Automorphisms of the Lie algebra of vector fields on affine \(n\)-space (Q2628341)

Summary: We study the vector fields \(\mathrm{Vec}(\mathbb A^n)\) on affine \(n\)-space \(\mathbb A^n\), the subspace \(\mathrm {Vec}^c(\mathbb A^n)\) of vector fields with constant divergence, and the subspace \(\mathrm {Vec}^0(\mathbb A^n)\) of vector fields with divergence zero, and we show that their automorphisms, as Lie algebras, are induced by the automorphisms of \(\mathbb A^n\): \[ \Aut(\mathbb A^n) \xrightarrow{~} \Aut_{\text{Lie}}(\mathrm {Vec}(\mathbb A^n)) \xrightarrow{~} \Aut_{\text{Lie}}(\mathrm {Vec}^0(\mathbb A^n)) \xrightarrow{~} \Aut_{\text{Lie}}(\mathrm {Vec}^0(\mathbb A^n)). \] This generalizes results of the second author obtained in dimension 2, see the second author, Lie subalgebras of vector fields and the Jacobian conjecture, \url{arxiv:1311.0232} (2013)]. The case of \(\mathrm {Vec}(\mathbb A^n)\) goes back to \textit{V. S. Kulikov} [Russ. Acad. Sci., Izv., Math. 41, No. 2, 351--365 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 5, 1086--1103 (1992; Zbl 0796.14008)]. 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 \(\mathrm {Vec}^0(\mathbb A^n)\) is isomorphic to the Lie algebra of the closed subgroup \(\mathrm {SAut}(\mathbb A^n) \subset\) \(\Aut(\mathbb A^n)\) of automorphisms with Jacobian determinant equal to 1.