Automorphism groups of Witt algebras

Jianzhi Han; Yucai Su

paper

Automorphism groups of Witt algebras

arXiv:1502.01441

Abstract

The automorphism groups $Aut A_{n}$ and $Aut W_{n}$ of the polynomial algebra $A_{n} = C [x_{1}, x_{2}, \dots, x_{n}]$ and the rank $n$ Witt algebra $W_{n} = Der A_{n}$ are studied in this paper. It is well-known that $Aut A_{n}$ for $n \geq 3$ and $Aut W_{n}$ for $n \geq 2$ are open. In the present paper, by characterizing the semigroup $End W_{n} ∖ {0}$ of nonzero endomorphisms of $W_{n}$ via the semigroup of the so-called Jacobi tuples, we establish an isomorphism between $Aut A_{n}$ and $Aut W_{n}$ for any positive integer $n$ . In particular, this enables us to work out the automorphism group $Aut W_{2}$ of $W_{2}$ .