Automorphisms of the Lie algebra of vector fields on affine n-Space

We study the vector fields Vec(A^n) on affine n-space A^n, the subspace Vec^c(A^n) of vector fields with constant divergence, and the subspace Vec^0(A^n) of vector fields with divergence zero, and we show that their automorphisms, as Lie algebras, are induced by the automorphisms of An: Aut(A^n) → Aut_Lie(Vec(A^n)) → Aut_Lie(Vec^c(A^n)) → Aut_Lie(Vec^0(A^n)): This generalizes results of the second author obtained in dimension 2. The case of Vec(A^n) goes back to Kulikov. This generalization is crucial in the context of infinite-dimensional algebraic groups, because Vec^c(A^n) is canonically isomorphic to the Lie algebra of Aut(A^n), and Vec^0(A^n) is isomorphic to the Lie algebra of the closed subgroup SAut(A^n) ⊂ Aut(A^n) of automorphisms with Jacobian determinant equal to 1.