The groups of automorphisms of the Lie algebras of polynomial vector fields with zero or constant divergence

Let Pn = K[x1,…, xn] be a polynomial algebra over a field K of characteristic zero and 𝔡𝔦𝔳n0 (respectively, 𝔡𝔦𝔳nc) be the Lie algebra of derivations of Pn with zero (respectively, constant) divergence. We prove that AutLie(𝔡𝔦𝔳n0)≈AutK−alg(Pn) (n ≥ 2) and AutLie(𝔡𝔦𝔳nc)≈AutK−alg(Pn). The Lie algebra 𝔡𝔦𝔳nc is a maximal Lie subalgebra of DerK(Pn). Minimal finite sets of generators are found for the Lie algebras 𝔡𝔦𝔳n0 and 𝔡𝔦𝔳nc.