Seminar Algebra and Geometry: Andriy Regeta (Universität Basel)

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

We show that every Lie algebra automorphisms of the vector fields VF(Aⁿ) of affine n-space Aⁿ, of the vector fields VF^c(Aⁿ) with constant divergence, and of the vector fields VF⁰(Aⁿ) with divergence zero is induced by an automorphism of Aⁿ.

As an immediate consequence, we get the following result due to Kulikov. If every injective endomorphism of the Lie algebra VF(Aⁿ) is an automorphism, then the Jacobian Conjecture holds in dimension n.

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