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.