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

作者：Kraft Hanspeter^*; Regeta Andriy

来源：Journal of the European Mathematical Society, 2017, 19(5): 1577-1588.

DOI：10.4171/JEMS/700

• Automorphisms
• vector fields
• Lie algebras
• 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 A(n):
Aut(A(n)) (->) over tilde Aut(Lie)(Vec(A(n))) (->) over tilde Aut(Lie)(Vec(c)(A(n))) (->) over tilde Aut(Lie)(Vec(0)(A(n)))
This generalizes results of the second author obtained in dimension 2 [Reg13]. The case of Vec(A(n)) goes back to Kulikov [Kul92].
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)) subset of Aut(A(n)) of automorphisms with Jacobian determinant equal to 1.

• 出版日期2017