Automorphisms of the Lie algebra of vector fields on affine nn-space

  • Hanspeter Kraft

    Universität Basel, Switzerland
  • Andriy Regeta

    Université de Grenoble I, France
Automorphisms of the Lie algebra of vector fields on affine $n$-space cover
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Abstract

We study the vector fields Vec(An)(\mathbb A^n) on affine nn-space An\mathbb A^n, the subspace Vecc(An)^c(\mathbb A^n) of vector fields with constant divergence, and the subspace Vec0(An)^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 An\mathbb A^n:

Aut(An)AutLie(Vec(An))AutLie(Vec0(An))AutLie(Vec0(An)).\mathrm {Aut}(\mathbb A^n) \xrightarrow{\sim} \mathrm {Aut_{Lie}}(\mathrm {Vec}(\mathbb A^n)) \xrightarrow{\sim} \mathrm {Aut_{Lie}}(\mathrm {Vec}^0(\mathbb A^n)) \xrightarrow{\sim} \mathrm {Aut_{Lie}}(\mathrm {Vec}^0(\mathbb A^n)).

This generalizes results of the second author obtained in dimension 2, see [Reg13]. The case of Vec(An)(\mathbb A^n) goes back to Kulikov [Kul92]. This generalization is crucial in the context of infinite-dimensional algebraic groups, because Vecc(An)^c(\mathbb A^n) is canonically isomorphic to the Lie algebra of Aut(An)(\mathbb A^n), and Vec0(An)^0(\mathbb A^n) is isomorphic to the Lie algebra of the closed subgroup SAut(An)(\mathbb A^n) \subset Aut(An)(\mathbb A^n) of automorphisms with Jacobian determinant equal to 1.

Cite this article

Hanspeter Kraft, Andriy Regeta, Automorphisms of the Lie algebra of vector fields on affine nn-space. J. Eur. Math. Soc. 19 (2017), no. 5, pp. 1577–1588

DOI 10.4171/JEMS/700