Hong DU
Journal of Systems Science and Complexity. 2004, 14(1): 143-146.
Let $F$ be a field of characteristic zero.
$W_n=F[t_1^{\pm 1}, t_2^{\pm 1},\cdots ,t_n^{\pm 1}]\frac \partial
{\partial t_1}+\cdots +F[t_1^{\pm 1}, t_2^{\pm 1},\cdots ,t_n^{\pm
1}]\frac
\partial {\partial t_n}$ is the Witt algebra over $F$,
$W_n^{+}=F[t_1, t_2\cdots ,t_n]\frac
\partial {\partial t_1}+\cdots +F[t_1, t_2\cdots
,t_n]\frac \partial {\partial t_n}$ is Lie subalgebra of $W_n.$ It is well known both
$W_n$ and $W_n^{+}$ are simple infinite dimensional Lie algebra. In Zhao's paper, it was
conjectured that $End(W_n^{+})-\{0\}=Aut(W_n^{+})$ and it was proved that the validity of
this conjecture implies the validity of the well-known Jacobian conjecture. In this short
note, we check the conjecture above for $n=1.$ We show
$End(W_1^{+})-\{0\}=Aut(W_1^{+}).$
