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Automorphisms of the Lie algebra of vector fields on affine n-space (1402.5012v1)

Published 20 Feb 2014 in math.AG

Abstract: We show that every Lie algebra automorphisms of the vector fields $Vec(An)$ of affine n-space $An$, of the vector fields $Vecc(An)$ with constant divergence, and of the vector fields $Vec0(An)$ with divergence zero is induced by an automorphism of $An$. This generalizes results of the second author obtained in dimension 2. The case of $Vec(An)$ 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 $Vec(An)$ is an automorphism, then the Jacobian Conjecture holds in dimension $n$.

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