The Lie algebra of polynomial vector fields and the Jacobian conjecture
The authors discuss polynomial maps \(\varphi:K^n\to K^n\) in relation to the Lie algebra of vector fields. Namely, let \(P\subset D\) be polynomial and formal vector fields of \(n\) variables. Probably the main result of the article in the following: every endomorphism of \(K[x_1,\dots, x_n]\) with invertible Jacobian is an automorphism (Jacobian conjecture) is true if every automorphism \(\varphi\) of the Lie algebra \(D\) with \(\varphi(P)\subset P\) satisfies \(\varphi(P)= P\). The authors generalize this also by changing \(P\) to a subalgebra \(A= \bigoplus_{\lambda\in H^*} H_\lambda\) where \(H_\lambda\) are root spaces for a Cartan subalgebra \(H\) in \(D\) (which has to be appropriately defined) \(P\) appears for \(H\) spanned by \(x_i\partial_{x_i}\).
scientific article; zbMATH DE number 1236107