The Lie algebra of polynomial vector fields and the Jacobian conjecture

Abstract

The Jacobian conjecture for polynomial maps ϕ:K ⁿ →K ⁿ is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra$\mathbb{D}$ of formal vector fields inn variables. To be precise, let$\mathbb{D}_0$ be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of$\mathbb{D}_0$,H _λ the root spaces corresponding to linear forms λ onH and$A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda$. Then every polynomial map ϕ:K ⁿ →K ⁿ with invertible Jacobian matrix is an automorphism if and only if every automorphism Φ of$\mathbb{D}$ with Φ(A)$\subseteq A$ satisfies Φ(A)=A.