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arXiv · 1402.5012

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

Hanspeter Kraft·Andriy Regeta
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Abstract

We show that every Lie algebra automorphisms of the vector fields $Vec(A^n)$ of affine n-space $A^n$, of the vector fields $Vec^c(A^n)$ with constant divergence, and of the vector fields $Vec^0(A^n)$ with divergence zero is induced by an automorphism of $A^n$. This generalizes results of the second author obtained in dimension 2. The case of $Vec(A^n)$ is due to Vladimir Bavula. As an immediate consequence, we get the following result which due to Viktor Kulikov. If every injective endomorphism of the simple Lie algebra $Vec(A^n)$ is an automorphism, then the Jacobian Conjecture holds in dimension $n$.

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Hanspeter Kraft, Andriy Regeta. 2014-02-20. Automorphisms of the Lie algebra of vector fields on affine n-space. https://arxiv.org/abs/1402.5012

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