Automorphisms of the Lie algebra of vector fields on affine $n$-space

Abstract

We study the vector fields Vec$({\mathbb{A}}^n)$ on affine $n$-space ${\mathbb{A}}^n$, the subspace Vec${}^c({\mathbb{A}}^n)$ of vector fields with constant divergence, and the subspace Vec${}^0({\mathbb{A}}^n)$ of vector fields with divergence zero, and we show that their automorphisms, as Lie algebras, are induced by the automorphisms of ${\mathbb{A}}^n$:

This generalizes results of the second author obtained in dimension 2, see [Reg13]. The case of Vec$({\mathbb{A}}^n)$ goes back to Kulikov [Kul92]. This generalization is crucial in the context of infinite-dimensional algebraic groups, because Vec${}^c({\mathbb{A}}^n)$ is canonically isomorphic to the Lie algebra of Aut$({\mathbb{A}}^n)$, and Vec${}^0({\mathbb{A}}^n)$ is isomorphic to the Lie algebra of the closed subgroup SAut$({\mathbb{A}}^n)\subset$ Aut$({\mathbb{A}}^n)$ of automorphisms with Jacobian determinant equal to 1.