Differential Structure of Étale Extensions of Polynomial Algebras
The Jacobian Conjecture (see [BCW]) asserts that an étale polynomial map F: ℂ n → ℂ n is an isomorphism. In the course of some work with Haboush on group actions [BH] we observed that this is easily proved if F is equivariant for a reductive group acting on the two ℂ n ’s and having a dense orbit and a fixed point. One is thus tempted to try to prove the Jacobian conjecture by “feeding” such group actions into the picture. For example, let G = SL n (ℂ) act linearly on the target ℂ n . We may assume that F(0) = 0. In order to make F equivariant for a G-action, we must make G act on the source ℂ n by “F º G º F −1”. Of course this begs the issue, since we don’t know that F −1 exists. However F−1 exists analytically near 0. In particular we can pull back the action of the Lie algebra sl n(ℂ) as vector fields on ℂ n . Moreover the fact that the Jacobian matrix of F has a polynomial inverse implies that sl n (ℂ) pulls back to polynomial vector fields on the source ℂ n .
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