Seminar Algebra and Geometry: Andriy Regeta (Universität Basel)

Lie subalgebras of Vector Fields and Jacobian Conjecture

We study Lie subalgebras L of the vector fields Vec^c(A²) of affine 2-space A² of constant divergence, and we classify those L which are isomorphic to the Lie algebra aff₂ of the group Aff₂(K) of affine transformations of A². We then show that the following statements are equivalent:
(a) The Jacobian Conjecture holds in dimension 2;
(b) All Lie subalgebras L ⊂ Vec^c(A²) isomorphic to aff₂ are conjugate under Aut(A²);
(c) All Lie subalgebras L ⊂ Vec^c(A²) isomorphic to aff₂ are algebraic.

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