Automorphism groups of Witt algebras

The automorphism groups ${\rm Aut\,}A_n$ and ${\rm Aut\,}W_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra $W_n={\rm Der\,}A_n$ are studied in this paper. It is well-known that ${\rm Aut\,}A_n$ for $n\ge3$ and ${\rm Aut\,}W_n$ for $n\ge2$ are open. In the present paper, by characterizing the semigroup ${\rm End\,}W_n\setminus\{0\}$ of nonzero endomorphisms of $W_n$ via the semigroup of the so-called Jacobi tuples, we establish an isomorphism between ${\rm Aut}\,A_n$ and ${\rm Aut\,}W_n$ for any positive integer $n$. In particular, this enables us to work out the automorphism group ${\rm Aut\,}W_2$ of $W_2$.