha D_k-\alpha_k x^{\alpha+\epsilon_i-\epsilon_k} D_i:k\ne i, 1\le j\le n\},$$ and $$X'_{r+s-1}\in{\rm span}_{\bbf}\{x^{\alpha-\epsilon_j}D_i: 1\leq j\leq n\}.$$

From the minimal conditions on $X$, we know that $\Delta(X-a_{\alpha, i}x^\alpha D_i)=0$. There is an element $a\in\ggg$ such that $$0=\Delta(X-a_{\alpha, i}x^\alpha D_i)=[a,  X-a_{\alpha, i}x^\alpha D_i],\,\,\,
X'=[a, X]=[a, a_{\alpha, i}x^\alpha D_i].$$ The statement follows from $X'_{r+s}\in [\ggg_{[0]}, a_{\alpha, i}x^\alpha D_i]$ and $X'_{r+s-1}\in [\ggg_{[-1]}, a_{\alpha, i}x^\alpha D_i]$.

\[X\_[r+s]{}-3\] If $(\alpha, 1), (\alpha+\epsilon_1-\epsilon_2, 1)\in {\rm supp}(X_{r+s})$, then $${\rm supp}(X'_{r+s})\subset   \{(\alpha+\epsilon_k-\epsilon_2, 1):  1\le k\le n\}.$$

The assertion follows directly from Lemma \[X\_[r+s]{}\].

As a consequence of Lemma \[X\_[r+s]{}\] and Lemma \[X\_[r+s]{}-3\], we have the following result on the structure of $X_{r+s}$ when $X'_{r+s}\neq 0$, which is crucial to our further discussion.

\[X\_[r+s]{}-6\] If $X'_{r+s}\ne0$, then there exist $\alpha\in A_n$ and $i\in\{1,2,\cdots, n\}$ such that $$\label{ca1}
{\rm supp}(X _{r+s})\subset  \{(\beta, i):  \beta\in A_n\},$$ or $$\label{ca2}
 {\rm supp}(X _{r+s})\subset   \{(\alpha,k):  1\leq k\leq n\}.$$

Let $\Upsilon$ be the set of all $k\in\{1,2,\cdots, n\}$ such that there exists some $\alpha\in A_n$ such that $(\alpha, k)\in {\rm supp}(X _{r+s})$.

Case 1: $|\Upsilon|=1$.

In this case, (\[ca1\]) holds.

Case 2: $|\Upsilon|>1$.

In this case, we may assume $(\alpha, 1), (\beta, 2)\in {\rm supp}(X _{r+s})$ without loss of generality. If $\beta\neq\alpha$, then it follows from Lemma \[X\_[r+s]{}\] that $$X'_{r+s}\in {\rm span}_{\bbf} \{x^\alpha D_k-\alpha_k x^{\alpha+\epsilon_1-\epsilon_k} D_1:k\ne 1\}\cap {\rm span}_{\bbf} \{x^\beta D_l-\beta_l x^{\beta+\epsilon_2-\epsilon_l} D_2:l\ne 2\}.$$ This implies that $$\beta=\alpha+\epsilon_1-\epsilon_2, \,\,\,(\alpha_1+1)\alpha_2\equiv 1\,(\text{mod}\,p),$$ and there exists some nonzero constant $c\in\bbf$ such that $$X'_{r+s}=c(x^\alpha D_2-\alpha_2 x^{\alpha+\epsilon_1-\epsilon_2} D_1).$$ This implies that $(\alpha+\epsilon_1-\epsilon_2, 1)\in {\rm supp}(X _{r+s})$ and $(\alpha, 2)\in{\rm supp}(X' _{r+s})$ by Lemma \[delta lem\]. It contradict with Lemma \[X\_[r+s]{}-3\]. Hence, (\[ca2\]) holds.

\[X\_[r+s]{}-7\] If there exist some $\alpha,\beta\in A_n$ and $i\neq j\in\{1,\cdots, n\}$ such that $$\{(\alpha,i), (\beta, j)\}\subset
 {\rm supp}(X _{r+s}),$$ then $X'_{r+s-1}=0$.

The assertion follows directly from Lemma \[X\_[r+s]{}\].

Let $$\mathscr{T}_1:={\rm span}\{I_1=\sum_{i=1}^n x_iD_i, h_j=x_jD_j+x_1D_j:2\le j\le n\},$$ and for $2\leq k\leq n$, set $$\mathscr{T}_k:={\rm span}\{I_k=x_kD_k+I_1, h_j=x_jD_j+x_1D_j, \mathbbm{h}_k=x_kD_k+x_1^2D_k :2\le j\le n,\, j\neq k\}.$$

\[I\_k\] We have $ \mathfrak{z}_{\ggg}(\mathscr{T}_k)=\mathscr{T}_k$ for any $k=1,2,\cdots,n$. Moreover, these $\mathscr{T}_k$ ($1\leq k\leq n$) are Cartan subalgebras of $\ggg$.

Define the following algebra isomorphisms $$\begin{aligned}
\psi_1:\, \mathfrak{A}_n &\longrightarrow &\mathfrak{A}_n\\
x_i&\longmapsto & x_i+(1-\delta_{i1})x_1,\quad 1\leq i\leq n,\end{aligned}$$ and for $2\leq k\leq n$, $$\begin{aligned}
\psi_k:\, \mathfrak{A}_n &\longrightarrow &\mathfrak{A}_n\\
x_i&\longmapsto & x_i+(1-\delta_{i1})x_1^{1+\delta_{ik}},\quad 1\leq i\leq n.\end{aligned}$$ Then it follows from [@Wi Theorem 2] that these algebra isomorphisms $\psi_k$ ($1\leq k\leq n$) induce the following Lie algebra isomorphisms, $$\begin{aligned}
\widetilde{\psi_k}:\, \ggg=\Der(\mathfrak{A}_n) &\longrightarrow & \ggg=\Der(\mathfrak{A}_n)\\
E&\longmapsto &\psi_k\circ E\circ\psi_k^{-1},\,\,\forall\,E\in\ggg.\end{aligned}$$ It follows from direct computation that $$\psi_1(x_1D_1)=x_1\big(D_1-\sum\limits_{j=2}^nD_j\big),\quad \psi_1(x_lD_l)=h_l,\,2\leq l\leq n,$$ and for $2\leq k\leq n$, $$\psi_k(x_1D_1)=x_1\big(D_1-\sum\limits_{\stackrel{j=2}{j\neq k}}^n D_j\big)-2x_1^2D_k,\,\,\, \psi_k(x_lD_l)=h_l,\, 2\leq l\leq n\,\,\text{and}\,\, l\neq k,\,\,\,\psi_k(x_kD_k)=\mathbbm{h}_k.$$ Therefore, $\psi_k(T)=\mathscr{T}_k$ for any $1\leq k\leq n$, so that these $\mathscr{T}_k$ ($1\leq k\leq n$) are Cartan subalgebras of $\ggg$. Moreover, $$\mathfrak{z}_{\ggg}(\mathscr{T}_k)=\mathfrak{z}_{\ggg}\big(\psi_k(T)\big)=\psi_k\big(\mathfrak{z}_{\ggg}(T)\big)=\psi_k(T)=\mathscr{T}_k.$$ We complete the proof.

\[X\_[r+s]{}-8\] If $ {\rm supp}(X _{r+s})\subset\{(\beta, 1): \beta\in A_n\}$, then $X'=c[I_1, X]$ for some $c\in\bbf$. In particular, if $X'\neq 0$, then $X_l=0$ for any $l<r+s-1$ with $p\nmid l$. Moreover, if there exists $(\beta, 1)\in {\rm supp}(X _{r+s})$ with $\beta_1<p-1$, then there exists some $c_k\in\bbf$ such that $X'=c_k[I_k, X],\,2\leq k\leq n.$

For any $1\leq k\leq n$ and any regular vector $\lambda\in\bbf^n$, let $$h_{\lambda}=
\begin{cases}
\lambda_1I_k+\sum\limits_{j=2}^n\lambda_jh_j, &\text{if}\,\,k=1, \vspace{1mm}\cr \lambda_1I_k+\sum\limits_{\stackrel{j=2}{j\neq k}}^n\lambda_jh_j+\lambda_k \mathbbm{h}_k, &\text{if}\,\,2\leq k\leq n.
\end{cases}$$ For $h_{\lambda}$ and $X$, it follows from Lemma \[delta lem\] and Corollary \[Try2\](2) that there exists $a\in\ggg$ such that $0=\Delta(h_{\lambda})=[a, h_{\lambda}]$ and $X'=[a, X]$. Then by Lemma \[I\_k\], there exist $a_i\in\bbf, 1\leq i\leq n$ such that $$a=
\begin{cases}
a_1I_k+\sum\limits_{j=2}^na_jh_j, &\text{if}\,\,k=1, \vspace{1mm}\cr a_1I_k+\sum\limits_{\stackrel{j=2}{j\neq k}}^na_jh_j+a_k \mathbbm{h}_k, &\text{if}\,\,2\leq k\leq n.
\end{cases}$$ Note that ${\rm supp}(X'_{r+s})\subseteq {\rm supp}(X _{r+s})$ and $X'_{r+s+1}=0$ by Lemma \[delta lem\]. If $a_j\ne0$, $[a, X]$ has a nonzero term $a_ja_{\beta, 1}x^{\beta}D_j$ or $a_ja_{\beta, 1}x^{\beta+\epsilon_1}D_j$. It follows that $a_j=0$ for $2\leq j\leq n$, i.e., $a=a_1I_k$. The assertion follows.

The result in Lemma \[X\_[r+s]{}-8\] does not need the assumption that $s$ is minimal for $X$.

\[key prop1\] Suppose ${\rm supp}(X _{r+s})\subset\{(\beta, 1): \beta\in A_n\}$. Then $X'=0$.

Assume $X'\neq 0$, we will deduce some contradictions in the following discussion.

Case 1: $n=1$.

In this case, It follows from Lemma \[X\_[r+s]{}-8\] that $X=X_0+X_{r+s-1}+X_{r+s}$. Since $\Delta(D_1)=0$ by Corollary \[Try2\](1) and $\mathfrak{z}_{\ggg}(D_1)=\bbf D_1$, there exists some $0\neq c\in\bbf$ such that $$X'=\Delta(X)=[cD_1, X].$$ This implies that $X'_{r+s}=0$ and $X'_{r+s-1}\neq 0$, so that $X_{r+s-1}\neq 0$. Consequently, $X'_{r+s-2}\neq 0$, since $\mathfrak{z}_{\ggg}(D_1)=\bbf D_1$ and $r+s-1\neq -1$. This contradicts with Lemma \[X’\].

Case 2: $n\geq 2$.

In this case, we first claim that $r+s<n(p-1)-1$. Indeed, if $r+s=n(p-1)-1$, then $X_{r+s}=a_{\tau, 1}x^{\tau}D_1$, where $0\neq a_{\tau, 1}\in\bbf$, $\tau=\sum_{j=1}^n(p-1)\varepsilon_j$. It follows from Lemma \[X\_[r+s]{}-8\] that $\Delta(D_j-X)=0$ for any $1\leq j\leq n$. Then for $X$ and $D_j-X$, there exists some $a_j\in\ggg$ such that $0=\Delta(D_j-X)=[a_j, D_j-X]$ and $$\label{jud1}
X'=\Delta(X)=[a_j, X]=[a_j, D_j],\quad 1\leq j\leq n.$$ The right-hand-side can not produce the term $x^{\tau}D_1$, which implies that $X'_{r+s}=0$. Hence, $X'_{r+s-1}\neq 0$. From Lemma \[X\_[r+s]{}\] we know that $X_{r+s-1}$ (also $X'_{r+s-1}$ with different coefficient) has a term $a_{\tau-\varepsilon_i, 1}x^{\tau-\varepsilon_i}D_1\ne0$ for some $i=1,2,\cdots, n$. Choose $j\in \{1,2,\cdots, n\}\setminus\{i\}$. It follows from Lemma \[X\_[r+s]{}-8\] that $\Delta(D_j-X)=0$. Then for $X$ and $D_j-X$, there exists some $a_j\in\ggg$ such that $0=\Delta(D_j-X)=[a_j, D_j-X]$ and $$\label{jud1'}
X'=\Delta(X)=[a_j, X]=[a_j, D_j].$$ The coefficient of the term $x^{\tau-\varepsilon_i}D_1$ on the right hand side of (\[jud1’\]) is $0$. This implies that $X'_{r+s-1}=0$, a contradiction. Therefore, $r+s<n(p-1)-1$.

According to the discussion above, and the assumption on $X$ at the beginning, we have $\Delta(X-x^{\tau}D_2)=0$. Then for $X$ and $X-x^{\tau}D_2$, there exists some $a\in\ggg$ such that $$0=\Delta(X-x^{\tau}D_2)=[a, X-x^{\tau}D_2]$$ and $$X'=\Delta(X)=[a, X]=[a, x^{\tau}D_2].$$ This implies that $r+s=n(p-1)-2$, $X'=X'_{r+s}$, and ${\rm supp}(X')\subset\{(\tau-\varepsilon_j, 2):1\leq j\leq n\}$. It contradicts with ${\rm supp}(X')\subset {\rm supp}(X)\subset \{(\beta, 1): \beta\in A_n\}$.

In conclusion, we have shown that $X'=0$. The proof is complete.

\[key prop2\] Suppose $ {\rm supp}(X _{r+s})\subset   \{(\alpha,k):  1\le k\le n\}$. Then $X'=0$.

According to the assumption, we can write $$X_{r+s}=c_1x^{\alpha}D_1+c_2x^{\alpha}D_2+\cdots+c_nx^{\alpha}D_n=x^{\alpha}(c_1D_1+c_2D_2+\cdots+c_nD_n),$$ where $c_i\in\bbf, 1\leq i\leq n$. We can assume that $c_1\neq 0$ without loss of generality. Let ${\mathfrak{B}}_n=\bbf[y_1,\cdots, y_n]/(y_1^p,\cdots, y_n^p)$ be the divided power algebra of $n$ variables $y_1,\cdots, y_n$, where $(y_1^p,\cdots, y_n^p)$ denotes the ideal of $\bbf[y_1,\cdots, y_n]$ generated by $y_i^p, 1\leq i\leq n$. Define the following algebra isomorphism $$\begin{aligned}
\varphi:\, \mathfrak{A}_n &\longrightarrow &\mathfrak{B}_n\\
x_i&\longmapsto & c_iy_1+(1-\delta_{i1})y_i,\quad 1\leq i\leq n.\end{aligned}$$ Then it induces the following Lie algebra isomorphism $$\begin{aligned}
\widetilde{\varphi}:\, \ggg=\Der(\mathfrak{A}_n) &\longrightarrow &\hhh:=\Der(\mathfrak{B}_n)\\
E&\longmapsto &\varphi\circ E\circ\varphi^{-1},\,\,\forall\,E\in\ggg.\end{aligned}$$

It follows from a direct computation that for any $\alpha\in A_n$ and $1\leq i\leq n$, we have $$\widetilde{\varphi}(x^{\alpha}D_i)=
\begin{cases}
\frac{1}{c_1}\prod\limits_{j=1}^n(c_jy_1+(1-\delta_{j1})y_j)^{\alpha_j}\big(\widetilde{D_1}-\sum\limits_{k=2}^n c_k\widetilde{D_k}\big), &\text{if}\,\,i=1, \vspace{1mm} \cr
\prod\limits_{j=1}^n(c_jy_1+(1-\delta_{j1})y_j)^{\alpha_j}\widetilde{D_i}, &\text{if}\,\,2\leq i\leq n,
\end{cases}$$ where $\widetilde{D_i}$ is a derivation on $\mathfrak{B}_n$ defined by $\widetilde{D_i}(y_j)=\delta_{ij}$ for $1\leq i, j\leq n$. The Lie algebra $\hhh$ is a free $\mathfrak{B}_n$-module of rank $n$ with basis $\widetilde{D_1},\cdots, \widetilde{D_n}$, and it has a natural $\bbz$-grading similar as the Lie algebra $\ggg$. In particular, $Y:=\widetilde{\varphi}(X)=Y_r+\cdots+Y_{r+s}$ with $Y_j\in\hhh_{[j]}$ for $r\leq j\leq r+s$, and $Y_r\neq 0, Y_{r+s}=\prod\limits_{j=1}^n(c_jy_1+(1-\delta_{j1})y_j)^{\alpha_j}\widetilde{D_1}\neq 0$.

Moreover, the above Lie algebra homomorphism $\widetilde{\varphi}$ and the 2-local derivation $\Delta$ on $\ggg$ induce a 2-local derivation $\widetilde{\Delta}$ on $\hhh$. Precise speaking, $$\begin{aligned}
\widetilde{\Delta}:\, \hhh=\Der(\mathfrak{B}_n) &\longrightarrow &\hhh=\Der(\mathfrak{B}_n)\\
E&\longmapsto &\widetilde{\varphi}\big(\Delta\big(\widetilde{\varphi}^{-1}(E)\big)\big),\,\,\forall\,E\in\hhh.\end{aligned}$$ Indeed, for any $E, F\in\hhh$, we have $\widetilde{\varphi}^{-1}(E), \widetilde{\varphi}^{-1}(F)\in\ggg$. Since $\Delta$ is a 2-local derivation on $\ggg$, there exists $D\in\ggg$ such that $$\Delta\big(\widetilde{\varphi}^{-1}(E)\big)=[D, \widetilde{\varphi}^{-1}(E)],\quad \Delta\big(\widetilde{\varphi}^{-1}(F)\big)=[D, \widetilde{\varphi}^{-1}(F)].$$ Hence, $$\widetilde{\Delta}(E)=\widetilde{\varphi}\big(\Delta\big(\widetilde{\varphi}^{-1}(E)\big)\big)=[\widetilde{\varphi}(D), E],\quad
\widetilde{\Delta}(F)=\widetilde{\varphi}\big(\Delta\big(\widetilde{\varphi}^{-1}(F)\big)\big)=[\widetilde{\varphi}(D), F].$$ This implies that $\widetilde{\Delta}$ is a 2-local derivation on $\hhh$.

Suppose $X'=\Delta(X)\neq 0$, then $$0\neq\widetilde{\varphi}(X')=\widetilde{\varphi}\big(\Delta\big(\widetilde{\varphi}^{-1}\widetilde{\varphi}(X)\big)\big)=\widetilde{\Delta}
\big(\widetilde{\varphi}(X)\big)=\widetilde{\Delta}(Y).$$ Without loss of generality, we can assume that $Y$ satisfies the same assumption as $X$. Then it follows from Proposition \[key prop1\] that $\widetilde{\Delta}(Y)=0$, a contradiction. Hence, $X'=0$. We complete the proof.

\[prop for W\] Suppose $\Delta$ is a 2-local derivation on $\ggg$ such that $$\Delta\big(\frakd^{(1)}_{\lambda}\big)=\Delta\Big(\sum\limits_{i=1}^nx_i^2D_i\Big)=0$$ for some regular vector $\lambda\in\bbf^n$. Then $\Delta=0$.

With Corollary \[Try2\](1) in mind, suppose $X':=\Delta(X)\neq 0$ for $X=X_r+\cdots+X_{r+s}$ satisfying the assumption stated in the paragraph before Lemma \[X’\], where $X_i\in\ggg_{[i]}$ for $r\leq i\leq r+s$, $ X_{r}\ne0$, $ X_{r+s}\ne0$. Then it follows from Lemma \[X’\] that $X'= X'_{r+s}+X'_{r+s-1}$ for $X'_j\in\ggg_{[j]}$ for $j=r+s-1, r+s$. We will deduce some contradictions in the following discussion.

Case 1: $X'_{r+s}\neq 0$.

In this case, without loss of generality, there exists some $\alpha\in A_n$ such that one of the following two subcases may happen by Corollary \[X\_[r+s]{}-6\].

Subcase 1: ${\rm supp}(X _{r+s})\subset  \{(\beta, 1):  \beta\in A_n\}$.

In this subcase, it follows from Proposition \[key prop1\] that $X'=0$, a contradiction.

Subcase 2: $ {\rm supp}(X _{r+s})\subset   \{(\alpha,k):  1\leq k\leq n\}$.

In this subcase, it follows from Proposition \[key prop2\] that $X'=0$, a contradiction.

Case 2: $X'_{r+s}=0$.

In this case, $X'_{r+s-1}\neq 0$. Thanks to Corollary \[X\_[r+s]{}-7\], without loss of generality, we may assume that $${\rm supp}(X _{r+s})\subset\{(\beta, 1): \beta\in A_n\}.$$ Then it follows from Proposition \[key prop1\] that $X'=0$, a contradiction.

In conclusion, we show that $\Delta=0$. The proof is complete.

We are now in the position to present the following main result in this section.

\[main thm\] Let $\ggg$ be the Jacobson-Witt algebra over an infinite field of characteristic $p>2$. Then every 2-local derivation on $\ggg$ is a derivation.

Let $\Delta$ be a 2-local derivation on $\ggg$. Take a regular vector $\lambda\in\bbf^n$. Then there exists an element $a\in\ggg$ such that $$\Delta\big(\frakd^{(1)}_{\lambda}\big)=\big[a, \frakd^{(1)}_{\lambda}\big],\quad \Delta\bigg(\sum\limits_{i=1}^n x_i^2D_i\bigg)=\bigg[a, \sum\limits_{i=1}^n x_i^2D_i\bigg].$$ Set $\Delta_1=\Delta-\ad a$. Then $\Delta_1$ is a 2-local derivation on $\ggg$ such that $$\Delta_1\big(\frakd^{(1)}_{\lambda}\big)=\Delta_1\bigg(\sum\limits_{i=1}^nx_i^2D_i\bigg)=0.$$ It follows from Proposition \[prop for W\] that $\Delta_1=0$. Thus $\Delta=\ad a$ is a derivation. The proof is complete.

\[exam\] Let $\ggg=W_1$ be the Witt algebra over a field $\bbf$ of characteristic $p=2$. Then $\ggg$ is a two dimensional solvable Lie algebra with a basis $\{e_{-1}, e_0\}$ and $[e_{-1}, e_0]=e_{-1}$. For $i=-1,0$, let $\mathbb{D}_{i}$ be a linear transformation on $\ggg$ with $\mathbb{D}_i(e_j)=\delta_{ij}e_{-1}$ for $j=-1,0$. It is easy to see that both $\mathbb{D}_{-1}$ and $\mathbb{D}_0$ are derivations on $\ggg$. Since $[\ggg,\ggg]=\bbf e_{-1}$, it follows that $\Der(\ggg)=\bbf \mathbb{D}_{-1}\oplus \bbf \mathbb{D}_{0}$. Let $$\begin{aligned}
\Delta:\, \ggg &\longrightarrow & \ggg\\
k_{-1}e_{-1}+k_0e_0&\mapsto &\begin{cases}
k_0e_{-1}, &\text{if}\,\,k_{-1}\neq 0, \vspace{1mm} \cr
0, &\text{if}\,\,k_{-1}=0.
\end{cases},\,\,\forall\,\,k_{-1}, k_0\in\bbf.\end{aligned}$$ In the following we will show that $\Delta$ is a 2-local derivation on $\ggg$, but not a derivation. For that, take any $x=a_{-1}e_{-1}+a_0e_0, y=b_{-1}e_{-1}+b_0e_0\in\ggg$. There are the following four cases.

Case 1: $a_{-1}=b_{-1}=0$.

In this case, take $D_{xy}=0$. Then $$\Delta(x)=D_{xy}(x)=0,\quad \Delta(y)=D_{xy}(y)=0.$$

Case 2: $a_{-1}\neq 0, b_{-1}=0$.

In this case, take $D_{xy}=\frac{a_0}{a_{-1}}\mathbb{D}_{-1}\in\Der(\ggg)$. Then $$\Delta(x)=D_{xy}(x)=a_0e_{-1},\quad \Delta(y)=D_{xy}(y)=0.$$

Case 3: $a_{-1}=0, b_{-1}\neq 0$.

In this case, take $D_{xy}=\frac{b_0}{b_{-1}}\mathbb{D}_{-1}\in\Der(\ggg)$. Then $$\Delta(x)=D_{xy}(x)=0,\quad \Delta(y)=D_{xy}(y)=b_0e_{-1}.$$

Case 4: $a_{-1}\neq 0$ and $b_{-1}\neq 0$.

In this case, take $D_{xy}=\mathbb{D}_0\in\Der(\ggg)$. Then

$$\Delta(x)=D_{xy}(x)=a_0e_{-1},\quad \Delta(y)=D_{xy}(y)=b_0e_{-1}.$$

Therefore, $\Delta$ is a 2-local derivation on $\ggg$. However, $$\Delta(e_{-1}+e_{0})=e_{-1}\neq \Delta(e_{-1})+\Delta(e_0).$$ Hence, $\Delta$ is not a derivation.

From Example \[exam\], we know that the assumption on the characteristic of the ground field in Theorem \[main thm\] is necessary.

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[^1]: This work is supported by the National Natural Science Foundation of China (Grant Nos. 11771279, 11671138 and 11871190) and NSERC (311907-2015)
